Calibrated Recommendation Systems (CRS) balance user preferences with constraints like diversity, fairness, and novelty to create inclusive recommendation lists. However, existing research often overlooks the mandatory inclusion of sponsored items, assuming unrestricted product selection. In practice, sponsored items, paid for by advertisers, must be included, which can conflict with CRS goals when advertisers' priorities misalign with system objectives. This paper addresses this gap by formulating CRS with sponsored items as a combinatorial optimization problem. We develop efficient approximation algorithms to generate the most calibrated recommendation lists while meeting sponsorship requirements.

Modern decision-making scenarios often involve data that is both high-dimensional and rich in higher-order contextual information, where existing bandits algorithms fail to generate effective policies. In response, we propose in this paper a generalized linear tensor bandits algorithm designed to tackle these challenges by incorporating low-dimensional tensor structures, and further derive a unified analytical framework of the proposed algorithm. Specifically, our framework introduces a convex optimization approach with the weakly decomposable regularizers, enabling it to not only achieve better results based on the tensor low-rankness structure assumption but also extend to cases involving other low-dimensional structures such as slice sparsity and low-rankness. The theoretical analysis shows that, compared to existing low-rankness tensor result, our framework not only provides better bounds but also has a broader applicability. Notably, in the special case of degenerating to low-rank matrices, our bounds still offer advantages in certain scenarios.

Developing efficient kernel methods for regression is popular in the past two decades. In this paper, utilizing boosting on kernel-based weak learners, we propose a novel kernel-based learning algorithm called kernel-based re-scaled boosting with truncation, dubbed as KReBooT. The proposed KReBooT benefits in controlling the structure and producing sparse estimators, and is near overfitting resistant. We conduct both theoretical analysis and numerical simulations to illustrate the excellent performance of KReBooT. Theoretically, we prove that KReBooT can achieve the optimal numerical convergence rate for nonlinear approximation. Furthermore, using a variant of Talagrand's concentration inequality, we provide fast learning rates for KReBooT, which is a new record of boosting-type algorithms. Numerically, we carry out several simulations to show the promising performance of KReBooT in terms of its good generalization, near over-fitting resistance and structure constraints.

In this paper, we study the generalization performance of global minima of empirical risk minimization (ERM) on over-parameterized deep ReLU nets. Using a novel deepening scheme for deep ReLU nets, we rigorously prove that there exist perfect global minima achieving optimal generalization error rates for numerous types of data under mild conditions. Since over-parameterization of deep ReLU nets is crucial to guarantee that the global minima of ERM can be realized by the widely used stochastic gradient descent (SGD) algorithm, our results present a potential way to fill the gap between optimization and generalization of deep learning.

Although robust tensor completion has been extensively studied, the effect of incorporating side information has not been explored. In this article, we fill this gap by developing a novel high-order robust tensor completion model that incorporates both latent and explicit side information. We base our model on the transformed t-product because the corresponding tensor tubal rank can characterize the inherent low-rank structure of a tensor. We study the effect of side information on sample complexity and prove that our model needs fewer observations than other tensor recovery methods when side information is perfect. This theoretically shows that informative side information is beneficial for learning. Extensive experimental results on synthetic and real data further demonstrate the superiority of the proposed method over several popular alternatives. In particular, we evaluate the performance of our solution based on two important applications, namely, link prediction in signed networks and rating prediction in recommender systems. We show that the proposed model, which manages to exploit side information in learning, outperforms other methods in the learning of such low-rank tensor data. Furthermore, when dealing with varying dimensions, we also design an online robust tensor completion with side information algorithm and validate its effectiveness using a real-world traffic dataset in the supplementary material. The source code is available athttps://github.com/yyyancy/RTCF.

Recently, transform-based tensor nuclear norm (TNN) methods have received increasing attention as a powerful tool for multi-dimensional visual data (color images, videos, and multispectral images, etc.) recovery. Especially, the redundant transform-based TNN achieves satisfactory recovery results, where the redundant transform along spectral mode can remarkably enhance the low-rankness of tensors. However, it suffers from expensive computational cost induced by the redundant transform. In this paper, we propose a learnable spatial-spectral transform-based TNN model for multi-dimensional visual data recovery, which not only enjoys better low-rankness capability but also allows us to design fast algorithms accompanying it. More specifically, we first project the large-scale original tensor to the small-scale intrinsic tensor via the learnable semi-orthogonal transforms along the spatial modes. Here, the semi-orthogonal transforms, serving as the key building block, can boost the spatial low-rankness and lead to a small-scale problem, which paves the way for designing fast algorithms. Secondly, to further boost the low-rankness, we apply the learnable redundant transform along the spectral mode to the small-scale intrinsic tensor. To tackle the proposed model, we apply an efficient proximal alternating minimization-based algorithm, which enjoys a theoretical convergence guarantee. Extensive experimental results on real-world data (color images, videos, and multispectral images) demonstrate that the proposed method outperforms state-of-the-art competitors in terms of evaluation metrics and running time.

