Multi-view clustering (MVC) seeks to uncover the intrinsic group structures embedded in multi-view data, which has attracted considerable attention in recent years. Existing approaches predominantly concentrate on incorporating suitable model priors to capture consistency across views. However, these explicit constraints often fail to hold in scenarios involving significant modal differences between views or the presence of noise, thereby limiting the efficacy of these methods in more complex contexts. To address these issues, this paper introduces BONE, a lightweight and interpretable MVC framework that Bridges Optimization and Neural networks for Efficient MVC. By leveraging learnable parameters to extract high-level features from low-level features derived through classical optimization, BONE integrates the consistency information across views without the need for explicit prior constraints, while eliminating the necessity for pre-training or post-processing. Extensive experiments show that BONE achieves clustering performance comparable to or even better than existing deep MVC methods, while using only 1% of the parameters, offering a new perspective for designing efficient MVC algorithms.

Nighttime flare removal is challenging due to the difficulty of acquiring real-world paired data. Existing methods, trained on synthetic pipelines, often struggle to generalize to real-world scenarios. A key limitation of these pipelines is their focus on single-flare scenes, whereas real-world conditions frequently involve more complex cases, such as multi-flare and composite flare scenarios, which are difficult to simulate effectively. This discrepancy significantly hampers model performance in practical applications. Through detailed analysis, we uncover a fundamental characteristic of flare degradation: regardless of whether the scene is synthetic single-flare, real-world single-flare, or multi-flare, the degradation information exhibits a similar distribution across frequency subbands—predominantly concentrated in the low-frequency region, with a minor presence in the high-frequency region. Notably, the severity of the glare effect correlates with an even stronger concentration in the low-frequency domain. This finding suggests that targeted frequency modeling can bridge the gap between synthetic and real-world domains, forming a principled approach to improving generalization. Building on this insight, we propose the Scale-Aware Frequency-Adaptive Guidance Network for Nighttime Flare Removal (SAFAformer), which integrates a Frequency-Adaptive Guidance Module (FAGM) and a Scale-Aware Transformer Block (SATB) to leverage frequency-domain properties during training. Extensive experiments demonstrate that SAFAformer achieves state-of-the-art performance in flare removal compared to existing methods. Our code and pre-trained models are available on GitHub for validation.

Scene Text Image Super-Resolution (STISR) methods improve recognition accuracy by refining text regions locally. However, most existing approaches are limited by short-range dependencies, hindering the modeling of long-range semantic relationships between characters, which reduces their effectiveness in complex text scenes. Moreover, current methods are predominantly evaluated on synthetic datasets, which do not adequately capture real-world challenges such as diverse text styles, complex backgrounds, and spatial distortions. To address these limitations, we propose a Dual-Guided Visual and Semantic (DGVS) framework. This innovative framework utilizes a recognizer-driven attention mechanism to decouple character sequences, effectively distinguishing text from background noise. Additionally, we incorporate a state-space model to establish global semantic reasoning links, enhancing the comprehension of contextual relationships within text images. Furthermore, we construct Real-World Text (RealWT), a novel real-world benchmark dataset that integrates diverse data sources, including online images and multi-device captures. This dataset incorporates factors like device variations, resolution differences, and degradation, offering a more realistic simulation of real-world conditions. It enables models to learn degradation patterns that closely mirror practical applications, offering a standardized benchmark for evaluating STISR performance. Extensive experiments demonstrate that our method outperforms existing approaches, as validated by evaluations on TextZoom and RealWT. Our dataset and code are available on https://github.com/yoursmith/sde-DGVS.

Sparse Principal Component Analysis (SPCA) is a powerful technique for dimensionality reduction and feature extraction in high-dimensional data, with applications spanning various fields such as computer vision, pattern recognition, and data mining. However, the computational intensity of SPCA presents a significant challenge, necessitating the development of efficient and robust algorithms. In this paper, we shed light on the SPCA problem and uncover intriguing structures that enable us to design an efficient algorithm, which we have named SPCA_ACC. Firstly, we identify a separable structure in this problem, which prompts us to draw on the Variable Projection (VP) strategy and generalize it to separable nonlinear problem in Stiefel manifold. This strategy projects out part of the parameters to obtain a reduced problems, allowing the SPCA_ACC algorithm to optimize in a lower-dimensional parameter space. Secondly, we resolve the coupling between different parameters of the SPCA problem in the optimization process on a fixed coordinate-sparsity manifold, which opens the way to the use of second-order Riemannian accelerated VP strategy. Moreover, we systematically analyze the advantages of using VP to solve the SPCA problem from a theoretical perspective, and confirm the local quadratic convergence of our algorithm. Numerical experiments on datasets of different sizes and types demonstrate that our method achieves rapid convergence and significantly reduces computational costs.

