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Bi-Level Optimization (BLO) is originated from the area of economic game theory and then introduced into the optimization community. BLO is able to handle problems with a hierarchical structure, involving two levels of optimization tasks, where one task is nested inside the other. In machine learning and computer vision fields, despite the different motivations and mechanisms, a lot of complex problems, such as hyper-parameter optimization, multi-task and meta learning, neural architecture search, adversarial learning and deep reinforcement learning, actually all contain a series of closely related subproblms. In this paper, we first uniformly express these complex learning and vision problems from the perspective of BLO. Then we construct a best-response-based single-level reformulation and establish a unified algorithmic framework to understand and formulate mainstream gradient-based BLO methodologies, covering aspects ranging from fundamental automatic differentiation schemes to various accelerations, simplifications, extensions and their convergence and complexity properties. Last but not least, we discuss the potentials of our unified BLO framework for designing new algorithms and point out some promising directions for future research. A list of important papers discussed in this survey, corresponding codes, and additional resources on BLOs are publicly available at: https://github.com/vis-opt-group/BLO. [ABSTRACT FROM AUTHOR]

Principal component analysis (PCA) and Kernel principal component analysis (KPCA) are fundamental methods in machine learning for dimensionality reduction. The former is a technique for finding this approximation in finite dimensions and the latter is often in an infinite dimensional reproducing Kernel Hilbert-space (RKHS). In this paper, we present a geometric framework for computing the principal linear subspaces in both (finite and infinite) situations as well as for the robust PCA case, that amounts to computing the intrinsic average on the space of all subspaces: the Grassmann manifold. Points on this manifold are defined as the subspaces spanned by $K$ K -tuples of observations. The intrinsic Grassmann average of these subspaces are shown to coincide with the principal components of the observations when they are drawn from a Gaussian distribution. We show similar results in the RKHS case and provide an efficient algorithm for computing the projection onto the this average subspace. The result is a method akin to KPCA which is substantially faster. Further, we present a novel online version of the KPCA using our geometric framework. Competitive performance of all our algorithms are demonstrated on a variety of real and synthetic data sets. [ABSTRACT FROM AUTHOR]

This paper revisits the temporal difference (TD) learning algorithm for the policy evaluation tasks in reinforcement learning. Typically, the performance of TD(0) and TD($\lambda$ λ ) is very sensitive to the choice of stepsizes. Oftentimes, TD(0) suffers from slow convergence. Motivated by the tight link between the TD(0) learning algorithm and the stochastic gradient methods, we develop a provably convergent adaptive projected variant of the TD(0) learning algorithm with linear function approximation that we term AdaTD(0). In contrast to the TD(0), AdaTD(0) is robust or less sensitive to the choice of stepsizes. Analytically, we establish that to reach an $\epsilon$ ε accuracy, the number of iterations needed is $\tilde{O}(\epsilon ^{-2}\ln ^4\frac{1}{\epsilon }/\ln ^4\frac{1}{\rho })$ O ˜ (ε - 2 ln 4 1 ε / ln 4 1 ρ) in the general case, where $\rho$ ρ represents the speed of the underlying Markov chain converges to the stationary distribution. This implies that the iteration complexity of AdaTD(0) is no worse than that of TD(0) in the worst case. When the stochastic semi-gradients are sparse, we provide theoretical acceleration of AdaTD(0). Going beyond TD(0), we develop an adaptive variant of TD($\lambda$ λ ), which is referred to as AdaTD($\lambda$ λ ). Empirically, we evaluate the performance of AdaTD(0) and AdaTD($\lambda$ λ ) on several standard reinforcement learning tasks, which demonstrate the effectiveness of our new approaches. [ABSTRACT FROM AUTHOR]