In this paper, we aim to build a novel bandits algorithm that is capable of fully harnessing the power of multi-dimensional data and the inherent non-linearity of reward functions to provide high-usable and accountable decision-making services. To this end, we introduce a generalized low-rank tensor contextual bandits model in which an action is formed from three feature vectors, and thus is represented by a tensor. In this formulation, the reward is determined through a generalized linear function applied to the inner product of the action’s feature tensor and a fixed but unknown parameter tensor with low-rank structure. To effectively achieve the trade-off between exploration and exploitation, we introduce an algorithm called “Generalized Low-Rank Tensor Exploration Subspace then Refine” (G-LowTESTR). This algorithm first collects data to explore the intrinsic low-rank tensor subspace information embedded in the scenario, and then converts the original problem into a lower-dimensional generalized linear contextual bandits problem. Rigorous theoretical analysis shows that the regret bound of G-LowTESTR is superior to those in vectorization and matricization cases. We conduct a series of synthetic and real data experiments to further highlight the effectiveness of G-LowTESTR, leveraging its ability to capitalize on the low-rank tensor structure for enhanced learning.

Online schemes and nonlocal similarity are two effective approaches for strengthening robust principal component analysis (RPCA) techniques in video denoising. However, their limitations are also evident. The online scheme is usually highly efficient but lacks consideration of regional appearance information, thus it cannot effectively handle videos with complex dynamics such as object movements. On the other hand, nonlocal similarity is used to better utilize regional information but incurs a heavy computational cost. Moreover, these two techniques are incompatible and challenging to work together. To overcome this barrier and harness the advantages of both approaches, this paper proposes a novel online nonlocal RPCA method. 1) A clustering based nonlocal strategy (ClusNonlocal) is adopted, which not only greatly reduces the computation cost, but also forms low-dimensional subspaces for online processing; 2) a new weighted RPCA model is proposed, which regards samples with different importances and improves the performance of subspace pursuit and video recovery; 3) a multi-level subspace updating scheme and weighted projection method is proposed, which keeps the performance of online video data processing at a high level at all time. A series of video denoising experiments are carried out to demonstrate the overall advantages of our procedure over several other ones, in terms of both visual quality and running speed.

Frequent directions (FDs), as a deterministic matrix sketching technique, have been proposed for tackling low-rank approximation problems. This method has a high degree of accuracy and practicality but experiences a lot of computational cost for large-scale data. Several recent works on the randomized version of FDs greatly improve the computational efficiency but unfortunately sacrifice some precision. To remedy such an issue, this article aims to find a more accurate projection subspace to further improve the efficiency and effectiveness of the existing FDs' techniques. Specifically, by utilizing the power of the block Krylov iteration and random projection technique, this article presents a fast and accurate FDs algorithm named r-BKIFD. The rigorous theoretical analysis shows that the proposed r-BKIFD has a comparable error bound with original FDs, and the approximation error can be arbitrarily small when the number of iterations is chosen appropriately. Extensive experimental results on both synthetic and real data further demonstrate the superiority of r-BKIFD over several popular FDs algorithms both in terms of computational efficiency and accuracy.

Low-tubal-rank tensor approximation has been proposed to analyze large-scale and multidimensional data. However, finding such an accurate approximation is challenging in the streaming setting, due to the limited computational resources. To alleviate this issue, this article extends a popular matrix sketching technique, namely, frequent directions (FDs), for constructing an efficient and accurate low-tubal-rank tensor approximation from streaming data based on the tensor singular value decomposition (t-SVD). Specifically, the new algorithm allows the tensor data to be observed slice by slice but only needs to maintain and incrementally update a much smaller sketch, which could capture the principal information of the original tensor. The rigorous theoretical analysis shows that the approximation error of the new algorithm can be arbitrarily small when the sketch size grows linearly. Extensive experimental results on both synthetic and real multidimensional data further reveal the superiority of the proposed algorithm compared with other sketching algorithms for getting low-tubal-rank approximation, in terms of both efficiency and accuracy.

The utilization of online optimization techniques is prevalent in many fields of artificial intelligence, enabling systems to continuously learn and adjust to their surroundings. This paper outlines a regularized online optimization problem, where the regularizer is defined on the average of the actions taken. The objective is to maximize the sum of rewards and the regularizer value while adhering to resource constraints, where the reward function is assumed to be DR-submodular. Both concave and DR-submodular regularizers are analyzed. Concave functions are useful in describing the impartiality of decisions, while DR-submodular functions can be employed to represent the overall effect of decisions on all relevant parties. We have developed two algorithms for each of the concave and DR-submodular regularizers. These algorithms are easy to implement, efficient, and produce sublinear regret in both cases. The performance of the proposed algorithms and regularizers has been verified through numerical experiments in the context of internet advertising.

It is known that the decomposition in low-rank and sparse matrices (L+S for short) can be achieved by several Robust PCA techniques. Besides the low rankness, the local smoothness (LSS) is a vitally essential prior for many real-world matrix data such as hyperspectral images and surveillance videos, which makes such matrices have low-rankness and local smoothness property at the same time. This poses an interesting question: Can we make a matrix decomposition in terms of L&LSS +S form exactly? To address this issue, we propose in this paper a new RPCA model based on three-dimensional correlated total variation regularization (3DCTV-RPCA for short) by fully exploiting and encoding the prior expression underlying such joint low-rank and local smoothness matrices. Specifically, using a modification of Golfing scheme, we prove that under some mild assumptions, the proposed 3DCTV-RPCA model can decompose both components exactly, which should be the first theoretical guarantee among all such related methods combining low rankness and local smoothness. In addition, by utilizing Fast Fourier Transform (FFT), we propose an efficient ADMM algorithm with a solid convergence guarantee for solving the resulting optimization problem. Finally, a series of experiments on both simulations and real applications are carried out to demonstrate the general validity of the proposed 3DCTV-RPCA model.