Color distortion and structural degradation in underwater images are classic challenges in underwater image enhancement. The core goal is to restore degraded images to high-quality images with both color and structure that conform to visual perception. However, in the traditional RGB space, these two issues are highly coupled, resulting in existing enhancement methods often neglecting one over the other. To address this challenge, we propose a guided diffusion model based on the principle of decoupling. Our key insight is that in perceptual color spaces such as HSV, color (H, S) and structure (V) are naturally separated. To exploit this property, we first design an adaptive perceptual guidance module, which analyzes the degraded HSV image and generates two orthogonal guidance signals: a color guide and a structure guide, which guide the denoising process of the diffusion model. To ensure that this decoupled guidance is faithfully implemented, we propose a corresponding decoupled loss optimization module, which uses independent loss functions to supervise the final output color and structure. By combining the forward decoupled guidance with the backward decoupled supervision, we construct a closed-loop optimization framework. This framework enables the model to collaboratively optimize color and structure under various degradation scenarios. Extensive experiments demonstrate that our proposed method outperforms existing state-of-the-art approaches in a variety of underwater scenes, particularly those degraded by color casts and haze. Furthermore, it exhibits superior performance on no-reference image quality assessment metrics. The source code is available at https://github.com/zy-world/DCD-UIE.

Retinex-based methods have become a general approach for solving low-light image enhancement (LLIE). However, traditional methods require post-processing of illumination (e.g., gamma correction), which lacks adaptability and disrupts the illumination structure. Retinex-based deep networks typically follow a ‘decomposition-adjustment-exposure control’ process, which is redundant and lacks robustness. One major issue is the inaccuracy in estimating and decomposing the initial illumination. Accurate initial illumination can prevent further post-processing instability. We propose IniRetinex, rethinking the Retinex-based LLIE method from the perspective of initialization. By using neural networks to provide reasonable initial illumination and solving for smooth illumination through optimization, higher performance LLIE is achieved. We construct a two-layer convolutional neural network to capture the low-frequency structure of the image, adaptively compensating for classical initial illumination and avoiding additional post-processing. The network requires no pre-training and can be implemented in an unsupervised manner with just a few iterations, making it highly efficient. Additionally, we propose a new illumination optimization strategy by introducing an additional proximal penalty term, improving illumination in areas with varying levels and enhancing image details. Extensive experiments on various low-light image datasets demonstrate that our method achieves state-of-the-art (SOTA) results on multiple benchmarks, offering higher stability and inference efficiency compared to current advanced methods.

Learning-based image dehazing algorithms have shown remarkable success in synthetic domains. However, real image dehazing is still in suspense due to computational resource constraints and the diversity of real-world scenes. Therefore, there is an urgent need for an algorithm that excels in both efficiency and adaptability to address real image dehazing effectively. This work proposes a Compression-and-Adaptation (CoA) computational flow to tackle these challenges from a divide-and-conquer perspective. First, model compression is performed in the synthetic domain to develop a compact dehazing parameter space, satisfying efficiency demands. Then, a bilevel adaptation in the real domain is introduced to be fearless in unknown real environments by aggregating the synthetic dehazing capabilities during the learning process. Leveraging a succinct design free from additional constraints, our CoA exhibits domain-irrelevant stability and model-agnostic flexibility, effectively bridging the model chasm between synthetic and real domains to further improve its practical utility. Extensive evaluations and analyses underscore the approach's superiority and effectiveness. The code is publicly available at https://github.com/fyxnl/COA.

