Text-to-image (T2I) models demonstrate strong generative capabilities and are increasingly used in design. However, their support for early exploratory ideation remains limited. Their linear, one-shot interaction paradigm aligns more closely with convergent, refinement-oriented stages of design. To address this gap, we present an interaction paradigm supporting early-stage ideation with T2I models, with a particular focus on novice designers. It introduces a dimension–attribute dictionary to guide prompt construction progressively and employs a dynamic, editable tree structure to help users organize and navigate their design space. Based on this paradigm, we developed a prototyping tool named InspirationGraph, focusing on the product design field. The results from a user study involving 24 participants highlight how this structured exploration approach supports divergent thinking and reduces cognitive load. We also uncover varying ideation patterns among designers and offer actionable insights into how T2I systems can be reimagined to better support the early-stage design.

Recent advances in LLMs have enabled new possibilities for creative content generation, yet their use in game design is often limited by poor integration across creative components, particularly for novice designers aiming to rapidly prototype playable concepts. Guided by the Elemental Tetrad framework, we present RPGAgent, an LLM-driven multi-agent system specifically designed to assist novice game creators in transforming a short story outline into a playable game. Specialized agents exchange structured data to generate coherent narrative, scene, and gameplay mechanics, ensuring structural correctness and consistency between story and world. By combining LLM-based generation with procedural content creation, the system offers a controllable and interpretable workflow. In a within-subjects study with 18 participants, RPGAgent outperformed a GPT-assisted baseline in both user experience and creative satisfaction during the prototyping of playable RPGs. These results demonstrate the potential of collaborative multi-agent frameworks for structured, AI-assisted game design.

In early-stage industrial design, teams generate essential but fragile process knowledge—semantic tags, sketches, exploration paths—that is rarely captured or reused but which may be useful at latter design stages, and AI could be used for this purpose. Yet existing AI creativity tools remain outcome-oriented, offering limited support for preserving, tracing, or recombining underlying reasoning. Our formative study (N=6) revealed persistent challenges in team–AI ideation across sessions and collaborators, including semantic–visual fragmentation, context loss, and cross-tool disruption. These insights inspired CoNode, a two-layer system that embeds AI nodes within a shared whiteboard through triplet workflows and augments them with workflow-level consolidation, reuse, and recombination via the CoSense module. We conducted a two-stage evaluation: User Study I (N=12) validates CoNode's foundational interaction paradigm layer, and User Study II (N=30) evaluates its process-oriented knowledge layer. Results show that CoNode significantly improves knowledge consolidation, reuse, and recombination, effectively facilitating the collaborative processes and demonstrating how generative AI can evolve process knowledge across collaborative rounds.

The sigmoid function, as a widely used activation function in neural networks, has gained much attention for its approximation and associated usage in edge devices. A recent study applied the Gaussian cumulative function to approximate the sigmoid function. Although this probabilistic method simplifies hardware implementation through a low-complexity binary search, it requires intensive random access memory (RAM) storage, and the search process is time-consuming. Besides, it targets minimizing the maximum mapping error rather than ensuring accurate approximation across all inputs. To address these issues, this article proposes a hardware-friendly and high-accuracy probabilistic-based sigmoid approximator. We first present that given an input, the output of a sigmoid function is strictly equivalent to the probability of a logistic random variable less than or equal to this input. Then, an indirect random variable quantizing strategy is exhibited to reduce memory usage and concurrently minimize precision loss. The latency for the proposed scheme is also optimized. Afterward, a resource-efficient and low-latency sigmoid approximator is developed on digital circuits. Finally, we derive an upper bound on the absolute error between the approximator's output and the true value. Experiments verify the usefulness of our scheme and showcase superior performance in approximation accuracy and resource cost.

Existing time-series forecasting methods often struggle to adapt to dynamic scenarios and lack flexibility in prediction. They typically require retraining the model when the prediction length or position changes. Moreover, these methods still face challenges in effectively capturing and utilizing time-position embeddings (PEs). To address these limitations, this article proposes a novel model called D2Vformer. Unlike conventional prediction methods that rely on fixed-length predictors, D2Vformer can directly handle scenarios with arbitrary prediction lengths. In addition, it significantly reduces training resource consumption and proves highly effective in real-world dynamic environments. In D2Vformer, the Date2Vec (D2V) module is devised to leverage timestamp information and feature sequences to generate time PEs. Subsequently, D2Vformer introduces an innovative fusion module that leverages an attention mechanism to capture the mapping between input and target time PEs, thereby enabling flexible prediction. Extensive experiments on six datasets demonstrate that D2V outperforms other time-PE methods, while D2Vformer surpasses state-of-the-art approaches in both fixed-length and arbitrary-length prediction tasks. The code for D2Vformer is available at: https://github.com/TeamofHaoWang/D2Vformer.

In urban transportation systems, missing data and noise contamination are almost inevitable. To address the challenge of imputing traffic data corrupted by noise and outliers in real-world scenarios, this paper proposes a novel algorithm based on spatiotemporal tensor completion. The proposed method transforms observed data into three-dimensional spatiotemporal tensors and utilizes tensor ring decomposition for data completion. Furthermore, spatial and temporal information is incorporated into the model by utilizing the graph Laplacian matrix. To handle outliers, they are treated as unknown parameters, and the ℓ0-norm is introduced to ensure their sparsity, thereby achieving the Spatio-Temporal Tensor Completion model with ℓ0-norm term (STTC-ℓ0). The solution to the model is derived using the alternating optimization framework with the alternating direction method of multipliers. Then, we discuss the convergence of the solution method. To further enhance the efficiency of our proposed method, we combine the unrolling algorithm with our iterative optimization model, creating a lightweight and efficient neural network tailored for tensor completion, called STTC-ℓ0-NN. Extensive experiments conducted on real datasets demonstrate the superiority of our proposed method over several state-of-the-art methods across various experimental scenarios. It is worth noting that STTC-ℓ0-NN reduces computational time by one to two orders of magnitude compared to existing methods while maintaining or even improving imputation accuracy. The code is available at https://github.com/TCCofWANG/STTC-L0-and-STTC-L0-NN.

This article presents a distributed k-winner-take-all (kWTA) with application in sealed-bid auctions with bidding price privacy protection. The proposed kWTA is in essence a distributed network of n agents which are arbitrarily connected. Let $\aleph _{i}$ be the set of neighbor agents of the ith agent, $u_{i}$ , $x_{i}$ , and $z_{i}$ are, respectively, its input, state variable, and output. The dynamics of the ith agent is given by $ ((dx_{i}(t))/dt) = \tau \left \{{{ z_{i}(x_{i}(t)) - (k/n) - \beta \sum _{j\in \aleph _{i}} (x_{i}(t) - x_{j}(t)) }}\right \}, z_{i}(x_{i}(t)) = h(u_{i}-x_{i}(t)), \text {for}~i = 1, \ldots , n$ where $\beta \gt 0$ , k is the number of winners and $h(\cdot)$ is the Heaviside function. By the theory of discontinuous dynamic systems, it is shown that the state equation for $d{\mathbf {x}}(t)/dt$ could be formulated as a gradient differential inclusion which minimizes the following nonsmooth convex function. $V({\mathbf {x}}) = \sum _{i=1}^{n} \max \{0, u_{i} - x_{i}\} + (k/n) \sum _{i=1}^{n} x_{i} + (\beta /2){\mathbf {x}}^{T} {\mathbf {L}} {\mathbf {x}}$ where ${\mathbf {x}} = (x_{1}, \ldots , x_{n})^{n}$ and ${\mathbf {L}} \in R^{n\times n}$ is the graph Laplacian matrix. A sufficient condition for $\beta $ is derived for the kWTA giving correct output and the condition is then applied in showing that ${\mathbf {z}}(t)$ converges to the correct output in finite-time. If $\beta \rightarrow \infty $ and $x_{1}(0) = \cdots = x_{n}(0)$ , we further show that $x_{1}(t) = \cdots = x_{n}(t)$ for $t \geq 0$ , and both ${\mathbf {z}}(t)$ and ${\mathbf {x}}(t)$ converge in finite-time. Besides, $x_{i}$ converges to $u_{\pi _{n-k+1}}$ (resp. $u_{\pi _{n-k}}$ ) if $x_{i}(0) \gg 1$ (resp. $x_{i}(0) = 0)$ for $i = 1, \ldots , n$ . If the input $u_{i}$ is set to be the bid price of the ith bidder and $k = 1$ , the proposed kWTA is able to determine both the winners and the clearing price for a sealed-bid first (resp. second) price auction in a distributed manner. Once ${\mathbf {z}}(t)$ and ${\mathbf {x}}(t)$ converge, each bidder can reveal from: 1) $z_{i}$ if he/she is a winner and 2) $x_{i}$ the clearing price. As bidders do not have to disclose their bidding prices during the winner (resp. the clearing price) determination process, the loosing (resp. winning) bidding price privacy can be protected in a sealed-bid first (resp. second) price auction. It is insofar the first application of an kWTA beyond the winner's determination.

In this brief, two fast discrete-time Wang kWTA (Fast Wang kWTA) algorithms are presented with an application in sealed-bid uniform price auctions. These algorithms can either be implemented in centralized or distributed manner. The structure of the Fast Wang kWTA is essentially the same as the original Wang k-winner-take-all (kWTA), except that our state update method is based on bisection method instead of gradient descent. By that, the number of iterations for getting correct output is largely reduced. Besides, the number is just a factor depended on the guess of the maximum input value. It is independent of the number of inputs, the number of winners, and the learning step size. The number of iterations is far smaller than the number required in the original Wang kWTA. In sequel, this Fast Wang kWTA is particularly suitable to be applied in solving the winner (resp. price) determination in real time and in distributed manner for a sealed-bid auction. In addition, the Fast Wang kWTA can ensure bidding price protection even if the communicated data are not encrypted and leaked.