As a domain-dependent prior knowledge, side information has been introduced into Robust Principal Component Analysis (RPCA) to alleviate its degenerate or suboptimal performance in some real applications. It has recently realized that the natural structural information can be better retained if the observed data is kept in the original tensor form rather than matricizing it or other order reduction means. Hence, studies on RPCA of tensor version have attracted more and more attentions. To share the merits from both direct tensor modeling and side information, we propose three models to deal with the problem of Tensor RPCA with side information based on tensor Singular Value Decomposition (t-SVD). To solve these models, we develop an efficient algorithm with convergence guarantee using the well-known alternating direction method of multiplier. Extensive experimental studies on both synthetic and real-world tensor data have been carried out to demonstrate the superiority of the proposed models over several other state-of-the-arts. Our code is released athttps://github.com/zsj9509/TPCPSF.

Hyperspectral image classification has two key characteristics: spatial dependency and class imbalance. The synthetic oversampling methods for solving the between-class imbalance has the problem of spatial information loss to some extent. However, the feature space optimization methods for solving data within-class complex distribution problems have not been studied extensively. To remedy such issue, we propose in this paper an improved spectral-spatial feature rotation based SVM ensemble method (SSFRoF). More precisely, in the data preprocessing stage and from the perspective of sample space optimization, we devise a spectral-spatial neighborhood based adaptive synthetic sampling algorithm (SSAS) that focuses adaptively on hard-to-learn samples, with artificial samples interpolated in the spectral-spatial neighborhood to further utilize the spatial information. In feature rotation stage and from the perspective of feature space optimization, we innovatively discover that the class overlapping problem can be handled by combining feature extraction, so as to achieve the goal of reducing the difficulty of imbalanced classification. Besides, the SVM and voting procedure are used in the ensembled classification stage. We then conduct a series of experiments to compare the proposed SSFRoS with several baselines in terms of seven metrics. The results show that SSFRoS has significant advantages in dealing with imbalanced hyperspectral image classification problem. In addition, a detailed ablation study shows that the lowest class recall can be improved by 46.67% and 5.26% on Indian Pines and Pavia University respectively, which demonstrates that using SSAS can enhance the recognition of hard-to-learn classes.

Hyperspectral (HS) imaging has been widely used in various real application problems. However, due to the hardware limitations, the obtained HS images usually have low spatial resolution, which could obviously degrade their performance. Through fusing a low spatial resolution HS image with a high spatial resolution auxiliary image (e.g., multispectral, RGB or panchromatic image), the so-called HS image fusion has underpinned much of recent progress in enhancing the spatial resolution of HS image. Nonetheless, a corresponding well registered auxiliary image cannot always be available in some real situations. To remedy this issue, we propose in this paper a newly single HS image super-resolution method based on a novel knowledge-driven deep unrolling technique. Precisely, we first propose a maximum a posterior based energy model with implicit priors, which can be solved by alternating optimization to determine an elementary iteration mechanism. We then unroll such iteration mechanism with an ingenious Transformer embedded convolutional recurrent neural network in which two structural designs are integrated. That is, the vision Transformer and 3D convolution learn the implicit spatial-spectral priors, and the recurrent hidden connections over iterations model the recurrence of the iterative reconstruction stages. Thus, an effective knowledge-driven, end-to-end and data-dependent HS image super-resolution framework can be successfully attained. Extensive experiments on three HS image datasets demonstrate the superiority of the proposed method over several state-of-the-art HS image super-resolution methods.

Background subtraction of videos has been a fundamental research topic in computer vision in the past decades. To alleviate the computation burden and enhance the efficiency, background subtraction from online compressive measurements has recently attracted much attention. However, current methods still have limitations. First, they are all based on matrix modeling, which breaks the spatial structure within video frames. Second, they generally ignore the complex disturbance within the background, which reduces the efficiency of the low-rank assumption. To alleviate this issue, we propose a tensor-based online compressive video reconstruction and background subtraction method, abbreviated as NIOTenRPCA, by explicitly modeling the background disturbance in different frames as nonidentical but correlated noise. By virtue of such sophisticated modeling, the proposed method can well adapt to complex video scenes and, thus, perform more robustly. Extensive experiments on a series of real-world video datasets have demonstrated the effectiveness of the proposed method compared with the existing state of the arts. The code of our method is released on the website: https://github.com/crystalzina/NIOTenRPCA.

In this paper, we study nonconvex tensor robust principal component analysis (RPCA) based on the $t$-SVD. We first propose an alternating projection method, i.e., APT, which converges linearly to the ground-truth under the incoherence conditions of tensors. However, as the projection to the low-rank tensor space in APT can be slow, we further propose to speedup such a process by utilizing the property of the tangent space of low-rank. The resulting algorithm, i.e., EAPT, is not only more efficient than APT but also keeps the linear convergence. Compared with existing tensor RPCA works, the proposed method, especially EAPT, is not only more effective due to the recovery guarantee and adaption in the transformed (frequency) domain but also more efficient due to faster convergence rate and lower iteration complexity. These benefits are also empirically verified both on synthetic data, and real applications, e.g., hyperspectral image denoising and video background subtraction.