We propose an online learning algorithm tailored for a class of machine learning models within a separable stochastic approximation framework. The central idea of our approach is to exploit the inherent separability in many models, recognizing that certain parameters are easier to optimize than others. This paper focuses on models where some parameters exhibit linear characteristics, which are common in machine learning applications. In our proposed algorithm, the linear parameters are updated using the recursive least squares (RLS) algorithm, akin to a stochastic Newton method. Subsequently, based on these updated linear parameters, the nonlinear parameters are adjusted using the stochastic gradient method (SGD). This dual-update mechanism can be viewed as a stochastic approximation variant of block coordinate gradient descent, where one subset of parameters is optimized using a second-order method while the other is handled with a first-order approach. We establish the global convergence of our online algorithm for non-convex cases in terms of the expected violation of first-order optimality conditions. Numerical experiments demonstrate that our method achieves significantly faster initial convergence and produces more robust performance compared to other popular learning algorithms. Additionally, our algorithm exhibits reduced sensitivity to learning rates and outperforms the recently proposedslimTrainalgorithm (Newman et al. 2022). For validation, the code has been made available on GitHub.

The identification of separable nonlinear models, prevalent in tasks such as signal analysis, image processing, time series analysis, and machine learning, presents a non-convex optimization challenge that necessitates the development of efficient identification algorithms. The Variable Projection (VP) algorithm has been proven to be quite effective for addressing these problems; however, traditional VP relying on the Hessian matrix and its inverse are highly time-consuming and unsuitable for complex, large-scale applications. This letter introduces a novel approach that employs the exponential moving average of gradient and gradient estimation bias to indirectly estimate the curvature of the objective landscape, proposing a Moving Average-based Variable Projection method (MAVP). The proposed algorithm utilizes only gradient information and can properly tackle the coupling relationships between different parameters during the optimization process, thereby achieving faster convergence. Numerical results on nonlinear time series analysis and image reconstruction demonstrate that the MAVP algorithm exhibits significant efficiency and effectiveness.

When capturing images under strong light sources at night, intense lens flare artifacts often appear, significantly degrading visual quality and impacting downstream computer vision tasks. Although transformer-based methods have achieved remarkable results in nighttime flare removal, they fail to adequately distinguish between flare and non-flare regions. This unified processing overlooks the unique characteristics of these regions, leading to suboptimal performance and unsatisfactory results in real-world scenarios. To address this critical issue, we propose a novel approach incorporating Location Prior Guidance (LPG) and a specialized flare removal model, LPFSformer. LPG is designed to accurately learn the location of flares within an image and effectively capture the associated glow effects. By employing Location Prior Injection (LPI), our method directs the model’s focus towards flare regions through the interaction of frequency and spatial domains. Additionally, to enhance the recovery of high-frequency textures and capture finer local details, we designed a Global Hybrid Feature Compensator (GHFC). GHFC aggregates different expert structures, leveraging the diverse receptive fields and CNN operations of each expert to effectively utilize a broader range of features during the flare removal process. Extensive experiments demonstrate that our LPFSformer achieves state-of-the-art flare removal performance compared to existing methods. Our code and a pre-trained LPFSformer have been uploaded to GitHub for validation.

Reflection removal is a crucial issue in image reconstruction, especially for high-definition images. Removing undesirable reflections can greatly enhance the performance of various visual systems, such as medical imaging, autonomous driving, and security surveillance. However, the resolution of existing reflection removal datasets is not high and the training data heavily relies on synthetic data, which hampers the performance of reflection removal methods and restricts the development of effective techniques tailored for high-definition images. Therefore, this paper introduces a new dataset, Real-world Reflection Removal in 4K (RR4K). This novel dataset, with its large capacity and high resolution of$6000\times 4000$pixels, represents a significant advancement in the field, ensuring a realistic and high quality benchmark. Furthermore, building upon the dataset, we propose an efficient method for single-image reflection removal, optimized for high-definition processing. This method employs the U-Net architecture, enhanced with large kernel distillation and scale-aware features, enabling it to effectively handle complex reflection scenarios while reducing computational demands. Comprehensive testing on the RR4K dataset and existing low-resolution datasets has demonstrated the method’s superior efficiency and effectiveness. We believe that our constructed RR4K dataset can better evaluate and design algorithms for removing undesirable reflection from real-world high-definition images. Our dataset and code are available athttps://github.com/jengchauwei/RR4K.