Extreme learning machine (ELM) is an effective and efficient neural model for universal approximation. However, its practical performance can degrade due to weight noise, node faults, and outliers. This brief introduces a robust ELM algorithm designed to address these issues and enhance network robustness. We first analyze the square error of the classic ELM, considering both weight noise and node faults. By integrating an outlier-resistant method, the maximum correntropy criterion (MCC), we derive a new objective function to bolster network resilience. This leads to the development of the robust fault-aware ELM (RFAELM) algorithm. The convergence property of RFAELM is rigorously proven. For validation, the proposed algorithm is evaluated in various noise and fault levels using eight different benchmark datasets. The simulation results, encompassing all imperfect conditions and datasets, verify the robustness and generalization of this new algorithm. Also, the new algorithm is compared with other robust ELM algorithms using different statistical measurements. The superior performance of RFAELM substantiates its significant improvement over existing algorithms.

Expressive human pose and shape estimation (a.k.a. 3D whole-body mesh recovery) involves the human body, hand, and expression estimation. Most existing methods have tack-led this task in a two-stage manner, first detecting the human body part with an off-the-shelf detection model and then in-ferring the different human body parts individually. Despite the impressive results achieved, these methods suffer from 1) loss of valuable contextual information via cropping, 2) introducing distractions, and 3) lacking inter-association among different persons and body parts, inevitably causing performance degradation, especially for crowded scenes. To address these issues, we introduce a novel ali-in-one-stage framework, AiOS, for multiple expressive human pose and shape recovery without an additional human detection step. Specifically, our method is built upon DETR, which treats multi-person whole-body mesh recovery task as a progressive set prediction problem with various sequential detection. We devise the decoder tokens and extend them to our task. Specifically, we first employ a human token to probe a hu-man location in the image and encode global features for each instance, which provides a coarse location for the later transformer block. Then, we introduce a joint-related token to probe the human joint in the image and encoder a fine-grained local feature, which collaborates with the global feature to regress the whole-body mesh. This straightfor-ward but effective model outperforms previous state-of-the-art methods by a 9% reduction in NMVE on AGORA, a 30% reduction in PVE on EHF, a 10% reduction in PVE on ARCTIC, and a 3% reduction in PVE on EgoBody.

Direct illumination rendering is an important technique in computer graphics. Precomputed radiance transfer algorithms can provide high quality rendering results in real time, but they can only support rigid models. On the other hand, ray tracing algorithms are flexible and can gracefully handle animated models. With NVIDIA RTX and the AI denoiser, we can use ray tracing algorithms to render visually appealing results in real time. Visually appealing though, they can deviate from the actual one considerably. We propose a visibility-boundary edge oriented infinite triangle bounding volume hierarchy (BVH) traversal algorithm to dynamically generate visibility in vector form. Our algorithm utilizes the properties of visibility-boundary edges and infinite triangle BVH traversal to maximize the efficiency of the vector form visibility generation. A novel data structure, temporal vectorized visibility, is proposed, which allows visibility in vector form to be shared across time and further increases the generation efficiency. Our algorithm can efficiently render close-to-reference direct illumination results. With the similar processing time, it provides a visual quality improvement around 10 dB in terms of peak signal-to-noise ratio (PSNR) w.r.t. the ray tracing algorithm reservoir-based spatiotemporal importance resampling (ReSTIR).

Subspace location estimators are a class of range-based source localization (SL) methods built upon the multidimensional similarity (MDS) theory. Since they are computationally lightweight while maintaining a reasonably good level of positioning accuracy, these techniques can be well-suited for the context of Internet of Things (IoT), where precise localization is necessary but the on-device computational resources turn out to be relatively limited. MDS analysis (MDSA), in signal processing terms, is the statistical process of disentangling the signal subspace components from their disturbance counterparts for an observed MDS matrix that measures the similarity among multiple source-sensor coordinate differences. A prominent drawback of traditional MDSA schemes devised under the assumption of Gaussian noise is their vulnerability to outliers in the available range-type data, which are frequently encountered in IoT SL applications due to adverse environmental factors like non line-of-sight signal propagation and interference. In this contribution, we use symmetric$\alpha $-stable$(S \alpha S)$distributions to systematically characterize the MDS matrix observation errors, thus accounting for the existence of outliers. To resist against$S \alpha S$disturbances, we cast MDSA as an$\ell _{p}$-norm-based robust low-rank approximation problem. We then develop a practical optimization solution by means of the alternating direction method of multipliers, for which we further conduct a theoretical analysis of convergence. Simulations and real-world experiments confirm the feasibility of our robust subspace positioning approach.

Over the past two decades, sparse unmixing (SU) has gained significant attention in the realm of hyperspectral imaging. The aims of SU are to seek a subset of spectral signatures and estimate their fractional abundances to represent each mixed spectral pixel. Conventional SU methods often employ the Frobenius norm and thus cannot work satisfactorily in the presence of non-Gaussian noise. Second, the ideal$\ell _{0}$-norm is usually substituted with its convex or nonconvex approximation in most existing algorithms, which may degrade the recovery performance. To address these issues, this article proposes a novel approach, termed sparse unmixing using$\ell _{0}$-norm constraint and log-cosh loss (SUNNING). We exploit the$\log $-$\cosh $function to minimize the fitting errors subject to three constraints, namely, nonnegativity, sum-to-one, and upper bounded$\ell _{0}$-norm. Then, we adopt the projected gradient descent (PGD) framework to solve such an optimization problem. SUNNING includes two alternating steps, gradient descent and nonconvex projection, where an optimality of the solution is guaranteed. Also, we prove the convergence of SUNNING, including the objective value and variable sequence. In addition, to attain higher unmixing accuracy, we exploit the spectral library pruning (SLP) strategy to eliminate inactive endmembers, yielding an improved SUNNING. Experimental results on synthetic and real-world datasets exhibit improved robustness and effectiveness of the suggested methods over the state-of-the-art algorithms. MATLAB code is available at:https://github.com/freeLix-YY/IEEE_TGRS2024_SparseUnmixing_SUNNING_demo

The effectiveness of an image classification system depends on the following two key components: 1) the feature learning module and 2) the classification module. A well-designed loss function can not only enhance the classification ability of the latter but also improve the feature extraction capabilities of the former. This article devises a novel hypersphere loss function, which enhances the intraclass compactness and interclass separability of feature vectors given by the feature learning module. Furthermore, a new generalized class center is introduced into the loss function to handle the inevitable variability in samples (such as illumination, background, blurriness, low resolution, etc.) within the same class. Then, an alternative learning strategy is employed to optimize trainable parameters and class centers. Specifically, we first fix the trainable parameters of the deep learning model and calculate class centers using the exponentially weighted moving average method. Subsequently, we fix the generalized class centers and update the model's trainable parameters using mini-batch stochastic gradient descent. The proposed algorithm is evaluated on a range of typical tasks, including standard image classification, face verification, object detection, and retail product checkout. The results demonstrate that our proposed algorithm outperforms several state-of-the-art approaches.

The dual neural network (DNN)-based k -winner-take-all (WTA) model is able to identify the k largest numbers from its m input numbers. When there are imperfections, such as non-ideal step function and Gaussian input noise, in the realization, the model may not output the correct result. This brief analyzes the influence of the imperfections on the operational correctness of the model. Due to the imperfections, it is not efficient to use the original DNN- k WTA dynamics for analyzing the influence. In this regard, this brief first derives an equivalent model to describe the dynamics of the model under the imperfections. From the equivalent model, we derive a sufficient condition for which the model outputs the correct result. Thus, we apply the sufficient condition to design an efficiently estimation method for the probability of the model outputting the correct result. Furthermore, for the inputs with uniform distribution, a closed form expression for the probability value is derived. Finally, we extend our analysis for handling non-Gaussian input noise. Simulation results are provided to validate our theoretical results.

Among many -winners-take-all ( WTA) models, the dual-neural network (DNN- WTA) model is with significantly less number of connections. However, for analog realization, noise is inevitable and affects the operational correctness of the WTA process. Most existing results focus on the effect of additive noise. This brief studies the effect of time-varying multiplicative input noise. Two scenarios are considered. The first one is the bounded noise case, in which only the noise range is known. Another one is for the general noise distribution case, in which we either know the noise distribution or have noise samples. For each scenario, we first prove the convergence property of the DNN- WTA model under multiplicative input noise and then provide an efficient method to determine whether a noise-affected DNN- WTA network performs the correct WTA process for a given set of inputs. With the two methods, we can efficiently measure the probability of the network performing the correct WTA process. In addition, for the case of the inputs being uniformly distributed, we derive two closed-form expressions, one for each scenario, for estimating the probability of the model having correct operation. Finally, we conduct simulations to verify our theoretical results.

Inspired by sparse learning, the Markowitz mean-variance model with a sparse regularization term is popularly used in sparse portfolio optimization. However, in penalty-based portfolio optimization algorithms, the cardinality level of the resultant portfolio relies on the choice of the regularization parameter. This brief formulates the mean-variance model as a cardinality ($\ell _{0}$-norm) constrained nonconvex optimization problem, in which we can explicitly specify the number of assets in the portfolio. We then use the alternating direction method of multipliers (ADMMs) concept to develop an algorithm to solve the constrained nonconvex problem. Unlike some existing algorithms, the proposed algorithm can explicitly control the portfolio cardinality. In addition, the dynamic behavior of the proposed algorithm is derived. Numerical results on four real-world datasets demonstrate the superiority of our approach over several state-of-the-art algorithms.