Ensemble learning methods, such as boosting, focus on producing a strong classifier based on numerous weak classifiers. In this paper, we develop a novel ensemble learning method called rescaled boosting with truncation (ReBooT) for binary classification by combining well-known rescaling and regularization ideas in boosting. Theoretically, we present some sufficient conditions for the convergence of ReBooT, derive an almost optimal numerical convergence rate, and deduce fast-learning rates in the framework of statistical learning theory. Experimentally, we conduct both toy simulations and four real-world data runs to show the power of ReBooT. Our results show that, compared with the existing boosting algorithms, ReBooT possesses better learning performance and interpretability in terms of solid theoretical guarantees, perfect structure constraints, and good prediction performance. History: Accepted by Ram Ramesh, Area Editor for Data Science & Machine Learning Funding: This work was supported by the National Natural Science Foundation of China [Grants 11971374, 61772374, and 61876133]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1224 .

This paper focuses on learning rate analysis of Nystrom regularization with sequential sub-sampling for $\tau$-mixing time series. Using a recently developed Banach-valued Bernstein inequality for $\tau$-mixing sequences and an integral operator approach based on second-order decomposition, we succeed in deriving almost optimal learning rates of Nystrom regularization with sequential sub-sampling for $\tau$-mixing time series. A series of numerical experiments are carried out to verify our theoretical results, showing the excellent learning performance of Nystrom regularization with sequential sub-sampling in learning massive time series data. All these results extend the applicable range of Nyström regularization from i.i.d. samples to non-i.i.d. sequences.

Low-rank tensor recovery (LRTR) is a natural extension of low-rank matrix recovery (LRMR) to high-dimensional arrays, which aims to reconstruct an underlying tensor from incomplete linear measurements M(X). However, LRTR ignores the error caused by quantization, limiting its application when the quantization is low-level. In this work, we take into account the impact of extreme quantization and suppose the quantizer degrades into a comparator that only acquires the signs of M(X). We still hope to recover X from these binary measurements. Under the tensor Singular Value Decomposition (t-SVD) framework, two recovery methods are proposedthe first is a tensor hard singular tube thresholding method; the second is a constrained tensor nuclear norm minimization method. These methods can recover a real n1 n2 n3 tensor X with tubal rank r from m random Gaussian binary measurements with errors decaying at a polynomial speed of the oversampling factor := m/((n1+ n2)n3r). To improve the convergence rate, we develop a new quantization scheme under which the convergence rate can be accelerated to an exponential function of . Numerical experiments verify our results, and the applications to real-world data demonstrate the promising performance of the proposed methods.

Tensor completion methods based on the tensor train (TT) have the issues of inaccurate weight assignment and ineffective tensor augmentation pre-processing. In this work, we propose a novel tensor completion approach via the element-wise weighted technique. Accordingly, a novel formulation for tensor completion and an effective optimization algorithm, called tensor completion by parallel weighted matrix factorization via tensor train (TWMac-TT), is proposed. In addition, we specifically consider the recovery quality of edge elements from adjacent blocks. Different from traditional reshaping and ket augmentation, we utilize a new tensor augmentation technique called overlapping ket augmentation, which can further avoid blocking artifacts. We then conduct extensive performance evaluations on synthetic data and several real image data sets. Our experimental results demonstrate that the proposed algorithm TWMac-TT outperforms several other competing tensor completion methods. The code is available athttps://github.com/yzcv/TWMac-TT-OKA

Compared with avid research activities of deep convolutional neural networks (DCNNs) in practice, the study of theoretical behaviors of DCNNs lags heavily behind. In particular, the universal consistency of DCNNs remains open. In this paper, we prove that implementing empirical risk minimization on DCNNs with expansive convolution (with zero-padding) is strongly universally consistent. Motivated by the universal consistency, we conduct a series of experiments to show that without any fully connected layers, DCNNs with expansive convolution perform not worse than the widely used deep neural networks with hybrid structure containing contracting (without zero-padding) convolutional layers and several fully connected layers.

Snapshot compressive imaging (SCI) is a new type of compressive imaging system that compresses multiple frames of images into a single snapshot measurement, which enjoys low cost, low bandwidth, and high-speed sensing rate. By applying the existing SCI methods to deal with hyperspectral images, however, could not fully exploit the underlying structures, and thereby demonstrate unsatisfactory reconstruction performance. To remedy such issue, this paper aims to propose a new effective method by taking advantage of two intrinsic priors of the hyperspectral images, namely deep image denoising and total variation (TV) priors. Specifically, we propose an optimization objective to utilize these two priors. By solving this optimization objective, our method is equivalent to incorporate a weighted FFDNet and a 2DTV or 3DTV denoiser into the plug-andplay framework. Extensive numerical experiments demonstrate the outperformance of the proposed method over several state-of-the-art alternatives. Additionally, we provide a detailed convergence analysis of the resulting plug-andplay algorithm under relatively weak conditions such as without using diminishing step sizes. The code is available at https://github.com/ucker/SCI-TVFFDNet.