The non-local attention has attracted widespread attention from researchers in enhancing the ability of deep single-image super-resolution methods to mine self-similarity information. However, non-local attention often faces problems of high computational complexity and inaccurate calculation of cross-correlation of deep features when capturing long-distance information. To solve these problems, we propose an innovative and efficient Weighted Adaptive Clustering Attention (WACA). Specifically, WACA consists of Sparse Adaptive Clustering Attention (SACA) and Weighted Residual Attention (WRA). SACA significantly reduces the noise signal in the process of feature correlation calculation with the help of asymmetric local sensitive hashing, and reduces the computational cost from quadratic to asymptotically linear with the sequence length. In addition, our designed WRA utilizes the high correlation characteristics of the self-similarity matrix between self-attention layers, and further significantly reduces the computational consumption of associated non-local information by co-optimizing the self-similarity matrix. To verify the effectiveness of WACA, we introduce corresponding modules on a residual backbone and construct a framework named Weighted Adaptive Clustering Network (WACN). Experimental results demonstrate that WACN has competitive performance in both quantitative and qualitative evaluations.

Self-similarity-based deep Single Image Super-Resolution (SISR) methods have gained popularity in recent years, especially with the integration of Non-Local Attention (NLA) in deep SISR. However, NLA suffers from the drawback of mixing relevant and irrelevant features, as it computes the response of each query by aggregating information from all non-local features. In this work, we propose a novel approach to exploit self-similarity more effectively using FlyHash, which is inspired by the fruit fly olfactory circuit and has exhibited outstanding performance in approximate similarity search. By limiting the association area of non-local features with FlyHash, we develop the Decoupled Non-Local Attention (DNLA) method, which tackles the difficulties of modeling a large amount of irrelevant non-local features while considerably lowering the computational complexity from quadratic to nearly linear. We verify the effectiveness of our DNLA approach with comprehensive ablation studies to show its capability of capturing nonlocal information for deep SISR. Furthermore, we build a deep Decoupled Non-Local Attention Network (DNLAN) with DNLA, which attains excellent results in both objective evaluation and subjective perception for SISR.

Single Image Super-Resolution (SISR) aims to reconstruct a high-resolution image from its corresponding low-resolution input. A common technique to enhance the reconstruction quality is Non-Local Attention (NLA), which leverages self-similar texture patterns in images. However, we have made a novel finding that challenges the prevailing wisdom. Our research reveals that NLA can be detrimental to SISR and even produce severely distorted textures. For example, when dealing with severely degrade textures, NLA may generate unrealistic results due to the inconsistency of non-local texture patterns. This problem is overlooked by existing works, which only measure the average reconstruction quality of the whole image, without considering the potential risks of using NLA. To address this issue, we propose a new perspective for evaluating the reconstruction quality of NLA, by focusing on the sub-pixel level that matches the pixel-wise fusion manner of NLA. From this perspective, we provide the approximate reconstruction performance upper bound of NLA, which guides us to design a concise yet effective Texture-Fidelity Strategy (TFS) to mitigate the degradation caused by NLA. Moreover, the proposed TFS can be conveniently integrated into existing NLA-based SISR models as a general building block. Based on the TFS, we develop a Deep Texture-Fidelity Network (DTFN), which achieves state-of-the-art performance for SISR. Our code and a pre-trained DTFN are available on GitHub†for verification.

Blind Super-Resolution (BlindSR) aims to reconstruct high-resolution (HR) images from low-resolution (LR) images without prior knowledge of the image degradation process. This is a challenging problem in real-world applications, where the degradation can be complex and unknown. Recent unsupervised learning-based BlindSR methods can estimate the image degradation in an unsupervised manner, but they suffer from limited adaptability to different types and intensities of degradation. They tend to capture the average level of degradation across all training samples, resulting in over-smoothing or over-sharpening effects for some images. As a result, the final reconstruction may exhibit the mean effect. Moreover, existing synthetic datasets do not reflect the real-world degradation scenarios, making it difficult to evaluate the performance of BlindSR methods. To address these issues, we propose a novel Degradation Intensity Estimation Module (DIEM) method, which can estimate the pixel-level degradation information of the input image more specifically and use it to guide image reconstruction. Furthermore, we construct a benchmark dataset under real scenarios, which is closer to the real-world BlindSR problem than existing synthetic datasets, and can provide a more reasonable evaluation of BlindSR methods. Extensive experimental results demonstrate that our DIEM-guided BlindSR method can achieve state-of-the-art image reconstruction results. Our code and pre-trained models have been uploaded to GitHub† for validation.