Deep neural networks boost many successful colorization methods, including automatic, interactive, and exemplar-based methods. Among them, interactive methods with global and/or local inputs are probably the most flexible to accurately add colors to a gray image. However, due to the sparseness of input-semantic correspondences, existing methods encounter difficulties in distributing inputs into correct regions. Moreover, they simply add or concatenate the features of different inputs to the network before color reconstruction, which cannot balance the influences of different inputs. To this end, we propose a novel interactive colorization network, which explicitly builds input-semantic correspondences with an attention mechanism and proposes a gated feature fusion module to balance the influences of global and local inputs. We further apply a differentiable histogram loss to impose a smooth impact of the global inputs. Extensive experiments demonstrate that our method can flexibly control the results and outperforms other state-of-the-art interactive methods.

Passive multistatic localization (PML) refers to locating a signal-reflecting/relaying target using the bistatic range and direct range measurements acquired by employing multiple spatially-separated transmitters and receivers, where the transmitter positions are unknown. In real-world applications, one of the major technical challenges faced in PML is the non-lineof-sight (NLOS) propagation of signals, and recent studies have turned to the concept of robust statistics to tackle such an issue. Continuing to delve into this research direction, here we address the discrepancy arising from the fact that the conventional robust statistical PML schemes inherently assume zero-centered error samples, which may not hold true when the positive NLOS biases are present. In contrast to the existing PML solutions, our proposal is based on the maximum correntropy criterion with variable center (MCC-VC), thereby taking into consideration the potential non-zero-centrality of error samples in the estimator derivation. Subsequently, we develop an alternating minimization algorithm to handle the nonconvex MCC-VC optimization problem in a way that can strike a fine balance between accuracy and computational efficiency. The superiority of our PML approach over its competitors is demonstrated via computer simulations

Reconstructing a 3D human body mesh from a monocular image is a challenging inverse problem because of occlusion and complicated human articulations. Recent deep learning-based methods have made significant progress in single-image human reconstruction. Most of these works are either model-based methods or model-free methods. However, model-based methods always suffer detail losses due to the limited parameter space, and model-free methods are hard to directly recover satisfactory results from images due to the use of a shared global feature for all vertices and the domain gap between 2D regular images and 3D irregular meshes. To resolve these issues, we propose a hybrid model, which combines the advantages of both model based approach and model-free approach to estimate a 3D human mesh in a coarse-to fine manner. Initially, we utilize a convolutional neural network (CNN) to estimate the parameters of a Skinned Multi-Person Linear Model (SMPL), which allows us to generate a coarse human mesh. After that, the vertex coordinates of the coarse human mesh are further refined by a graph convolutional neural network (GCN). Unlike previous GCN-based methods, whose vertex coordinates are recovered from a shared global feature, we propose a LOcal CorRespondence-Aware (LOCRA) module to extract local special features for each vertex. To make the local features related to the human pose, we also add a keypoint-related loss to supervise the training process of the LOCRA module. Experiments demonstrate that our hybrid model with the LOCRA module outperforms existing methods on multiple public benchmarks.

Sparse index tracking, as one of the passive investment strategies, is to track a benchmark financial index via constructing a portfolio with a few assets in a market index. It can be considered as parameter learning in an adaptive system, in which we periodically update the selected assets and their investment percentages based on the sliding window approach. However, many existing algorithms for sparse index tracking cannot explicitly and directly control the number of assets or the tracking error. This article formulates sparse index tracking as two constrained optimization problems and then proposes two algorithms, namely, nonnegative orthogonal matching pursuit with projected gradient descent (NNOMP-PGD) and alternating direction method of multipliers for$\ell _{0}$-norm (ADMM-$\ell _{0}$). The NNOMP-PGD aims at minimizing the tracking error subject to the number of selected assets less than or equal to a predefined number. With the NNOMP-PGD, investors can directly and explicitly control the number of selected assets. The ADMM-$\ell _{0}$aims at minimizing the number of selected assets subject to the tracking error that is upper bounded by a preset threshold. It can directly and explicitly control the tracking error. The convergence of the two proposed algorithms is also presented. With our algorithms, investors can explicitly and directly control the number of selected assets or the tracking error of the resultant portfolio. In addition, numerical experiments demonstrate that the proposed algorithms outperform the existing approaches.

From the feature representation's point of view, the feature learning module of a convolutional neural network (CNN) is to transform an input pattern into a feature vector. This feature vector is then multiplied with a number of output weight vectors to produce softmax scores. The common training objective in CNNs is based on the softmax loss, which ignores the intra-class compactness. This brief proposes a constrained center loss (CCL)-based algorithm to extract robust features. The training objective of a CNN consists of two terms, softmax loss and CCL. The aim of the softmax loss is to push the feature vectors from different classes apart. Meanwhile, the CCL aims at clustering the feature vectors such that the feature vectors from the same classes are close together. Instead of using stochastic gradient descent (SGD) algorithms to learn all the connection weights and the cluster centers at the same time. Our CCL-based algorithm is based on the alternative learning strategy. We first fix the connection weights of the CNN and update the cluster centers based on an analytical formula, which can be implemented based on the minibatch concept. We then fix the cluster centers and update the connection weights for a number of SGD minibatch iterations. We also propose a simplified CCL (SCCL) algorithm. Experiments are performed on six commonly used benchmark datasets. The results demonstrate that the two proposed algorithms outperform several state-of-the-art approaches.

For decades, adding fault/noise during training by gradient descent has been a technique for getting a neural network (NN) tolerant to persistent fault/noise or getting an NN with better generalization. In recent years, this technique has been readvocated in deep learning to avoid overfitting. Yet, the objective function of such fault/noise injection learning has been misinterpreted as the desired measure (i.e., the expected mean squared error (mse) of the training samples) of the NN with the same fault/noise. The aims of this article are: 1) to clarify the above misconception and 2) investigate the actual regularization effect of adding node fault/noise when training by gradient descent. Based on the previous works on adding fault/noise during training, we speculate the reason why the misconception appears. In the sequel, it is shown that the learning objective of adding random node fault during gradient descent learning (GDL) for a multilayer perceptron (MLP) is identical to the desired measure of the MLP with the same fault. If additive (resp. multiplicative) node noise is added during GDL for an MLP, the learning objective is not identical to the desired measure of the MLP with such noise. For radial basis function (RBF) networks, it is shown that the learning objective is identical to the corresponding desired measure for all three fault/noise conditions. Empirical evidence is presented to support the theoretical results and, hence, clarify the misconception that the objective function of a fault/noise injection learning might not be interpreted as the desired measure of the NN with the same fault/noise. Afterward, the regularization effect of adding node fault/noise during training is revealed for the case of RBF networks. Notably, it is shown that the regularization effect of adding additive or multiplicative node noise (MNN) during training an RBF is reducing network complexity. Applying dropout regularization in RBF networks, its effect is the same as adding MNN during training.

The objective of compressive sampling is to determine a sparse vector from an observation vector. This brief describes an analog neural method to achieve the objective. Unlike previous analog neural models which either resort to the$\ell _{1}$-norm approximation or are with local convergence only, the proposed method avoids any approximation of the$\ell _{1}$-norm term and is probably capable of leading to the optimum solution. Moreover, its computational complexity is lower than that of the other three comparison analog models. Simulation results show that the error performance of the proposed model is comparable to several state-of-the-art digital algorithms and analog models and that its convergence is faster than that of the comparison analog neural models.

User-guided colorization can predict the colors of a grayscale image according to user inputs, including exemplar images, local inputs and global inputs. Global inputs-based methods are probably the easiest ones to use, but are hard to distribute the input colors into correct regions, due to the lack of color-semantic correspondences. In this paper, we propose a novel histogram-guided semantic-aware colorization method, which explicitly builds the correspondences between global colors and local features with an attention mechanism and uses a differentiable histogram loss to impose the histogram of the results. Our method starts with a semantic-aware subnetwork to build the color-semantic correspondences, followed by a colorization subnetwork to reconstruct the color channels. Experiments demonstrate that our method can effectively control the results with the input histogram. Extensive visual, numerical and user study comparisons show that our method outperforms other global input-based state-of-the-art methods in color naturalness and consistency.

The dual neural network-based$k$-winner-take-all (DNN-$k$WTA) is an analog neural model that is used to identify the$k$largest numbers from$n$inputs. Since threshold logic units (TLUs) are key elements in the model, offset voltage drifts in TLUs may affect the operational correctness of a DNN-$k$WTA network. Previous studies assume that drifts in TLUs follow some particular distributions. This brief considers that only the drift range, given by$[-\Delta, \Delta]$, is available. We consider two drift cases: time-invariant and time-varying. For the time-invariant case, we show that the state of a DNN-$k$WTA network converges. The sufficient condition to make a network with the correct operation is given. Furthermore, for uniformly distributed inputs, we prove that the probability that a DNN-$k$WTA network operates properly is greater than$(1-2\Delta)^{n}$. The aforementioned results are generalized for the time-varying case. In addition, for the time-invariant case, we derive a method to compute the exact convergence time for a given data set. For uniformly distributed inputs, we further derive the mean and variance of the convergence time. The convergence time results give us an idea about the operational speed of the DNN-$k$WTA model. Finally, simulation experiments have been conducted to validate those theoretical results.