In this paper, we develop a quadrature framework for large-scale kernel machines via a numerical integration representation. Considering that the integration domain and measure of typical kernels, e.g., Gaussian kernels, arc-cosine kernels, are fully symmetric, we leverage a numerical integration technique, deterministic fully symmetric interpolatory rules, to efficiently compute quadrature nodes and associated weights for kernel approximation. Thanks to the full symmetric property, the applied interpolatory rules are able to reduce the number of needed nodes while retaining a high approximation accuracy. Further, we randomize the above deterministic rules by the classical Monte-Carlo sampling and control variates techniques with two merits: 1) The proposed stochastic rules make the dimension of the feature mapping flexibly varying, such that we can control the discrepancy between the original and approximate kernels by tuning the dimnension. 2) Our stochastic rules have nice statistical properties of unbiasedness and variance reduction. In addition, we elucidate the relationship between our deterministic/stochastic interpolatory rules and current typical quadrature based rules for kernel approximation, thereby unifying these methods under our framework. Experimental results on several benchmark datasets show that our methods compare favorably with other representative kernel approximation based methods. [ABSTRACT FROM AUTHOR]

Variance reduction is popular in accelerating gradient descent and stochastic gradient descent for optimization problems defined on both euclidean space and Riemannian manifold. This paper further improves on existing variance reduction methods for non-convex Riemannian optimization, including R-SVRG and R-SRG/R-SPIDER by providing a unified framework for batch size adaptation. Such framework is more general than the existing works by considering retraction and vector transport and mini-batch stochastic gradients. We show that the adaptive-batch variance reduction methods require lower gradient complexities for both general non-convex and gradient dominated functions, under both finite-sum and online optimization settings. Moreover, under the new framework, we complete the analysis of R-SVRG and R-SRG, which is currently missing in the literature. We prove convergence of R-SVRG with much simpler analysis, which leads to curvature-free complexity bounds. We also show improved results for R-SRG under double-loop convergence, which match the optimal complexities as the R-SPIDER. In addition, we prove the first online complexity results for R-SVRG and R-SRG. Lastly, we discuss the potential of adapting batch size for non-smooth, constrained and second-order Riemannian optimizers. Extensive experiments on a variety of applications support the analysis and claims in the paper. [ABSTRACT FROM AUTHOR]

The non-negative matrix factorization (NMF) algorithm represents the original image as a linear combination of a set of basis images. This image representation method is in line with the idea of “parts constitute a whole” in human thinking. The existing deep NMF performs deep factorization on the coefficient matrix. In these methods, the basis images used to represent the original image is essentially obtained by factorizing the original images once. To extract features reflecting the deep localization characteristics of images, a novel deep NMF architecture based on underlying basis images learning is proposed for the first time. The architecture learns the underlying basis images by deep factorization on the basis images matrix. The deep factorization architecture proposed in this paper has strong interpretability. To implement this architecture, this paper proposes a deep non-negative basis matrix factorization algorithm to obtain the underlying basis images. Then, the objective function is established with an added regularization term, which directly constrains the basis images matrix to obtain the basis images with good local characteristics, and a regularized deep non-negative basis matrix factorization algorithm is proposed. The regularized deep nonlinear non-negative basis matrix factorization algorithm is also proposed to handle pattern recognition tasks with complex data. This paper also theoretically proves the convergence of the algorithm. Finally, the experimental results show that the deep NMF architecture based on the underlying basis images learning proposed in this paper can obtain better recognition performance than the other state-of-the-art methods. [ABSTRACT FROM AUTHOR]

This paper explores the non-convex composition optimization consisting of inner and outer finite-sum functions with a large number of component functions. This problem arises in important applications such as nonlinear embedding and reinforcement learning. Although existing approaches such as stochastic gradient descent (SGD) and stochastic variance reduced gradient (SVRG) descent can be applied to solve this problem, their query complexities tend to be high, especially when the number of inner component functions is large. Therefore, to significantly improve the query complexity of current approaches, we have devised the stochastic composition via variance reduction (SCVR). What's more, we analyze the query complexity under different numbers of inner function and outer function. Based on different kinds of estimation of inner component function, we also present the SCVRII algorithm, though the order of query complexities are the same with SCVR. Additionally, we propose an extension to handle the mini-batch cases, which improve the query complexity under the optimal mini-batch size. The experimental results validate our proposed algorithms and theoretical analyses. [ABSTRACT FROM AUTHOR]