In this paper, the generative model is used to introduce the structural properties of the signal to replace the common sparse hypothesis, and a non-convex compressed sensing sparse deviation model based on the generative model (ℓq-Gen) is proposed. By establishing ℓqvariant of the restricted isometry property (q-RIP) and Set-Restricted Eigenvalue Condition (q-S-REC), the error upper bound of the optimal decoder is derived when the recovered signal is within the sparse deviation range of the generator. Furthermore, it is proved that the Gaussian matrix satisfying a certain number of measurements is sufficient to ensure a good recovery for the generating function with high probability. Finally, a series of experiments are carried out to verify the effectiveness and superiority of the ℓq-Gen model.

It is known that boosting can be interpreted as an optimization technique to minimize an underlying loss function. Specifically, the underlying loss being minimized by the traditional AdaBoost is the exponential loss, which proves to be very sensitive to random noise/outliers. Therefore, several boosting algorithms, e.g., LogitBoost and SavageBoost, have been proposed to improve the robustness of AdaBoost by replacing the exponential loss with some designed robust loss functions. In this article, we present a new way to robustify AdaBoost, that is, incorporating the robust learning idea of self-paced learning (SPL) into the boosting framework. Specifically, we design a new robust boosting algorithm based on the SPL regime, that is, SPLBoost, which can be easily implemented by slightly modifying off-the-shelf boosting packages. Extensive experiments and a theoretical characterization are also carried out to illustrate the merits of the proposed SPLBoost.

This paper focuses on the recovery of low-tubal-rank tensors from binary measurements under the frame of tensor Singular Value Decomposition. We show that the direction of a tubal-rank-r tensor X ∈ ℝn1×n2×n3can be approximated from Ω((n1+ n2)n3r) random Gaussian measurements. In addition, incorporating nonadaptive thresholds in the measurements, it is proved that the full X can be recovered. As we will see, under this nonadaptive measurement scheme, recovery errors decay at the rate of polynomial of the oversampling factor λ := m/(n1+ n2)n3r, i.e., O(λ-1/6). In order to obtain faster decay rate, we introduce a recursive strategy which generates thresholds according to previous estimates for each iteration. Under this quantization scheme, An iterative recovery algorithm is proposed which establishes recovery errors decaying at the rate of exponent of λ. Numerical experiments are conducted to demonstrate our results.

The recently proposed Tensor Nuclear Norm (TNN) minimization has been widely used for tensor completion. However, previous works didn’t consider the structural difference between the observed data and missing data, which widely exists in many applications. In this paper, we propose to incorporate a constraint item on the missing values into low-tubal-rank tensor completion to promote the structural hypothesis of the missing values such as sparsity. Theoretically, the proposed model has lower recovery error than classical model, and the target tensor can be recovered exactly with overwhelming probability provided low-tubal-rankness on whole area and sparsity on missing area. Algorithmically, an efficient algorithm by Alternating Direction Method of Multiplier (ADMM) is presented. Extensive experiments on both synthetic and real-world data demonstrate its superiority compared with several state-of-the-art methods.

It is known that frequent directions (FD) is a popular deterministic matrix sketching technique for low rank approximation. However, FD and its randomized variants usually meet high computational cost or computational instability in dealing with large-scale datasets, which limits their use in practice. To remedy such issues, this paper aims at improving the efficiency and effectiveness of FD. Specifically, by utilizing the power of Block Krylov Iteration and count sketch techniques, we propose a fast and accurate FD algorithm dubbed as BKICS-FD. We derive the error bound of the proposed BKICS-FD and then carry out extensive numerical experiments to illustrate its superiority over several popular FD algorithms, both in terms of computational speed and accuracy.

Hyperspectral (HS) imaging has shown its superiority in many real applications. However, it is usually difficult to obtain high-resolution (HR) HS images through existing imaging techniques due to the hardware limitations. To improve the spatial resolution of HS images, this article proposes an effective HS-multispectral (HS-MS) image fusion method by combining the ideas of nonlocal low-rank tensor modeling and spectral unmixing. To be more precise, instead of unfolding the HS image into a matrix as done in the literature, we directly represent it as a tensor, then a designed nonlocal Tucker decomposition is used to model its underlying spatial-spectral correlation and the spatial self-similarity. The MS image serves mainly as a data constraint to maintain spatial consistency. To further reduce the spectral distortions in spatial enhancement, endmembers, and abundances from the spectral are used for spectral regularization. An efficient algorithm based on the alternating direction method of multipliers (ADMM) is developed to solve the resulting model. Extensive experiments on four HS image data sets demonstrate the superiority of the proposed method over several state-of-the-art HS-MS image fusion methods.

The total variation (TV) is a powerful regularization term encoding the local smoothness prior structure underlying images. By combining the TV regularization term with low rank prior, the 3D total variation (3DTV) regularizer has achieved advanced performance in general hyperspectral image (HSI) processing tasks. Intrinsically, 3DTV assumes i.i.d. sparsity structures on all bands of the gradient maps calculated along the spectrum and space of an HSI. This, however, largely deviates from the real-world cases, where the gradient maps generally have different while correlated gradient map structures across all bands. To alleviate this issue, we propose an enhanced 3DTV (E-3DTV) regularization term beyond the conventional. Instead of imposing sparsity on gradient maps themselves, the new term calculates sparsity on the subspace bases on gradient maps along all bands of an HSI, which naturally encodes the correlation and difference among all these bands, and thus more faithfully reflects the insightful configurations of an HSI. The E-3DTV term can easily replace the conventional 3DTV term and be embedded into an HSI processing model to ameliorate its performance. We made such attempts on two typical related tasks: HSI denoising and compressed sensing. The superiority of our proposed method is substantiated by extensive experiments on synthetic and real HSI data, visually and quantitatively on both tasks, as compared with current state-of-the-arts. The code of our algorithm is released athttps://github.com/andrew-pengjj/Enhanced-3DTV.git.