Recent methods based on deep learning have shown promise in converting grayscale images to colored ones. However, most of them only allow limited user inputs (no inputs, only global inputs, or only local inputs), to control the output colorful images. The possible difficulty lies in how to differentiate the influences of different inputs. To solve this problem, we propose a two-stage deep colorization method allowing users to control the results by flexibly setting global inputs and local inputs. The key steps include enabling color themes as global inputs by extracting K mean colors and generating K-color maps to define a global theme loss, and designing a loss function to differentiate the influences of different inputs without causing artifacts. We also propose a color theme recommendation method to help users choose color themes. Based on the colorization model, we further propose an image compression scheme, which supports variable compression ratios in a single network. Experiments on colorization show that our method can flexibly control the colorized results with only a few inputs and generate state-of-the-art results. Experiments on compression show that our method achieves much higher image quality at the same compression ratio when compared to the state-of-the-art methods.

Image colorization recovers a colorful image from a grayscale one. Trained by large-scale datasets, recent deep neural networks based methods can produce impressive colorful images. However, they usually directly train a single network using training images of fixed resolution. It is hard for such a single network to learn the features of different scales for colorization. Moreover, they are prone to generate color bleedings and blurry details around objects boundaries. To address these problems, we propose a novel edge-aware multi-scale progressive network (EMSPN). The key idea is to train a series of multi-scale networks in a progressive manner, so that the network in finer scales can leverage the outputs of its previous scale. In addition, we also propose an edge-map loss to effectively prevent bleedings and blurs around the image edges. Experimental results show that our work outperforms existing methods and achieves state-of-the-art results.

Over decades, gradient descent has been applied to develop learning algorithm to train a neural network (NN). In this brief, a limitation of applying such algorithm to train an NN with persistent weight noise is revealed. Let V(w) be the performance measure of an ideal NN. V(w) is applied to develop the gradient descent learning (GDL). With weight noise, the desired performance measure (denoted as J(w) ) is E[V(~w)|w] , where ~w is the noisy weight vector. Applying GDL to train an NN with weight noise, the actual learning objective is clearly not V(w) but another scalar function L(w) . For decades, there is a misconception that L(w) = J(w) , and hence, the actual model attained by the GDL is the desired model. However, we show that it might not: 1) with persistent additive weight noise, the actual model attained is the desired model as L(w) = J(w) ; and 2) with persistent multiplicative weight noise, the actual model attained is unlikely the desired model as L(w) ≠ J(w) . Accordingly, the properties of the models attained as compared with the desired models are analyzed and the learning curves are sketched. Simulation results on 1) a simple regression problem and 2) the MNIST handwritten digit recognition are presented to support our claims.

Colorization methods using deep neural networks have become a recent trend. However, most of them do not allow user inputs, or only allow limited user inputs (only global inputs or only local inputs), to control the output colorful images. The possible reason is that it's difficult to differentiate the influence of different kind of user inputs in network training. To solve this problem, we propose a novel deep colorization method allowing inputting global and local inputs simultaneously or individually, which is not supported in previous deep colorization methods. The key steps include designing a neural network model that can appropriately combine the different inputs, and designing an appropriate loss function that can differentiate the influence of different inputs. Experimental results show that our method can magnificently control the colorized results and generate state-of-art results.

Abstract Dual paraboloid mapping is an approach for environment mapping. Its major advantage is its fast map generation speed. For graphics applications, when filtering is needed, the filtering tool would naturally be mipmapping. However, directly applying mipmapping to dual paraboloid mapping would give us three problems. They are the discontinuity across the dual paraboloid map boundary, the non‐uniform sampling problem and the depth testing issue. We propose three approaches to solve these problems. Our approaches are based on some closed form equations derived via theoretical analysis. Using these equations, we modify the coordinates involved during the rendering process. In other words, these problems are handled just by using dual paraboloid maps and mipmaps differently, instead of fundamentally altering their data structures. Consequently, we are fixing the problems without damaging the map generation speed advantage. Applying all three approaches, we improve the rendering quality of dual paraboloid map mipmaps to a level equivalent to that of cubemap mipmaps, while preserving its fast map generation speed advantage. This gives dual paraboloid map mipmaps the potential to be a better choice than cubemap mipmaps for the devices with less computational power. The effectiveness and the efficiency of the proposed approaches are demonstrated using a glossy reflection application and an omnidirectional soft shadow generation application.

This brief presents analytical results on the effect of additive weight/bias noise on a Boltzmann machine (BM), in which the unit output is in {-1, 1} instead of {0, 1}. With such noise, it is found that the state distribution is yet another Boltzmann distribution but the temperature factor is elevated. Thus, the desired gradient ascent learning algorithm is derived, and the corresponding learning procedure is developed. This learning procedure is compared with the learning procedure applied to train a BM with noise. It is found that these two procedures are identical. Therefore, the learning algorithm for noise-free BMs is suitable for implementing as an online learning algorithm for an analog circuit-implemented BM, even if the variances of the additive weight noise and bias noise are unknown.

This paper studies the effects of uniform input noise and Gaussian input noise on the dual neural network-based WTA (DNN- WTA) model. We show that the state of the network (under either uniform input noise or Gaussian input noise) converges to one of the equilibrium points. We then derive a formula to check if the network produce correct outputs or not. Furthermore, for the uniformly distributed inputs, two lower bounds (one for each type of input noise) on the probability that the network produces the correct outputs are presented. Besides, when the minimum separation amongst inputs is given, we derive the condition for the network producing the correct outputs. Finally, experimental results are presented to verify our theoretical results. Since random drift in the comparators can be considered as input noise, our results can be applied to the random drift situation.

A commonly used measurement model for locating a mobile source is time-difference-of-arrival (TDOA). As each TDOA measurement defines a hyperbola, it is not straightforward to compute the mobile source position due to the nonlinear relationship in the measurements. This brief exploits the Lagrange programming neural network (LPNN), which provides a general framework to solve nonlinear constrained optimization problems, for the TDOA-based localization. The local stability of the proposed LPNN solution is also analyzed. Simulation results are included to evaluate the localization accuracy of the LPNN scheme by comparing with the state-of-the-art methods and the optimality benchmark of Cramér-Rao lower bound.

In this paper, the effect of input noise, output node stochastic, and recurrent state noise on the Wang $k$ WTA is analyzed. Here, we assume that noise exists at the recurrent state $y(t)$ and it can either be additive or multiplicative. Besides, its dynamical change (i.e., $dy/dt$ ) is corrupted by noise as well. In sequel, we model the dynamics of $y(t)$ as a stochastic differential equation and show that the stochastic behavior of $y(t)$ is equivalent to an Ito diffusion. Its stationary distribution is a Gibbs distribution, whose modality depends on the noise condition. With moderate input noise and very small recurrent state noise, the distribution is single modal and hence $y(\infty )$ has high probability varying within the input values of the $k$ and $k+1$ winners (i.e., correct output). With small input noise and large recurrent state noise, the distribution could be multimodal and hence $y(\infty )$ could have probability varying outside the input values of the $k$ and $k+1$ winners (i.e., incorrect output). In this regard, we further derive the conditions that the $k$ WTA has high probability giving correct output. Our results reveal that recurrent state noise could have severe effect on Wang $k$ WTA. But, input noise and output node stochastic could alleviate such an effect.

In the training stage of radial basis function (RBF) networks, we need to select some suitable RBF centers first. However, many existing center selection algorithms were designed for the fault-free situation. This brief develops a fault tolerant algorithm that trains an RBF network and selects the RBF centers simultaneously. We first select all the input vectors from the training set as the RBF centers. Afterward, we define the corresponding fault tolerant objective function. We then add an -norm term into the objective function. As the -norm term is able to force some unimportant weights to zero, center selection can be achieved at the training stage. Since the -norm term is nondifferentiable, we formulate the original problem as a constrained optimization problem. Based on the alternating direction method of multipliers framework, we then develop an algorithm to solve the constrained optimization problem. The convergence proof of the proposed algorithm is provided. Simulation results show that the proposed algorithm is superior to many existing center selection algorithms.

The original Summed Area Table (SAT) structure is designed for handling 2D rectangular data. Due to the nature of spherical functions, the SAT structure cannot handle cube maps directly. This paper proposes a new SAT structure for cube maps and develops the corresponding lookup algorithm. Our formulation starts by considering a cube map as part of an auxiliary 3D function defined in a 3D rectangular space. We interpret the 2D integration process over the cube map surface as a 3D integration over the auxiliary 3D function. One may suggest that we can create a 3D SAT for this auxiliary function, and then use the 3D SAT to achieve the 3D integration. However, it is not practical to generate or store this special 3D SAT directly. This 3D SAT has some nice properties that allow us to store it in a storage-friendly data structure, namely Summed Area Cube Map (SACM). A SACM can be stored in a standard cube map texture. The lookup algorithm of our SACM structure can be implemented efficiently on current graphics hardware. In addition, the SACM structure inherits the favorable properties of the original SAT structure.

The major limitation of the Lagrange programming neural network (LPNN) approach is that the objective function and the constraints should be twice differentiable. Since sparse approximation involves nondifferentiable functions, the original LPNN approach is not suitable for recovering sparse signals. This paper proposes a new formulation of the LPNN approach based on the concept of the locally competitive algorithm (LCA). Unlike the classical LCA approach which is able to solve unconstrained optimization problems only, the proposed LPNN approach is able to solve the constrained optimization problems. Two problems in sparse approximation are considered. They are basis pursuit (BP) and constrained BP denoise (CBPDN). We propose two LPNN models, namely, BP-LPNN and CBPDN-LPNN, to solve these two problems. For these two models, we show that the equilibrium points of the models are the optimal solutions of the two problems, and that the optimal solutions of the two problems are the equilibrium points of the two models. Besides, the equilibrium points are stable. Simulations are carried out to verify the effectiveness of these two LPNN models.