This paper focuses on communication-efficient federated learning problem, and develops a novel distributed quantized gradient approach, which is characterized by adaptive communications of the quantized gradients. Specifically, the federated learning builds upon the server-worker infrastructure, where the workers calculate local gradients and upload them to the server; then the server obtain the global gradient by aggregating all the local gradients and utilizes it to update the model parameter. The key idea to save communications from the worker to the server is to quantize gradients as well as skip less informative quantized gradient communications by reusing previous gradients. Quantizing and skipping result in ‘lazy’ worker-server communications, which justifies the term Lazily Aggregated Quantized (LAQ) gradient. Theoretically, the LAQ algorithm achieves the same linear convergence as the gradient descent in the strongly convex case, while effecting major savings in the communication in terms of transmitted bits and communication rounds. Empirically, extensive experiments using realistic data corroborate a significant communication reduction compared with state-of-the-art gradient- and stochastic gradient-based algorithms. [ABSTRACT FROM AUTHOR]

Stochastic gradient descent (SGD) has become the method of choice for training highly complex and nonconvex models since it can not only recover good solutions to minimize training errors but also generalize well. Computational and statistical properties are separately studied to understand the behavior of SGD in the literature. However, there is a lacking study to jointly consider the computational and statistical properties in a nonconvex learning setting. In this paper, we develop novel learning rates of SGD for nonconvex learning by presenting high-probability bounds for both computational and statistical errors. We show that the complexity of SGD iterates grows in a controllable manner with respect to the iteration number, which sheds insights on how an implicit regularization can be achieved by tuning the number of passes to balance the computational and statistical errors. As a byproduct, we also slightly refine the existing studies on the uniform convergence of gradients by showing its connection to Rademacher chaos complexities. [ABSTRACT FROM AUTHOR]

Identifying statistical dependence between the features and the label is a fundamental problem in supervised learning. This paper presents a framework for estimating dependence between numerical features and a categorical label using generalized Gini distance, an energy distance in reproducing kernel Hilbert spaces (RKHS). Two Gini distance based dependence measures are explored: Gini distance covariance and Gini distance correlation. Unlike Pearson covariance and correlation, which do not characterize independence, the above Gini distance based measures define dependence as well as independence of random variables. The test statistics are simple to calculate and do not require probability density estimation. Uniform convergence bounds and asymptotic bounds are derived for the test statistics. Comparisons with distance covariance statistics are provided. It is shown that Gini distance statistics converge faster than distance covariance statistics in the uniform convergence bounds, hence tighter upper bounds on both Type I and Type II errors. Moreover, the probability of Gini distance covariance statistic under-performing the distance covariance statistic in Type II error decreases to 0 exponentially with the increase of the sample size. Extensive experimental results are presented to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

A dynamically weighted directed network (DWDN) is frequently encountered in various big data-related applications like a terminal interaction pattern analysis system (TIPAS) concerned in this study. It consists of large-scale dynamic interactions among numerous nodes. As the involved nodes increase drastically, it becomes impossible to observe their full interactions at each time slot, making a resultant DWDN High Dimensional and Incomplete (HDI). An HDI DWDN, in spite of its incompleteness, contains rich knowledge regarding involved nodes’ various behavior patterns. To extract such knowledge from an HDI DWDN, this paper proposes a novel Alternating direction method of multipliers (ADMM)-based Nonnegative Latent-factorization of Tensors (ANLT) model. It adopts three-fold ideas: a) building a data density-oriented augmented Lagrangian function for efficiently handling an HDI tensor's incompleteness and nonnegativity; b) splitting the optimization task in each iteration into an elaborately designed subtask series where each one is solved based on the previously solved ones following the ADMM principle to achieve fast convergence; and c) theoretically proving that its convergence is guaranteed with its efficient learning scheme. Experimental results on six DWDNs from real applications demonstrate that the proposed ANLT outperforms state-of-the-art models significantly in both computational efficiency and prediction accuracy for missing links of an HDI DWDN. Hence, this study proposes a novel and efficient approach to large-scale DWDN representation. [ABSTRACT FROM AUTHOR]