We propose a framework to learn deep generative models via \textbf{V}ariational \textbf{Gr}adient Fl\textbf{ow} (VGrow) on probability spaces. The evolving distribution that asymptotically converges to the target distribution is governed by a vector field, which is the negative gradient of the first variation of the $f$-divergence between them. We prove that the evolving distribution coincides with the pushforward distribution through the infinitesimal time composition of residual maps that are perturbations of the identity map along the vector field. The vector field depends on the density ratio of the pushforward distribution and the target distribution, which can be consistently learned from a binary classification problem. Connections of our proposed VGrow method with other popular methods, such as VAE, GAN and flow-based methods, have been established in this framework, gaining new insights of deep generative learning. We also evaluated several commonly used divergences, including Kullback-Leibler, Jensen-Shannon, Jeffreys divergences as well as our newly discovered “logD” divergence which serves as the objective function of the logD-trick GAN. Experimental results on benchmark datasets demonstrate that VGrow can generate high-fidelity images in a stable and efficient manner, achieving competitive performance with state-of-the-art GANs.

MOTIVATION: A large number of recent genome-wide association studies (GWASs) for complex phenotypes confirm the early conjecture for polygenicity, suggesting the presence of large number of variants with only tiny or moderate effects. However, due to the limited sample size of a single GWAS, many associated genetic variants are too weak to achieve the genome-wide significance. These undiscovered variants further limit the prediction capability of GWAS. Restricted access to the individual-level data and the increasing availability of the published GWAS results motivate the development of methods integrating both the individual-level and summary-level data. How to build the connection between the individual-level and summary-level data determines the efficiency of using the existing abundant summary-level resources with limited individual-level data, and this issue inspires more efforts in the existing area. RESULTS: In this study, we propose a novel statistical approach, LEP, which provides a novel way of modeling the connection between the individual-level data and summary-level data. LEP integrates both types of data by LEveraging Pleiotropy to increase the statistical power of risk variants identification and the accuracy of risk prediction. The algorithm for parameter estimation is developed to handle genome-wide-scale data. Through comprehensive simulation studies, we demonstrated the advantages of LEP over the existing methods. We further applied LEP to perform integrative analysis of Crohn's disease from WTCCC and summary statistics from GWAS of some other diseases, such as Type 1 diabetes, Ulcerative colitis and Primary biliary cirrhosis. LEP was able to significantly increase the statistical power of identifying risk variants and improve the risk prediction accuracy from 63.39% (±0.58%) to 68.33% (±0.32%) using about 195 000 variants. AVAILABILITY AND IMPLEMENTATION: The LEP software is available at https://github.com/daviddaigithub/LEP. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Rain streaks removal is an important issue in outdoor vision systems and has recently been investigated extensively. In this paper, we propose a novel video rain streak removal approach FastDeRain, which fully considers the discriminative characteristics of rain streaks and the clean video in the gradient domain. Specifically, on the one hand, rain streaks are sparse and smooth along the direction of the raindrops, whereas on the other hand, clean videos exhibit piecewise smoothness along the rain-perpendicular direction and continuity along the temporal direction. Theses smoothness and continuity results in the sparse distribution in the different directional gradient domain, respectively. Thus, we minimize 1) the ℓ1 norm to enhance the sparsity of the underlying rain streaks, 2) two ℓ1 norm of unidirectional Total Variation (TV) regularizers to guarantee the anisotropic spatial smoothness, and 3) an ℓ1 norm of the time-directional difference operator to characterize the temporal continuity. A split augmented Lagrangian shrinkage algorithm (SALSA) based algorithm is designed to solve the proposed minimization model. Experiments conducted on synthetic and real data demonstrate the effectiveness and efficiency of the proposed method. According to comprehensive quantitative performance measures, our approach outperforms other state-of-the-art methods, especially on account of the running time. The code of FastDeRain can be downloaded at https://github.com/TaiXiangJiang/FastDeRain.

Low-rank matrix estimation arises in a number of statistical and machine learning tasks. In particular, the coefficient matrix is considered to have a low-rank structure in multivariate linear regression and multivariate quantile regression. In this paper, we propose a method called penalized matrix least squares approximation (PMLSA) toward a unified yet simple low-rank matrix estimate. Specifically, PMLSA can transform many different types of low-rank matrix estimation problems into their asymptotically equivalent least-squares forms, which can be efficiently solved by a popular matrix fast iterative shrinkage-thresholding algorithm. Furthermore, we derive analytic degrees of freedom for PMLSA, with which a Bayesian information criterion (BIC)-type criterion is developed to select the tuning parameters. The estimated rank based on the BIC-type criterion is verified to be asymptotically consistent with the true rank under mild conditions. Extensive experimental studies are performed to confirm our assertion.