Many existing results on fault-tolerant algorithms focus on the single fault source situation, where a trained network is affected by one kind of weight failure. In fact, a trained network may be affected by multiple kinds of weight failure. This paper first studies how the open weight fault and the multiplicative weight noise degrade the performance of radial basis function (RBF) networks. Afterward, we define the objective function for training fault-tolerant RBF networks. Based on the objective function, we then develop two learning algorithms, one batch mode and one online mode. Besides, the convergent conditions of our online algorithm are investigated. Finally, we develop a formula to estimate the test set error of faulty networks trained from our approach. This formula helps us to optimize some tuning parameters, such as RBF width.

Fault tolerance is one interesting property of artificial neural networks. However, the existing fault models are able to describe limited node fault situations only, such as stuck-at-zero and stuck-at-one. There is no general model that is able to describe a large class of node fault situations. This paper studies the performance of faulty radial basis function (RBF) networks for the general node fault situation. We first propose a general node fault model that is able to describe a large class of node fault situations, such as stuck-at-zero, stuck-at-one, and the stuck-at level being with arbitrary distribution. Afterward, we derive an expression to describe the performance of faulty RBF networks. An objective function is then identified from the formula. With the objective function, a training algorithm for the general node situation is developed. Finally, a mean prediction error (MPE) formula that is able to estimate the test set error of faulty networks is derived. The application of the MPE formula in the selection of basis width is elucidated. Simulation experiments are then performed to demonstrate the effectiveness of the proposed method.

Abstract In vector graphics, gradient meshes represent an image object by one or more regularly connected grids. Every grid point has attributes as the position, colour and gradients of these quantities specified. Editing the attributes of an existing gradient mesh (such as the colour gradients) is not only non‐intuitive but also time‐consuming. To facilitate user‐friendly colour editing, we develop an optimization‐based colour transfer method for gradient meshes. The key idea is built on the fact that we can approximate a colour transfer operation on gradient meshes with a linear transfer function. In this paper, we formulate the approximation as an optimization problem, which aims to minimize the colour distribution of the example image and the transferred gradient mesh. By adding proper constraints, i.e. image gradients, to the optimization problem, the details of the gradient meshes can be better preserved. With the linear transfer function, we are able to edit the colours and colour gradients of the mesh points automatically, while preserving the structure of the gradient mesh. The experimental results show that our method can generate pleasing recoloured gradient meshes.

The dual neural network (DNN)-based k -winner-take-all ( k WTA) model is an effective approach for finding the k largest inputs from n inputs. Its major assumption is that the threshold logic units (TLUs) can be implemented in a perfect way. However, when differential bipolar pairs are used for implementing TLUs, the transfer function of TLUs is a logistic function. This brief studies the properties of the DNN- kWTA model under this imperfect situation. We prove that, given any initial state, the network settles down at the unique equilibrium point. Besides, the energy function of the model is revealed. Based on the energy function, we propose an efficient method to study the model performance when the inputs are with continuous distribution functions. Furthermore, for uniformly distributed inputs, we derive a formula to estimate the probability that the model produces the correct outputs. Finally, for the case that the minimum separation ∆min of the inputs is given, we prove that if the gain of the activation function is greater than 1/4∆min max(ln 2n, 2 ln 1 - ϵ/ϵ ), then the network can produce the correct outputs with winner outputs greater than 1-ϵ and loser outputs less than ϵ, where ϵ is the threshold less than 0.5.

Many existing pre-computed radiance transfer (PRT) approaches for all-frequency lighting store the information of a 3D object in the pre-vertex manner. To preserve the fidelity of high frequency effects, the 3D object must be tessellated densely. Otherwise, rendering artifacts due to interpolation may appear. This paper presents an all-frequency lighting algorithm for direct illumination based on a new visibility representation which approximates a visibility function using a sequence of 3D vectors. The algorithm is able to construct the visibility function of an on-screen pixel on-the-fly. Hence even though the 3D object is not tessellated densely, the rendering artifacts can be suppressed greatly. Besides, a summed area table based rendering algorithm, which is able to handle the integration over a non-axis aligned polygon, is developed. Using our approach, we can rotate lighting environment, change view point, and adjust the shininess of the 3D object in a real-time manner. Experimental results show that our approach can render plausible all-frequency lighting effects for direct illumination in real-time, especially for specular shadows, which are difficult for other methods to obtain.

Abstract Soft shadows play an important role in photo‐realistic rendering. Although there are many efficient soft shadow algorithms, most of them focus on the one‐side light source situation, where a planar light source is on the outside of the scene. In fact, in many situations, such as games, light sources are omnidirectional. They may be surrounded by a number of 3D objects. This paper proposes a soft shadow algorithm for the omnidirectional situation. We develop a concentric spherical representation to model the behaviour of omnidirectional light sources. To provide better rendering results, a novel summed area table based filtering scheme for spherical functions is proposed. In addition, we utilize unicube mapping, which samples the spherical space more uniformly, to further improve the filtering quality.

Editing a photo-realistic gradient mesh is a tough task. Even only editing the colors of an existing gradient mesh can be exhaustive and time-consuming. To facilitate user-friendly color editing, we develop an example-based color transfer method for gradient meshes, which borrows the color characteristics of an example image to a gradient mesh. We start by exploiting the constraints of the gradient mesh, and accordingly propose a linear-operator-based color transfer framework. Our framework operates only on colors and color gradients of the mesh points and preserves the topological structure of the gradient mesh. Bearing the framework in mind, we build our approach on PCA-based color transfer. After relieving the color range problem, we incorporate a fusion-based optimization scheme to improve color similarity between the reference image and the recolored gradient mesh. Finally, a multi-swatch transfer scheme is provided to enable more user control. Our approach is simple, effective, and much faster than color transferring the rastered gradient mesh directly. The experimental results also show that our method can generate pleasing recolored gradient meshes.

Recently, an analog neural network model, namely Wang's kWTA, was proposed. In this model, the output nodes are defined as the Heaviside function. Subsequently, its finite time convergence property and the exact convergence time are analyzed. However, the discovered characteristics of this model are based on the assumption that there are no physical defects during the operation. In this brief, we analyze the convergence behavior of the Wang's kWTA model when defects exist during the operation. Two defect conditions are considered. The first one is that there is input noise. The second one is that there is stochastic behavior in the output nodes. The convergence of the Wang's kWTA under these two defects is analyzed and the corresponding energy function is revealed.

Fault tolerance is an interesting topic in neural networks. However, many existing results on this topic focus only on the situation of a single fault source. In fact, a trained network may be affected by multiple fault sources. This brief studies the performance of faulty radial basis function (RBF) networks that suffer from multiplicative weight noise and open weight fault concurrently. We derive a mean prediction error (MPE) formula to estimate the generalization ability of faulty networks. The MPE formula provides us a way to understand the generalization ability of faulty networks without using a test set or generating a number of potential faulty networks. Based on the MPE result, we propose methods to optimize the regularization parameter, as well as the RBF width.

Improving fault tolerance of a neural network has been studied for more than two decades. Various training algorithms have been proposed in sequel. The on-line node fault injection-based algorithm is one of these algorithms, in which hidden nodes randomly output zeros during training. While the idea is simple, theoretical analyses on this algorithm are far from complete. This paper presents its objective function and the convergence proof. We consider three cases for multilayer perceptrons (MLPs). They are: (1) MLPs with single linear output node; (2) MLPs with multiple linear output nodes; and (3) MLPs with single sigmoid output node. For the convergence proof, we show that the algorithm converges with probability one. For the objective function, we show that the corresponding objective functions of cases (1) and (2) are of the same form. They both consist of a mean square errors term, a regularizer term, and a weight decay term. For case (3), the objective function is slight different from that of cases (1) and (2). With the objective functions derived, we can compare the similarities and differences among various algorithms and various cases.

Injecting weight noise during training is a simple technique that has been proposed for almost two decades. However, little is known about its convergence behavior. This paper studies the convergence of two weight noise injection-based training algorithms, multiplicative weight noise injection with weight decay and additive weight noise injection with weight decay. We consider that they are applied to multilayer perceptrons either with linear or sigmoid output nodes. Let w(t) be the weight vector, let V(w) be the corresponding objective function of the training algorithm, let α >; 0 be the weight decay constant, and let μ(t) be the step size. We show that if μ(t)→ 0, then with probability one E[||w(t)||2(2)] is bound and lim(t) → ∞ ||w(t)||2 exists. Based on these two properties, we show that if μ(t)→ 0, Σtμ(t)=∞, and Σtμ(t)(2) <; ∞, then with probability one these algorithms converge. Moreover, w(t) converges with probability one to a point where ∇wV(w)=0.

A k-winner-take-all (kWTA) network is able to find out the k largest numbers from n inputs. Recently, a dual neural network (DNN) approach was proposed to implement the kWTA process. Compared to the conventional approach, the DNN approach has much less number of interconnections. A rough upper bound on the convergence time of the DNN-kWTA model, which is expressed in terms of input variables, was given. This brief derives the exact convergence time of the DNN-kWTA model. With our result, we can study the convergence time without spending excessive time to simulate the network dynamics. We also theoretically study the statistical properties of the convergence time when the inputs are uniformly distributed. Since a nonuniform distribution can be converted into a uniform one and the conversion preserves the ordering of the inputs, our theoretical result is also valid for nonuniformly distributed inputs.

Injecting weight noise during training has been a simple strategy to improve the fault tolerance of multilayer perceptrons (MLPs) for almost two decades, and several online training algorithms have been proposed in this regard. However, there are some misconceptions about the objective functions being minimized by these algorithms. Some existing results misinterpret that the prediction error of a trained MLP affected by weight noise is equivalent to the objective function of a weight noise injection algorithm. In this brief, we would like to clarify these misconceptions. Two weight noise injection scenarios will be considered: one is based on additive weight noise injection and the other is based on multiplicative weight noise injection. To avoid the misconceptions, we use their mean updating equations to analyze the objective functions. For injecting additive weight noise during training, we show that the true objective function is identical to the prediction error of a faulty MLP whose weights are affected by additive weight noise. It consists of the conventional mean square error and a smoothing regularizer. For injecting multiplicative weight noise during training, we show that the objective function is different from the prediction error of a faulty MLP whose weights are affected by multiplicative weight noise. With our results, some existing misconceptions regarding MLP training with weight noise injection can now be resolved.