The graph-information-based fuzzy clustering has shown promising results in various datasets. However, its performance is hindered when dealing with high-dimensional data due to challenges related to redundant information and sensitivity to the similarity matrix design. To address these limitations, this article proposes an implicit fuzzy k-means (FKMs) model that enhances graph-based fuzzy clustering for high-dimensional data. Instead of explicitly designing a similarity matrix, our approach leverages the fuzzy partition result obtained from the implicit FKMs model to generate an effective similarity matrix. We employ a projection-based technique to handle redundant information, eliminating the need for specific feature extraction methods. By formulating the fuzzy clustering model solely based on the similarity matrix derived from the membership matrix, we mitigate issues, such as dependence on initial values and random fluctuations in clustering results. This innovative approach significantly improves the competitiveness of graph-enhanced fuzzy clustering for high-dimensional data. We present an efficient iterative optimization algorithm for our model and demonstrate its effectiveness through theoretical analysis and experimental comparisons with other state-of-the-art methods, showcasing its superior performance.

Blind super-resolution (BlindSR) has recently attracted attention in the field of remote sensing. Due to the lack of paired data, most works assume that the acquired remote sensing images are high-resolution (HR) and use predefined degradation models to synthesize low-resolution (LR) images for training and evaluation. However, these acquired remote sensing images are often degraded by various factors, which still require super-resolution reconstruction to meet practical needs. Using them as ground truth images will limit the model’s ability to restore fine details, resulting in blurry and noisy reconstructions. To overcome these limitations, we propose an unsupervised degradation-aware network which transforms natural images into the degraded domain as real-world remote sensing images. It uses natural images containing rich texture information as a reference for fine-grained restoration of the network, enabling the network to produce clearer reconstructions. Furthermore, we discovered the remarkable capability of patch-wise discriminator to perceive the degradation type of different regions within the acquired remote sensing image. Inspired by this finding, we design a novel degradation representation module (DRM) that can estimate the degradation information from LR images and guide the network to perform adaptive restoration. Comprehensive experimental results demonstrate that our proposed unsupervised blind super-resolution framework (UDASR) achieves state-of-the-art restoration performance. Our code and pre-trained models have been uploaded to GitHub† for validation.

Recent developments in the field of non-local attention (NLA) have led to a renewed interest in self-similarity-based single image super-resolution (SISR). Researchers usually use the NLA to explore non-local self-similarity (NSS) in SISR and achieve satisfactory reconstruction results. However, a surprising phenomenon that the reconstruction performance of the standard NLA is similar to that of the NLA with randomly selected regions prompted us to revisit NLA. In this paper, we first analyzed the attention map of the standard NLA from different perspectives and discovered that the resulting probability distribution always has full support for every local feature, which implies a statistical waste of assigning values to irrelevant non-local features, especially for SISR which needs to model long-range dependence with a large number of redundant non-local features. Based on these findings, we introduced a concise yet effective soft thresholding operation to obtain high-similarity-pass attention (HSPA), which is beneficial for generating a more compact and interpretable distribution. Furthermore, we derived some key properties of the soft thresholding operation that enable training our HSPA in an end-to-end manner. The HSPA can be integrated into existing deep SISR models as an efficient general building block. In addition, to demonstrate the effectiveness of the HSPA, we constructed a deep high-similarity-pass attention network (HSPAN) by integrating a few HSPAs in a simple backbone. Extensive experimental results demonstrate that HSPAN outperforms state-of-the-art approaches on both quantitative and qualitative evaluations. Our code and a pre-trained model were uploaded to GitHub (https://github.com/laoyangui/HSPAN) for validation.

Separable nonlinear models (SNLMs) are of great importance in system modeling, signal processing, and machine learning because of their flexible structure and excellent description of nonlinear behaviors. The online identification of such models is quite challenging, and previous related work usually ignores the special structure where the estimated parameters can be partitioned into a linear and a nonlinear part. In this brief, we propose an efficient first-order recursive algorithm for SNLMs by introducing the variable projection (VP) step. The proposed algorithm utilizes the recursive least-squares method to eliminate the linear parameters, resulting in a reduced function. Then, the stochastic gradient descent (SGD) algorithm is employed to update the parameters of the reduced function. By considering the tight coupling relationship between linear parameters and nonlinear parameters, the proposed first-order VP algorithm is more efficient and robust than the traditional SGD algorithm and alternating optimization algorithm. More importantly, since the proposed algorithm just uses the first-order information, it is easier to apply it to large-scale models. Numerical results on examples of different sizes confirm the effectiveness and efficiency of the proposed algorithm.