Multiple-view triangulation by $\ell _\infty$ minimisation has become established in computer vision. State-of-the-art $\ell _\infty$ triangulation algorithms exploit the quasiconvexity of the cost function to derive iterative update rules that deliver the global minimum. Such algorithms, however, can be computationally costly for large problem instances that contain many image measurements, e.g., from web-based photo sharing sites or long-term video recordings. In this paper, we prove that $\ell _\infty$ triangulation admits a coreset approximation scheme, which seeks small representative subsets of the input data called coresets. A coreset possesses the special property that the error of the $\ell _\infty$ solution on the coreset is within known bounds from the global minimum. We establish the necessary mathematical underpinnings of the coreset algorithm, specifically, by enacting the stopping criterion of the algorithm and proving that the resulting coreset gives the desired approximation accuracy. On large-scale triangulation problems, our method provides theoretically sound approximate solutions. Iterated until convergence, our coreset algorithm is also guaranteed to reach the true optimum. On practical datasets, we show that our technique can in fact attain the global minimiser much faster than current methods. [ABSTRACT FROM AUTHOR]

In real-world applications, it is important for machine learning algorithms to be robust against data outliers or corruptions. In this paper, we focus on improving the robustness of a large class of learning algorithms that are formulated as low-rank semi-definite programming (SDP) problems. Traditional formulations use the square loss, which is notorious for being sensitive to outliers. We propose to replace this with more robust noise models, including the $\ell _1$ ℓ 1 -loss and other nonconvex losses. However, the resultant optimization problem becomes difficult as the objective is no longer convex or smooth. To alleviate this problem, we design an efficient algorithm based on majorization-minimization. The crux is on constructing a good optimization surrogate, and we show that this surrogate can be efficiently obtained by the alternating direction method of multipliers (ADMM). By properly monitoring ADMM's convergence, the proposed algorithm is empirically efficient and also theoretically guaranteed to converge to a critical point. Extensive experiments are performed on four machine learning applications using both synthetic and real-world data sets. Results show that the proposed algorithm is not only fast but also has better performance than the state-of-the-arts. [ABSTRACT FROM AUTHOR]

Kernel-based methods for support vector machines (SVM) have shown highly advantageous performance in various applications. However, they may incur prohibitive computational costs for large-scale sample datasets. Therefore, data reduction (reducing the number of support vectors) appears to be necessary, which gives rise to the topic of the sparse SVM. Motivated by this problem, the sparsity constrained kernel SVM optimization has been considered in this paper in order to control the number of support vectors. Based on the established optimality conditions associated with the stationary equations, a Newton-type method is developed to handle the sparsity constrained optimization. This method is found to enjoy the one-step convergence property if the starting point is chosen to be close to a local region of a stationary point, thereby leading to a super-high computational speed. Numerical comparisons with several powerful solvers demonstrate that the proposed method performs exceptionally well, particularly for large-scale datasets in terms of a much lower number of support vectors and shorter computational time. [ABSTRACT FROM AUTHOR]

Kernel methods have achieved tremendous success in the past two decades. In the current big data era, data collection has grown tremendously. However, existing kernel methods are not scalable enough both at the training and predicting steps. To address this challenge, in this paper, we first introduce a general sparse kernel learning formulation based on the random feature approximation, where the loss functions are possibly non-convex. In order to reduce the scale of random features required in experiment, we also use that formulation based on the orthogonal random feature approximation. Then we propose a new asynchronous parallel doubly stochastic algorithm for large scale sparse kernel learning (AsyDSSKL). To the best our knowledge, AsyDSSKL is the first algorithm with the techniques of asynchronous parallel computation and doubly stochastic optimization. We also provide a comprehensive convergence guarantee to AsyDSSKL. Importantly, the experimental results on various large-scale real-world datasets show that, our AsyDSSKL method has the significant superiority on the computational efficiency at the training and predicting steps over the existing kernel methods. [ABSTRACT FROM AUTHOR]