Boosting is a learning scheme that combines weak learners to produce a strong composite learner, with the underlying intuition that one can obtain accurate learner by combining "rough" ones. This paper aims at developing a new boosting strategy, called rescaled boosting (RBoosting), to accelerate the numerical convergence rate and, consequently, improve learning performances of the original boosting. Our studies show that RBoosting possesses the almost optimal numerical convergence rate in the sense that, up to a logarithmic factor, it can reach the minimax nonlinear approximation rate. We then use RBoosting to tackle classification problems and deduce corresponding statistical consistency and tight generalization error estimates. A series of theoretical and experimental results shows that RBoosting outperforms boosting in terms of generalization.

The low-rank tensor factorization (LRTF) technique has received increasing attention in many computer vision applications. Compared with the traditional matrix factorization technique, it can better preserve the intrinsic structure information and thus has a better low-dimensional subspace recovery performance. Basically, the desired low-rank tensor is recovered by minimizing the least square loss between the input data and its factorized representation. Since the least square loss is most optimal when the noise follows a Gaussian distribution, -norm-based methods are designed to deal with outliers. Unfortunately, they may lose their effectiveness when dealing with real data, which are often contaminated by complex noise. In this paper, we consider integrating the noise modeling technique into a generalized weighted LRTF (GWLRTF) procedure. This procedure treats the original issue as an LRTF problem and models the noise using a mixture of Gaussians (MoG), a procedure called MoG GWLRTF. To extend the applicability of the model, two typical tensor factorization operations, i.e., CANDECOMP/PARAFAC factorization and Tucker factorization, are incorporated into the LRTF procedure. Its parameters are updated under the expectation-maximization framework. Extensive experiments indicate the respective advantages of these two versions of MoG GWLRTF in various applications and also demonstrate their effectiveness compared with other competing methods.

Rain streaks removal is an important issue of the outdoor vision system and has been recently investigated extensively. In this paper, we propose a novel tensor based video rain streaks removal approach by fully considering the discriminatively intrinsic characteristics of rain streaks and clean videos, which needs neither rain detection nor time-consuming dictionary learning stage. In specific, on the one hand, rain streaks are sparse and smooth along the raindrops direction, and on the other hand, the clean videos possess smoothness along the rain-perpendicular direction and global and local correlation along time direction. We use the l1 norm to enhance the sparsity of the underlying rain, two unidirectional Total Variation (TV) regularizers to guarantee the different discriminative smoothness, and a tensor nuclear norm and a time directional difference operator to characterize the exclusive correlation of the clean video along time. Alternation direction method of multipliers (ADMM) is employed to solve the proposed concise tensor based convex model. Experiments implemented on synthetic and real data substantiate the effectiveness and efficiency of the proposed method. Under comprehensive quantitative performance measures, our approach outperforms other state-of-the-art methods.

The RPCA model has achieved good performances in various applications. However, two defects limit its effectiveness. Firstly, it is designed for dealing with data in matrix form, which fails to exploit the structure information of higher order tensor data in some pratical situations. Secondly, it adopts L1-norm to tackle noise part which makes it only valid for sparse noise. In this paper, we propose a tensor RPCA model based on CP decomposition and model data noise by Mixture of Gaussians (MoG). The use of tensor structure to raw data allows us to make full use of the inherent structure priors, and MoG is a general approximator to any blends of consecutive distributions, which makes our approach capable of regaining the low dimensional linear subspace from a wide range of noises or their mixture. The model is solved by a new proposed algorithm inferred under a variational Bayesian framework. The superiority of our approach over the existing state-of-the-art approaches is demonstrated by extensive experiments on both of synthetic and real data.

In this letter, we consider the problem of compressive sensing of hyperspectral images (HSIs). We propose a novel tensor-based approach by modeling the global spatial-spectral correlation and local smoothness properties hidden in HSIs. Specifically, we use the tensor Tucker decomposition to describe the global spatial-spectral correlation among all HSI bands, and a weighted 3-D total variation to characterize the local smooth structure in both spatial and spectral modes. We then design an efficient algorithm to solve the resulting optimization problem by using the alternating direction method of multipliers. Experimental results on several HSI data sets demonstrate improved reconstruction performance of the proposed approach, as compared with other competing approaches.

L2-rescale boosting (L2-RBoosting) is a variant of L2-Boosting, which can essentially improve the generalization performance of L2-Boosting. The key feature of L2-RBoosting lies in introducing a shrinkage degree to rescale the ensemble estimate in each iteration. Thus, the shrinkage degree determines the performance of L2-RBoosting. The aim of this paper is to develop a concrete analysis concerning how to determine the shrinkage degree in L2-RBoosting. We propose two feasible ways to select the shrinkage degree. The first one is to parameterize the shrinkage degree and the other one is to develop a data-driven approach. After rigorously analyzing the importance of the shrinkage degree in L2-RBoosting, we compare the pros and cons of the proposed methods. We find that although these approaches can reach the same learning rates, the structure of the final estimator of the parameterized approach is better, which sometimes yields a better generalization capability when the number of sample is finite. With this, we recommend to parameterize the shrinkage degree of L2-RBoosting. We also present an adaptive parameter-selection strategy for shrinkage degree and verify its feasibility through both theoretical analysis and numerical verification. The obtained results enhance the understanding of L2-RBoosting and give guidance on how to use it for regression tasks.