Cube mapping is widely used in many graphics applications due to the availability of hardware support. However, it does not sample the spherical surface evenly. Recently, a uniform spherical mapping, isocube mapping, was proposed. It exploits the six-face structure used in cube mapping and samples the spherical surface evenly. Unfortunately, some texels in isocube mapping are not rectilinear. This nonrectilinear property may degrade the filtering quality. This paper proposes a novel spherical mapping, namely unicube mapping. It has the advantages of cube mapping (exploitation of hardware and rectilinear structure) and isocube mapping (evenly sampling pattern). In the implementation, unicube mapping uses a simple function to modify the lookup vector before the conventional cube map lookup process. Hence, unicube mapping fully exploits the cube map hardware for real-time filtering and lookup. More importantly, its rectilinear partition structure allows a direct and real-time acquisition of the texture environment. This property facilitates dynamic environment mapping in a real time manner.

Environment sampling is a popular technique for rendering scenes with distant environment illumination. However, the temporal consistency of animations synthesized under dynamic environment sequences has not been fully studied. This paper addresses this problem and proposes a novel method, namely spatiotemporal sampling, to fully exploit both the temporal and spatial coherence of environment sequences. Our method treats an environment sequence as a spatiotemporal volume and samples the sequence by stratifying the volume adaptively. For this purpose, we first present a new metric to measure the importance of each stratified volume. A stratification algorithm is then proposed to adaptively suppress the abrupt temporal and spatial changes in the generated sampling patterns. The proposed method is able to automatically adjust the number of samples for each environment frame and produce temporally coherent sampling patterns. Comparative experiments demonstrate the capability of our method to produce smooth and consistent animations under dynamic environment sequences.

In this paper, the problem of index assignments for quantizers with uniform encoders and channel-optimized decoders for M-PSK schemes is studied. The analytical expressions of MSD (Mean-Squared Distortion) for such quantizers with the natural, zigzag and NBC-Gray mappings, respectively, have been derived. An interesting result is that at a wide range of MSD levels, the zigzag mapping can offer a significant gain in SNR over the other two mappings when used in such quantizers, especially when the alphabet size M gets larger, which is agreed by simulation results.

In the last two decades, many online fault/noise injection algorithms have been developed to attain a fault tolerant neural network. However, not much theoretical works related to their convergence and objective functions have been reported. This paper studies six common fault/noise-injection-based online learning algorithms for radial basis function (RBF) networks, namely 1) injecting additive input noise, 2) injecting additive/multiplicative weight noise, 3) injecting multiplicative node noise, 4) injecting multiweight fault (random disconnection of weights), 5) injecting multinode fault during training, and 6) weight decay with injecting multinode fault. Based on the Gladyshev theorem, we show that the convergence of these six online algorithms is almost sure. Moreover, their true objective functions being minimized are derived. For injecting additive input noise during training, the objective function is identical to that of the Tikhonov regularizer approach. For injecting additive/multiplicative weight noise during training, the objective function is the simple mean square training error. Thus, injecting additive/multiplicative weight noise during training cannot improve the fault tolerance of an RBF network. Similar to injective additive input noise, the objective functions of other fault/noise-injection-based online algorithms contain a mean square error term and a specialized regularization term.

The weight-decay technique is an effective approach to handle overfitting and weight fault. For fault-free networks, without an appropriate value of decay parameter, the trained network is either overfitted or underfitted. However, many existing results on the selection of decay parameter focus on fault-free networks only. It is well known that the weight-decay method can also suppress the effect of weight fault. For the faulty case, using a test set to select the decay parameter is not practice because there are huge number of possible faulty networks for a trained network. This paper develops two mean prediction error (MPE) formulae for predicting the performance of faulty radial basis function (RBF) networks. Two fault models, multiplicative weight noise and open weight fault, are considered. Our MPE formulae involve the training error and trained weights only. Besides, in our method, we do not need to generate a huge number of faulty networks to measure the test error for the fault situation. The MPE formulae allow us to select appropriate values of decay parameter for faulty networks. Our experiments showed that, although there are small differences between the true test errors (from the test set) and the MPE values, the MPE formulae can accurately locate the appropriate value of the decay parameter for minimizing the true test error of faulty networks.

This paper proposes a novel multiscale spherical radial basis function (MSRBF) representation for all-frequency lighting. It supports the illumination of distant environment as well as the local illumination commonly used in practical applications, such as games. The key is to define a multiscale and hierarchical structure of spherical radial basis functions (SRBFs) with basis functions uniformly distributed over the sphere. The basis functions are divided into multiple levels according to their coverage (widths). Within the same level, SRBFs have the same width. Larger width SRBFs are responsible for lower frequency lighting while the smaller width ones are responsible for the higher frequency lighting. Hence, our approach can achieve the true all-frequency lighting that is not achievable by the single-scale SRBF approach. Besides, the MSRBF approach is scalable as coarser rendering quality can be achieved without reestimating the coefficients from the raw data. With the homogeneous form of basis functions, the rendering is highly efficient. The practicability of the proposed method is demonstrated with real-time rendering and effective compression for tractable storage.

We propose a novel reaction diffusion (RD) simulator to evolve image-resembling mazes. The evolved mazes faithfully preserve the salient interior structures in the source images. Since it is difficult to control the generation of desired patterns with traditional reaction diffusion, we develop our RD simulator on a different computational platform, cellular neural networks. Based on the proposed simulator, we can generate the mazes that exhibit both regular and organic appearance, with uniform and/or spatially varying passage spacing. Our simulator also provides high controllability of maze appearance. Users can directly and intuitively "paint" to modify the appearance of mazes in a spatially varying manner via a set of brushes. In addition, the evolutionary nature of our method naturally generates maze without any obvious seam even though the input image is a composite of multiple sources. The final maze is obtained by determining a solution path that follows the user-specified guiding curve. We validate our method by evolving several interesting mazes from different source images.

Omnidirectional videos are usually mapped to planar domain for encoding with off-the-shelf video compression standards. However, existing work typically neglects the effect of the sphere-to-plane mapping. In this paper, we show that by carefully designing the mapping, we can improve the visual quality, stability and compression efficiency of encoding omnidirectional videos. Here we propose a novel mapping scheme, known as the rhombic dodecahedron map (RD map) to represent data over the spherical domain. By using a family of skew great circles as the subdivision kernel, the RD map not only produces a sampling pattern with very low discrepancy, it can also support a highly efficient data indexing mechanism over the spherical domain. Since the proposed map is quad-based, geodesic-aligned, and of very low area and shape distortion, we can reliably apply 2-D wavelet-based and DCT-based encoding methods that are originally designated to planar perspective videos. At the end, we perform a series of analysis and experiments to investigate and verify the effectiveness of the proposed method; with its ultra-fast data indexing capability, we show that we can playback omnidirectional videos with very high frame rates on conventional PCs with GPU support.

An illumination adjustable image (IAI) contains a large number of prerecorded images under various light directions. Relighting a scene under complicated lighting conditions can be achieved from the IAI. Using the radial basis function (RBF) approach to represent an IAI is proven to be more efficient than using the spherical harmonic approach. However, to represent high-frequency lighting effects, we need to use many RBFs. Hence, the relighting speed could be very slow. This brief investigates a partial reconstruction scheme for relighting an IAI based on the locality of RBFs. Compared with the conventional RBF and spherical harmonics (SH) approaches, the proposed scheme has a much faster relighting speed under the similar distortion performance.

In this paper, an objective function for training a functional link network to tolerate multiplicative weight noise is presented. Basically, the objective function is similar in form to other regularizer-based functions that consist of a mean square training error term and a regularizer term. Our study shows that under some mild conditions the derived regularizer is essentially the same as a weight decay regularizer. This explains why applying weight decay can also improve the fault-tolerant ability of a radial basis function (RBF) with multiplicative weight noise. In accordance with the objective function, a simple learning algorithm for a functional link network with multiplicative weight noise is derived. Finally, the mean prediction error of the trained network is analyzed. Simulated experiments on two artificial data sets and a real-world application are performed to verify theoretical result.

In classical training methods for node open fault, we need to consider many potential faulty networks. When the multinode fault situation is considered, the space of potential faulty networks is very large. Hence, the objective function and the corresponding learning algorithm would be computationally complicated. This paper uses the Kullback-Leibler divergence to define an objective function for improving the fault tolerance of radial basis function (RBF) networks. With the assumption that there is a Gaussian distributed noise term in the output data, a regularizer in the objective function is identified. Finally, the corresponding learning algorithm is developed. In our approach, the objective function and the learning algorithm are computationally simple. Compared with some conventional approaches, including weight-decay-based regularizers, our approach has a better fault-tolerant ability. Besides, our empirical study shows that our approach can improve the generalization ability of a fault-free RBF network.

In this paper, we present an example-based colorization technique robust to illumination differences between grayscale target and color reference images. To achieve this goal, our method performs color transfer in an illumination-independent domain that is relatively free of shadows and highlights. It first recovers an illumination-independent intrinsic reflectance image of the target scene from multiple color references obtained by web search. The reference images from the web search may be taken from different vantage points, under different illumination conditions, and with different cameras. Grayscale versions of these reference images are then used in decomposing the grayscale target image into its intrinsic reflectance and illumination components. We transfer color from the color reflectance image to the grayscale reflectance image, and obtain the final result by relighting with the illumination component of the target image. We demonstrate via several examples that our method generates results with excellent color consistency.