Self-similarity is valuable to the exploration of non-local textures in single image super-resolution (SISR). Researchers usually assume that the importance of non-local textures is positively related to their similarity scores. In this paper, we surprisingly found that when repairing severely damaged query textures, some non-local textures with low-similarity which are closer to the target can provide more accurate and richer details than the high-similarity ones. In these cases, low-similarity does not mean inferior but is usually caused by different scales or orientations. Utilizing this finding, we proposed a Global Learnable Attention (GLA) to adaptively modify similarity scores of non-local textures during training instead of only using a fixed similarity scoring function such as the dot product. The proposed GLA can explore non-local textures with low-similarity but more accurate details to repair severely damaged textures. Furthermore, we propose to adopt Super-Bit Locality-Sensitive Hashing (SB-LSH) as a preprocessing method for our GLA. With the SB-LSH, the computational complexity of our GLA is reduced from quadratic to asymptotic linear with respect to the image size. In addition, the proposed GLA can be integrated into existing deep SISR models as an efficient general building block. Based on the GLA, we constructed a Deep Learnable Similarity Network (DLSN), which achieves state-of-the-art performance for SISR tasks of different degradation types (e.g., blur and noise). Our code and a pre-trained DLSN have been uploaded to GitHub†for validation.

Many fuzzy subspace clustering methods have been proposed for high-dimensional image data with rich structural information. However, since these methods do not fully exploit the subspace information in each cluster, their performance on image clustering is still not promising. In this work, we propose to find soft partitions directly based on the construction of subspaces. For each cluster, we use a bilinear orthogonal subspace to represent it. Then, through the reconstruction error of a sample in the subspace corresponding to a cluster, a new membership measure for the sample to the cluster is established. Furthermore, the graph regularization is imposed on these bilinear subspaces to preserve the local relational or manifold information of the image data in the original space. Altogether, we get a clustering model considering not only the subspace information but also the manifold information in image data. An efficient optimization algorithm is proposed to our model, and its theoretical convergence and time complexity are presented correspondingly. The proposed method is a one-stage clustering model that does not require vectorized image data, thereby reducing the computational burden while maintaining the structural relationship between pixels in the image. Competitive experimental results on benchmark datasets show that our model can converge quickly with strong clustering performance, which confirms the efficiency and superiority of the proposed method compared to other state-of-the-art fuzzy clustering methods.

Images captured under low-light environments typically have poor visibility, affecting many advanced computer vision tasks. In recent years, there have been some low-light image enhancement models based on deep learning, but they have not been able to effectively mine the deep multiscale features in the image, resulting in poor generalization performance and instability of the model. The disadvantages are mainly reflected in the color distortion, color unsaturation and artifacts. Current methods unable to adjust the exposure effectively, resulting in uneven exposure or partial overexposure. To address these issues, we propose an end-to-end low-light image enhancement model, which is called multiscale low-light image enhancement network with illumination constraint (MLLEN-IC), to achieve preferable generalization ability and stable performance. On the one hand, we use the squeeze-and-excitation-Res2Net block (SE-Res2block) as a base unit to enhance the model’s ability by extracting deep multiscale features. On the other hand, to make the model more adaptable in low-light image enhancement tasks, we calculate the illumination constraint by the low-light itself to prevent overexposure, uneven exposure, and unsaturated colors. Extensive experiments are conducted to demonstrate MLLEN-IC not only adjusts light levels, but also has a more natural visual effect, and avoids problems such as color distortion, artifacts, and uneven exposure. In particular, MLLEN-IC has pretty generalization and stability performance. The source code and supplementary are available athttps://github.com/CCECfgd/MLLEN-IC.

The broad learning system (BLS) is an emerging flat network, which has demonstrated its outstanding performance in classification and regression problems. The regularization plays an important role in the performance of the BLS. In real applications, since the BLS network is usually expanded dynamically, a predetermined regularization parameter may reduce the performance of the network. Using a fixed regularization in some cases, the classification accuracy of the BLS decreases dramatically when we expand the network. To alleviate this problem, we propose a method that automatically finds appropriate regularization parameters for different datasets, which is based on the weighted generalized cross-validation (WGCV). The experimental results indicate that the WGCV method improves the performance of the BLS, and alleviates the accuracy decrease of the incremental learning algorithm.