Because of the limitations of matrix factorization, such as losing spatial structure information, the concept of tensor factorization has been applied for the recovery of a low dimensional subspace from high dimensional visual data. Generally, the recovery is achieved by minimizing the loss function between the observed data and the factorization representation. Under different assumptions of the noise distribution, the loss functions are in various forms, like L1 and L2 norms. However, real data are often corrupted by noise with an unknown distribution. Then any specific form of loss function for one specific kind of noise often fails to tackle such real data with unknown noise. In this paper, we propose a tensor factorization algorithm to model the noise as a Mixture of Gaussians (MoG). As MoG has the ability of universally approximating any hybrids of continuous distributions, our algorithm can effectively recover the low dimensional subspace from various forms of noisy observations. The parameters of MoG are estimated under the EM framework and through a new developed algorithm of weighted low-rank tensor factorization (WLRTF). The effectiveness of our algorithm are substantiated by extensive experiments on both of synthetic data and real image data.

In this paper, we propose a novel approach to hyperspectral image super-resolution by modeling the global spatial-and-spectral correlation and local smoothness properties over hyperspectral images. Specifically, we utilize the tensor nuclear norm and tensor folded-concave penalty functions to describe the global spatial-and-spectral correlation hidden in hyperspectral images, and 3D total variation (TV) to characterize the local spatial-and-spectral smoothness across all hyperspectral bands. Then, we develop an efficient algorithm for solving the resulting optimization problem by combing the local linear approximation (LLA) strategy and alternative direction method of multipliers (ADMM). Experimental results on one hyperspectral image dataset illustrate the merits of the proposed approach.

The mixed l 2 / l p (0 < p ≤ 1) norm minimisation method with partially known support for recovering block‐sparse signals is studied. The authors mainly extend this work on block‐sparse compressed sensing by incorporating some known part of the block support information as a priori and establish sufficient restricted p ‐isometry property ( p ‐RIP) conditions for exact and robust recovery. The authors’ theoretical results show it is possible to recover the block‐sparse signals via l 2 / l p minimisation from reduced number of measurements by applying the partially known support. The authors also derive a lower bound on necessary random Gaussian measurements for the p ‐RIP conditions to hold with high possibility. Finally, a series of numerical experiments are carried out to illustrate that fewer measurements with smaller p are needed to reconstruct the signal.

Background subtraction has been a fundamental and widely studied task in video analysis, with a wide range of applications in video surveillance, teleconferencing, and 3D modeling. Recently, motivated by compressive imaging, background subtraction from compressive measurements (BSCM) is becoming an active research task in video surveillance. In this paper, we propose a novel tensor-based robust principal component analysis (TenRPCA) approach for BSCM by decomposing video frames into backgrounds with spatial-temporal correlations and foregrounds with spatio-temporal continuity in a tensor framework. In this approach, we use 3D total variation to enhance the spatio-temporal continuity of foregrounds, and Tucker decomposition to model the spatio-temporal correlations of video background. Based on this idea, we design a basic tensor RPCA model over the video frames, dubbed as the holistic TenRPCA model. To characterize the correlations among the groups of similar 3D patches of video background, we further design a patch-group-based tensor RPCA model by joint tensor Tucker decompositions of 3D patch groups for modeling the video background. Efficient algorithms using the alternating direction method of multipliers are developed to solve the proposed models. Extensive experiments on simulated and real-world videos demonstrate the superiority of the proposed approaches over the existing state-of-the-art approaches.

Many computer vision problems can be posed as learning a low-dimensional subspace from high dimensional data. The low rank matrix factorization (LRMF) represents a commonly utilized subspace learning strategy. Most of the current LRMF techniques are constructed on the optimization problem using L_1 norm and L_2 norm, which mainly deal with Laplacian and Gaussian noise, respectively. To make LRMF capable of adapting more complex noise, this paper proposes a new LRMF model by assuming noise as Mixture of Exponential Power (MoEP) distributions and proposes a penalized MoEP model by combining the penalized likelihood method with MoEP distributions. Such setting facilitates the learned LRMF model capable of automatically fitting the real noise through MoEP distributions. Each component in this mixture is adapted from a series of preliminary super-or sub-Gaussian candidates. An Expectation Maximization (EM) algorithm is also designed to infer the parameters involved in the proposed PMoEP model. The advantage of our method is demonstrated by extensive experiments on synthetic data, face modeling and hyperspectral image restoration.

In this paper, we propose a new sparsity regularizer for measuring the low-rank structure underneath a tensor. The proposed sparsity measure has a natural physical meaning which is intrinsically the size of the fundamental Kronecker basis to express the tensor. By embedding the sparsity measure into the tensor completion and tensor robust PCA frameworks, we formulate new models to enhance their capability in tensor recovery. Through introducing relaxation forms of the proposed sparsity measure, we also adopt the alternating direction method of multipliers (ADMM) for solving the proposed models. Experiments implemented on synthetic and multispectral image data sets substantiate the effectiveness of the proposed methods.

In this paper, we develop a novel coordinate descent fuzzy twin SVM (CDFTSVM) for classification. The proposed CDFTSVM not only inherits the advantages of twin SVM but also leads to a rapid and robust classification results. Specifically, our CDFTSVM has two distinguished advantages: (1) An effective fuzzy membership function is produced for removing the noise incurred by the contaminant inputs. (2) A coordinate descent strategy with shrinking by active set is used to deal with the computational complexity brought by the high dimensional input. In addition, a series of simulation experiments are conducted to verify the performance of the CDFTSVM, which further supports our previous claims.