Even though the temporal information is lost, a still picture of moving animals hints at their motion. In this paper, we infer motion cycle of animals from the "motion snapshots" (snapshots of different individuals) captured in a still picture. By finding the motion path in the graph connecting motion snapshots, we can infer the order of motion snapshots with respect to time, and hence the motion cycle. Both "half-cycle" and "full-cycle" motions can be inferred in a unified manner. Therefore, we can animate a still picture of a moving animal group by morphing among the ordered snapshots. By refining the pose, morphology, and appearance consistencies, smooth and realistic animal motion can be synthesized. Our results demonstrate the applicability of the proposed method to a wide range of species, including birds, fishes, mammals, and reptiles.

Discrete wavelet transform (DWT) has been heavily studied and developed in various scientific and engineering fields. Its multiresolution and locality nature facilitates applications requiring progressiveness and capturing high-frequency details. However, when dealing with enormous data volume, its performance may drastically reduce. On the other hand, with the recent advances in consumer-level graphics hardware, personal computers nowadays usually equip with a graphics processing unit (GPU) based graphics accelerator which offers SIMD-based parallel processing power. This paper presents a SIMD algorithm that performs the convolution-based DWT completely on a GPU, which brings us significant performance gain on a normal PC without extra cost. Although the forward and inverse wavelet transforms are mathematically different, the proposed algorithm unifies them to an almost identical process that can be efficiently implemented on GPU. Different wavelet kernels and boundary extension schemes can be easily incorporated by simply modifying input parameters. To demonstrate its applicability and performance, we apply it to wavelet-based geometric design, stylized image processing, texture-illuminance decoupling, and JPEG2000 image encoding

This paper proposes a novel six-face spherical map, isocube, that fully utilizes the cubemap hardware built in most GPUs. Unlike the cubemap, the proposed isocube uniformly samples the unit sphere (uniformly distributed), and all samples span the same solid angle (equally important). Its mapping computation contains only a small overhead. By feeding the cubemap hardware with the six-face isocube map, the isocube can exploit all built-in texturing operators tailored for the cubemap and achieve a very high frame rate. In addition, we develop an anisotropic filtering that compensates aliasing artifacts due to texture magnification. This filtering technique extends the existing hardware anisotropic filtering and can be applied not only to the proposed isocube, but also to other texture mapping applications.

This paper presents a novel preamble-based synchronization scheme for orthogonal frequency division multiplexing systems over time-variant multipath fading channels. A new timing metric is derived with the use of a local synchronizing sequence (LSS). We propose one specifically designed training sequence which consists of two segments of equal length where each segment is constructed from a different pseudo-noise (PN) sequence. The advantages of using the LSS are twofold: multipath effects are reduced in the new timing metric and the metric trajectory is impulse-like. The new algorithm is shown to provide excellent performance in timing estimation even in severe time-variant fading channels. As for frequency synchronization, a two-step approach handles both fractional and integer frequency offset providing a large frequency acquisition range without loss of estimation accuracy.

This paper presents a new receive scheme applied to orthogonal frequency-division multiplexing (OFDM) systems for reducing inter-carrier interference (ICI) caused by frequency offset at the receiver. The proposed scheme uses a class of time domain second order polynomial Nyquist window on the receive side. The signal to interference plus noise ratio (SINR) and bit error ratio (BER) performances of the receiver are analyzed in this paper. We also propose a method to find the optimum window parameters to maximize the SINR of the receiver thus providing an adaptive receiving capability. The results show that the proposed method that enables the use of unity roll-off factor provides better SINR and BER than conventional receive methods whose roll-off factors are generally less than one. The new second order polynomial window is shown to provide better performance than raised cosine and "better than" Nyquist window.

This paper presents a mean square error (MSE) analysis of the recursive least square (RLS) algorithm for system identification over wide-sense stationary uncorrelated scattering (WSSUS) fading channels. A new sum-of-sinusoids (SOS) model for modeling fading channel is proposed. Unlike most of RLS analyses, our analysis takes the correlation of the inverse of correlation matrix into account and hence yields an improved recursive formula for the MSE, which can be computed recursively with the information of the second order statistics of fading channels. The steady state MSE of RLS is derived for Clarke's model. It is shown that the MSE analysis using the SOS model have much better agreement with experimental results than the AR model.

In image-based relighting, a pixel is associated with a number of sampled radiance values. This paper presents a two-level compression method. In the first level, the plenoptic property of a pixel is approximated by a spherical radial basis function (SRBF) network. That means that the spherical plenoptic function of each pixel is represented by a number of SRBF weights. In the second level, we apply a wavelet-based method to compress these SRBF weights. To reduce the visual artifact due to quantization noise, we develop a constrained method for estimating the SRBF weights. Our proposed approach is superior to JPEG, JPEG2000, and MPEG. Compared with the spherical harmonics approach, our approach has a lower complexity, while the visual quality is comparable. The real-time rendering method for our SRBF representation is also discussed.

Recursive least square (RLS) is an efficient approach to neural network training. However, in the classical RLS algorithm, there is no explicit decay in the energy function. This will lead to an unsatisfactory generalization ability for the trained networks. In this paper, we propose a generalized RLS (GRLS) model which includes a general decay term in the energy function for the training of feedforward neural networks. In particular, four different weight decay functions, namely, the quadratic weight decay, the constant weight decay and the newly proposed multimodal and quartic weight decay are discussed. By using the GRLS approach, not only the generalization ability of the trained networks is significantly improved but more unnecessary weights are pruned to obtain a compact network. Furthermore, the computational complexity of the GRLS remains the same as that of the standard RLS algorithm. The advantages and tradeoffs of using different decay functions are analyzed and then demonstrated with examples. Simulation results show that our approach is able to meet the design goals: improving the generalization ability of the trained network while getting a compact network.

Spherical harmonic (SH) basis functions have been widely used for representing spherical functions in modeling various illumination properties. They can compactly represent low-frequency spherical functions. However, when the unconstrained least square method is used for estimating the SH coefficients of a hemispherical function, the magnitude of these SH coefficients could be very large. Hence, the rendering result is very sensitive to quantization noise (introduced by modern texture compression like S3TC, IEEE half float data type on GPU, or other lossy compression methods) in these SH coefficients. Our experiments show that, as the precision of SH coefficients is reduced, the rendered images may exhibit annoying visual artifacts. To reduce the noise sensitivity of the SH coefficients, this paper first discusses how the magnitude of SH coefficients affects the rendering result when there is quantization noise. Then, two fast fitting methods for estimating the noise-resistant SH coefficients are proposed. They can effectively control the magnitude of the estimated SH coefficients and, hence, suppress the rendering artifacts. Both statistical and visual results confirm our theory.

The ability to change illumination is a crucial factor in image-based modeling and rendering. Image-based relighting offers such capability. However, the tradeoff is the enormous increase of storage requirement. In this paper, we propose a compression scheme that effectively reduces the data volume while maintaining the real-time relighting capability. The proposed method is based on principal component analysis (PCA). A block-wise PCA is used to practically process the huge input data. The output of PCA is a set of eigenimages and the corresponding relighting coefficients. By dropping those low-energy eigenimages, the data size is drastically reduced. To further compress the data, eigenimages left are compressed using transform coding and quantization while the relighting coefficients are compressed using uniform quantization. We also suggest the suitable target bit rate for each phase of the compression method in order to preserve the visual quality. Finally, we propose a real-time engine that relights images from the compressed data.

In the area of image-based modeling and rendering (IBMR), a new scene under a novel illumination condition is synthesized by interpolating reference images. However, for good quality rendering tremendous reference images under various illumination conditions are required. We present a two-level compression method for reference images. In the first level, the spherical harmonic transform is used to approximate the plenoptic function of each pixel In the second level, we use the embedded zerotrees wavelet (EZW) method for removing the spatial redundancy in spherical coefficients produced from the first level process. Simulation results show our approach is much superior than compressing reference images with JPEG.

The ability to change illumination is a crucial factor in image-based modeling and rendering. Image-based relighting offers such capability. However, the trade-off is the enormous increase of storage requirement. In this paper, we propose a compression scheme that effectively reduces the data volume while maintaining the real-time relighting capability. The proposed method is based on principal component analysis (PCA). A block-wise PCA is used to practically process the huge input data. The output of PCA is a set of eigenimages and the corresponding relighting coefficients. By dropping those low-energy eigenimages, the data size is drastically reduced. To further compress the data, eigenimages left are compressed using transform coding and quantization while the relighting coefficients are compressed using uniform quantization. We also suggest the suitable target bit rate for each phase of the compression method in order to preserve the visual quality. Finally, we propose real-time engine that relights images from the compressed data.

The global extended Kalman filtering (EKF) algorithm for recurrent neural networks (RNNs) is plagued by the drawback of high computational cost and storage requirement. In this paper, we present a local EKF training-pruning approach that can solve this problem. In particular, the by-products, obtained along with the local EKF training, can be utilized to measure the importance of the network weights. Comparing with the original global approach, the proposed local approach results in much lower computational cost and storage requirement. Hence, it is more practical in solving real world problems. Simulation showed that our approach is an effective joint-training-pruning method for RNNs under online operation.