Deep neural networks have achieved breakthrough improvement in various application fields. Nevertheless, they usually suffer from a time-consuming training process because of the complicated structures of neural networks with a huge number of parameters. As an alternative, a fast and efficient discriminative broad learning system (BLS) is proposed, which takes the advantages of flat structure and incremental learning. The BLS has achieved outstanding performance in classification and regression problems. However, the previous studies ignored the reason why the BLS can generalize well. In this article, we focus on the interpretation from the viewpoint of the frequency domain. We discover the existence of the frequency principle in BLS, i.e., the BLS preferentially captures low-frequency components quickly and then fits the high frequencies during the incremental process of adding feature nodes and enhancement nodes. The frequency principle may be of great inspiration for expanding the application of BLS.

Many inverse problems in machine learning, system identification, and image processing include nuisance parameters, which are important for the recovering of other parameters. Separable nonlinear optimization problems fall into this category. The special separable structure in these problems has inspired several efficient optimization strategies. A well-known method is the variable projection (VP) that projects out a subset of the estimated parameters, resulting in a reduced problem that includes fewer parameters. The expectation maximization (EM) is another separated method that provides a powerful framework for the estimation of nuisance parameters. The relationships between EM and VP were ignored in previous studies, though they deal with a part of parameters in a similar way. In this article, we explore the internal relationships and differences between VP and EM. Unlike the algorithms that separate the parameters directly, the hierarchical identification algorithm decomposes a complex model into several linked submodels and identifies the corresponding parameters. Therefore, this article also studies the difference and connection between the hierarchical algorithm and the parameter-separated algorithms like VP and EM. In the numerical simulation part, Monte Carlo experiments are performed to further compare the performance of different algorithms. The results show that the VP algorithm usually converges faster than the other two algorithms and is more robust to the initial point of the parameters.

Stationarity is fundamental for time-series modeling and prediction. In this article, we focus on the radial basis function network-based autoregressive (RBF-AR) models which have been widely used in practical applications. Compared to previous work, we give a less-restrictive sufficient condition for the asymptotic stationarity of the RBF-AR model. The parameter estimation of the RBF-AR model is converted to the optimization of a variable projection functional with constraints of stationarity to always derive a stationary model. The constrained evolutionary algorithm is used to solve the optimization problem. Numerical results demonstrate the effectiveness of the proposed method.

The separable nonlinear least-squares (SNLLS) problems considered in this article frequently appear in a wide range of research fields, such as machine learning, computer vision, system identification, and signal processing. The variable projection algorithm proposed by Golub and Pereyra, which reduces the dimension of the parameters by projecting the linear parameters out of the problem, is quite valuable in solving SNLLS problems. Previous implementations of the variable projection algorithm are based on matrix factorization. In this article, we propose an iterative implementation of the variable projection algorithm. Compared with previous implementations based on matrix decomposition, the proposed method can effectively avoid suffering from large condition number of the matrix or even matrix decomposition failure when dealing with ill-posed SNLLS problems. Numerical experiments on real-world data and synthetic data show the efficiency and robustness of the proposed iterative variable projection algorithm.

Abstract In this paper, a variable projection approach based on the BFGS (Broyden–Fletcher–Goldfarb–Shanno) method for image reconstruction problems is proposed, which is an alternative to the common alternating minimisation scheme. The image restoration is expressed as a nonlinear least‐squares problem with reduced parameter space. To improve the efficiency of the algorithm, the BFGS method is proposed to be used to optimise the reduced objective function. The large‐scale problem considered in this paper is projected on to a small Krylov subspace using Lanczos bidiagonalisation. The regularisation parameter is selected by a weighted generalised cross validation criterion. Numerical examples demonstrate the efficiency and effectiveness of the proposed algorithm.

We propose, in this paper, a framework for time series and nonlinear system modeling, called the basis function matrix-based flexible coefficient autoregressive (BFM-FCAR) model. It has very flexible nonlinear structure. We show that many famous nonlinear time series models can be derived under this framework by choosing the proper basis function matrices. Some probabilistic properties (the conditions of geometrical ergodicity) of the BFM-FCAR model are investigated. Taking advantage of the model structure, we present an efficient parameter estimation algorithm for the proposed framework by using the variable projection method. Finally, we show how new models are generated from the proposed framework.

Separable nonlinear least squares (SNLLS) problems have attracted interest in a wide range of research fields such as machine learning, computer vision, and signal processing. During the past few decades, several algorithms, including the joint optimization algorithm, alternated least squares (ALS) algorithm, embedded point iterations (EPI) algorithm, and variable projection (VP) algorithms, have been employed for solving SNLLS problems in the literature. The VP approach has been proven to be quite valuable for SNLLS problems and the EPI method has been successful in solving many computer vision tasks. However, no clear explanations about the intrinsic relationships of these algorithms have been provided in the literature. In this paper, we give some insights into these algorithms for SNLLS problems. We derive the relationships among different forms of the VP algorithms, EPI algorithm and ALS algorithm. In addition, the convergence and robustness of some algorithms are investigated. Moreover, the analysis of the VP algorithm generates a negative answer to Kaufman's conjecture. Numerical experiments on the image restoration task, fitting the time series data using the radial basis function network based autoregressive (RBF-AR) model, and bundle adjustment are given to compare the performance of different algorithms.

In this article, we study the recursive algorithms for a class of separable nonlinear models (SNLMs) in which the parameters can be partitioned into a linear part and a nonlinear part. Such models are very common in machine learning, system identification, and signal processing. Utilizing the special structure of the SNLMs, we propose a recursive variable projection (RVP) algorithm, in which at each recursion, the linear parameters of the model are eliminated, and the nonlinear parameters are updated by the recursive Levenberg-Marquart algorithm. Then, based on the updated nonlinear parameters, the linear parameters are updated by the recursive least-squares algorithm. According to a convergence analysis of the RVP algorithm, the parameter estimation error is mean-square bounded. Numerical examples confirm the satisfactory performance of the proposed algorithm.

In this paper, we consider the term selection problem for a class of separable nonlinear models. The strategy is a two-step process in which the nonlinear parameters of the model are first optimized by a variable projection method, and then the least absolute shrinkage and selection operator are adopted to obtain a sparse solution by picking out the critical terms automatically. This process may be repeated several times. The proposed algorithm is tested on parameter estimation problems for an exponential model and a neural network-based model. The numerical results show that the proposed algorithm can pick out the appropriate terms from the overparameterized model and the obtained parsimonious model performs better than other methods.

In the previous work, the parameters of radial basis function network based autoregressive (RBF-AR) models are estimated offline and no longer updated afterward. In this letter, an adaptive learning algorithm is proposed for the RBF-AR models. The proposed strategy is that the nonlinear parameters are previously determined by an off-line variable projection method; and once new samples are available, the linear parameters are updated. The linear adaptive algorithm adopted in this letter is the multi-innovation least squares method, due to its high performance. The simulation results show that with the adaption of the linear parameters, the prediction performance of the RBF-AR models may be significantly improved, which demonstrates the effectiveness of the proposed algorithm.

Separable nonlinear models are very common in various research fields, such as machine learning and system identification. The variable projection (VP) approach is efficient for the optimization of such models. In this paper, we study various VP algorithms based on different matrix decompositions. Compared with the previous method, we use the analytical expression of the Jacobian matrix instead of finite differences. This improves the efficiency of the VP algorithms. In particular, based on the modified Gram-Schmidt (MGS) method, a more robust implementation of the VP algorithm is introduced for separable nonlinear least-squares problems. In numerical experiments, we compare the performance of five different implementations of the VP algorithm. Numerical results show the efficiency and robustness of the proposed MGS method-based VP algorithm.

For a class of nonlinear least squares problems, it is usually very beneficial to separate the variables into a linear and a nonlinear part and take full advantage of reliable linear least squares techniques. Consequently, the original problem is turned into a reduced problem which involves only nonlinear parameters. We consider in this paper four separated algorithms for such problems. The first one is the variable projection (VP) algorithm with full Jacobian matrix of Golub and Pereyra. The second and third ones are VP algorithms with simplified Jacobian matrices proposed by Kaufman and Ruano et al. respectively. The fourth one only uses the gradient of the reduced problem. Monte Carlo experiments are conducted to compare the performance of these four algorithms. From the results of the experiments, we find that: 1) the simplified Jacobian proposed by Ruano et al. is not a good choice for the VP algorithm; moreover, it may render the algorithm hard to converge; 2) the fourth algorithm perform moderately among these four algorithms; 3) the VP algorithm with the full Jacobian matrix perform more stable than that of the VP algorithm with Kuafman's simplified one; and 4) the combination of VP algorithm and Levenberg-Marquardt method is more effective than the combination of VP algorithm and Gauss-Newton method.