The image-based modeling and rendering (IBMR) approaches allow the time complexity of synthesizing novel images to be independent of scene complexity. Unfortunately, illumination control (relighting) is no longer trivial under the image-based framework. To relight the image-based scenery, the scene must be captured under various illumination conditions. This drastically increases the data volume. Hence, data compression is a must. In this paper, we describe a compression algorithm for an illumination-adjustable image representation. We focus on compressing constant-viewpoint images. A divide-and-conquer approach is proposed. The compression algorithm consists of three parts which exploit the intrapixel, interpixel, and interchannel data correlations. Experimental result shows that the proposed method not just effectively compresses the data but also outperforms standard image and video coding methods.

Ant routing is a method for network routing in agent technology. Although its effectiveness and efficiency have been demonstrated and reported in the literature, its properties have not yet been well studied. This paper presents some preliminary analysis on an ant algorithm in regard to its population growing property and jumping behavior. Results conclude that as long as the value max, {i/spl Omega//sub j/|} is known, the practitioner is able to design the algorithm parameters, such as the number of agents being created for each request, k, and the maximum allowable number of jumps of an agent, in order to meet the network constraint.

Image-based modeling and rendering has been demonstrated as a cost-effective and efficient approach to virtual reality applications. The computational model that most image-based techniques are based on is the plenoptic function. Since the original formulation of the plenoptic function does not include illumination, most previous image-based virtual reality applications simply assume that the illumination is fixed. We propose a formulation of the plenoptic function, called the plenoptic illumination function, which explicitly specifies the illumination component. Techniques based on this new formulation can be extended to support relighting as well as view interpolation. To relight images with various illumination configurations, we also propose a local illumination model, which utilizes the rules of image superposition. We demonstrate how this new formulation can be applied to extend two existing image-based representations, panorama representation such as QuickTime VR and two-plane parameterization, to support relighting with trivial modifications. The core of this framework is compression, and we therefore show how to exploit two types of data correlation, the intra-pixel and the inter-pixel correlations, in order to achieve a manageable storage size.

Recursive least squares (RLS)-based algorithms are a class of fast online training algorithms for feedforward multilayered neural networks (FMNNs). Though the standard RLS algorithm has an implicit weight decay term in its energy function, the weight decay effect decreases linearly as the number of learning epochs increases, thus rendering a diminishing weight decay effect as training progresses. In this paper, we derive two modified RLS algorithms to tackle this problem. In the first algorithm, namely, the true weight decay RLS (TWDRLS) algorithm, we consider a modified energy function whereby the weight decay effect remains constant, irrespective of the number of learning epochs. The second version, the input perturbation RLS (IPRLS) algorithm, is derived by requiring robustness in its prediction performance to input perturbations. Simulation results show that both algorithms improve the generalization capability of the trained network.

The training of neural networks using the extended Kalman filter (EKF) algorithm is plagued by the drawback of high computational complexity and storage requirement that may become prohibitive even for networks of moderate size. In this paper, we present a local EKF training and pruning approach that can solve this problem. In particular, the by-products obtained along with the local EKF training can be utilized to measure the importance of the network weights. Comparing with the original global approach, the proposed local EKF training and pruning approach results in a much lower computational complexity and storage requirement. Hence, it is more practical in solving real world problems. The performance of the proposed algorithm is demonstrated on one medium- and one large-scale problems, namely, sunspot data prediction and handwritten digit recognition.

Pruning a neural network to a reasonable smaller size, and if possible to give a better generalization, has long been investigated. Conventionally the common technique of pruning is based on considering error sensitivity measure, and the nature of the problem being solved is usually stationary. In this article, we present an adaptive pruning algorithm for use in a nonstationary environment. The idea relies on the use of the extended Kalman filter (EKF) training method. Since EKF is a recursive Bayesian algorithm, we define a weight-importance measure in term of the sensitivity of a posteriori probability. Making use of this new measure and the adaptive nature of EKF, we devise an adaptive pruning algorithm called adaptive Bayesian pruning. Simulation results indicate that in a noisy nonstationary environment, the proposed pruning algorithm is able to remove network redundancy adaptively and yet preserve the same generalization ability.

In this paper, the regularization of employing the forgetting recursive least square (FRLS) training technique on feedforward neural networks is studied. We derive our result from the corresponding equations for the expected prediction error and the expected training error. By comparing these error equations with other equations obtained previously from the weight decay method, we have found that the FRLS technique has an effect which is identical to that of using the simple weight decay method. This new finding suggests that the FRLS technique is another on-line approach for the realization of the weight decay effect. Besides, we have shown that, under certain conditions, both the model complexity and the expected prediction error of the model being trained by the FRLS technique are better than the one trained by the standard RLS method.

Recently we have proposed a simple circuit of winner-take-all (WTA) neural network. Assuming no external input, we have derived an analytic equation for its network response time. In this paper, we further analyze the network response time for a class of winner-take-all circuits involving self-decay and show that the network response time of such a class of WTA is the same as that of the simple WTA model.

In the use of extended Kalman filter approach in training and pruning a feedforward neural network, one usually encounters the problems on how to set the initial condition and how to use the result obtained to prune a neural network. In this paper, some cues on the setting of the initial condition will be presented with a simple example illustrated. Then based on three assumptions--1) the size of training set is large enough; 2) the training is able to converge; and 3) the trained network model is close to the actual one, an elegant equation linking the error sensitivity measure (the saliency) and the result obtained via extended Kalman filter is devised. The validity of the devised equation is then testified by a simulated example.

In combining principal and minor components analysis, a parallel extraction method based on the recursive least square algorithm is suggested to extract the principal components of the input vectors. After the extraction, the error covariance matrix obtained in the learning process is used to perform minor components analysis. The minor components found are then pruned so as to achieve a higher compression ratio. Simulation results show that both the convergent speed and the compression ratio are improved, which in turn indicate that our method effectively combines the extraction of the principal components and the pruning of the minor components.

Pruning is one of the effective techniques for improving the generalization error of neural networks. Existing pruning techniques are derived mainly from the viewpoint of energy minimization, which is commonly used in gradient-based learning methods. In recurrent networks, extended Kalman filter (EKF)-based training has been shown to be superior to gradient-based learning methods in terms of speed. This article explains a pruning procedure for recurrent neural networks using EKF training. The sensitivity of a posterior probability is used as a measure of the importance of a weight instead of error sensitivity since posterior probability density is readily obtained from this training method. The pruning procedure is tested using three problems: (1) the prediction of a simple linear time series, (2) the identification of a nonlinear system, and (3) the prediction of an exchange-rate time series. Simulation results demonstrate that the proposed pruning method is able to reduce the number of parameters and improve the generalization ability of a recurrent network.

Forgetting learning is an incremental learning rule in associative memories. With it, the recent learning items can be encoded, and the old learning items will be forgotten. In this article, we analyze the storage behavior of bidirectional associative memory (BAM) under the forgetting learning. That is, “Can the most recent k learning item be stored as a fixed point?” Also, we discuss how to choose the forgetting constant in the forgetting learning such that the BAM can correctly store as many as possible of the most recent learning items. Simulation is provided to verify the theoretical analysis.

In the transmission of vector quantized data, the vector quantizer and the communication system are usually designed separately. With such an approach, the channel noise results in significant degradations in the performance of the vector quantizer. To solve this problem, we should properly create the mapping from the codebook of the quantizer to the channel signal set of the communication system. This paper proposes a new approach to construct such a mapping based on the ordering property of the self-organizing feature map (SOFM). We use the neighborhood structure of the SOFM and the neighborhood structure of the channel signal set to construct the mapping. Simulation results confirm that the proposed approach is robust with respect to channel noise.

In this paper, a bidirectional associative memory (BAM) model with second-order connections, namely second-order bidirectional associative memory (SOBAM), is first reviewed. The stability and statistical properties of the SOBAM are then examined. We use an example to illustrate that the stability of the SOBAM is not guaranteed. For this result, we cannot use the conventional energy approach to estimate its memory capacity. Thus, we develop the statistical dynamics of the SOBAM. Given that a small number of errors appear in the initial input, the dynamics shows how the number of errors varies during recall. We use the dynamics to estimate the memory capacity, the attraction basin, and the number of errors in the retrieved items. Extension of the results to higher-order bidirectional associative memories is also discussed.

This paper presents an algorithm to form a topographic map resembling to the self-organizing map. The idea stems on defining an energy function which reveals the local correlation between neighboring neurons. The larger the value of the energy function, the higher the correlation of the neighborhood neurons. On this account, the proposed algorithm is defined as the gradient ascent of this energy function. Simulations on two-dimensional maps are illustrated.

The stability, capacity and statistical dynamics of second-order bidirectional associative memory (BAM) are presented here. We first use an example to illustrate that the state of second-order BAR I may converge to limited cycles. When error in the retrieved pairs is not allowed, a lower bound of memory capacity is derived. That is O(min(n/sup 2//(log n),p/sup 2//(log p))) where n and p are the dimensions of the library pairs. Since the state of second-order BAM may converge to limited cycles, the conventional method cannot be used to estimate its memory capacity when small errors in the retrieval pairs are allowed. Hence, the statistical dynamics of second-order BAM is introduced: starting with an initial state close to the library pairs, how the confidence interval of the number of errors changes during recalling. From the dynamics, the attraction basin, memory capacity, and final error in the retrieval pairs can be estimated. Also, some numerical results are given. Finally, an extension of the results to higher-order BAM is discussed.>

Borrowing the idea of the perceptron, bidirectional learning (BL) is proposed to enhance the recall performance of bidirectional associative memory (BAM). By modifying the proof of convergence of the perceptron, the author has proved that BL yields one of the solution connection matrices within a finite number of iterations (if the solutions exist). According to the above convergence of BL, the capacity of BAM with BL is larger than or equal to that with any other learning rule. Hence, BL can be considered as an optimum learning rule for BAM in the sense of capacity. Simulations show that BL greatly improves the capacity and the error correction capability of BAM.>