The iterative closest point (ICP) algorithm and its variants are a fundamental technique for rigid registration between two point sets, with wide applications in different areas from robotics to 3D reconstruction. The main drawbacks for ICP are its slow convergence, as well as its sensitivity to outliers, missing data, and partial overlaps. Recent work such as Sparse ICP achieves robustness via sparsity optimization at the cost of computational speed. In this paper, we propose a new method for robust registration with fast convergence. First, we show that the classical point-to-point ICP can be treated as a majorization-minimization (MM) algorithm, and propose an Anderson acceleration approach to speed up its convergence. In addition, we introduce a robust error metric based on the Welsch’s function, which is minimized efficiently using the MM algorithm with Anderson acceleration. On challenging datasets with noises and partial overlaps, we achieve similar or better accuracy than Sparse ICP while being at least an order of magnitude faster. Finally, we extend the robust formulation to point-to-plane ICP, and solve the resulting problem using a similar Anderson-accelerated MM strategy. Our robust ICP methods improve the registration accuracy on benchmark datasets while being competitive in computational time. [ABSTRACT FROM AUTHOR]

Clustering is a long-standing important research problem, however, remains challenging when handling large-scale image data from diverse sources. In this paper, we present a novel Binary Multi-View Clustering (BMVC) framework, which can dexterously manipulate multi-view image data and easily scale to large data. To achieve this goal, we formulate BMVC by two key components: compact collaborative discrete representation learning and binary clustering structure learning, in a joint learning framework. Specifically, BMVC collaboratively encodes the multi-view image descriptors into a compact common binary code space by considering their complementary information; the collaborative binary representations are meanwhile clustered by a binary matrix factorization model, such that the cluster structures are optimized in the Hamming space by pure, extremely fast bit-operations. For efficiency, the code balance constraints are imposed on both binary data representations and cluster centroids. Finally, the resulting optimization problem is solved by an alternating optimization scheme with guaranteed fast convergence. Extensive experiments on four large-scale multi-view image datasets demonstrate that the proposed method enjoys the significant reduction in both computation and memory footprint, while observing superior (in most cases) or very competitive performance, in comparison with state-of-the-art clustering methods. [ABSTRACT FROM AUTHOR]

Incomplete multi-view clustering (IMVC) optimally combines multiple pre-specified incomplete views to improve clustering performance. Among various excellent solutions, the recently proposed multiple kernel $k$ k -means with incomplete kernels (MKKM-IK) forms a benchmark, which redefines IMVC as a joint optimization problem where the clustering and kernel matrix imputation tasks are alternately performed until convergence. Though demonstrating promising performance in various applications, we observe that the manner of kernel matrix imputation in MKKM-IK would incur intensive computational and storage complexities, over-complicated optimization and limitedly improved clustering performance. In this paper, we first propose an Efficient and Effective Incomplete Multi-view Clustering (EE-IMVC) algorithm to address these issues. Instead of completing the incomplete kernel matrices, EE-IMVC proposes to impute each incomplete base matrix generated by incomplete views with a learned consensus clustering matrix. Moreover, we further improve this algorithm by incorporating prior knowledge to regularize the learned consensus clustering matrix. Two three-step iterative algorithms are carefully developed to solve the resultant optimization problems with linear computational complexity, and their convergence is theoretically proven. After that, we theoretically study the generalization bound of the proposed algorithms. Furthermore, we conduct comprehensive experiments to study the proposed algorithms in terms of clustering accuracy, evolution of the learned consensus clustering matrix and the convergence. As indicated, our algorithms deliver their effectiveness by significantly and consistently outperforming some state-of-the-art ones. [ABSTRACT FROM AUTHOR]

Tailoring nearest neighbors algorithms to boosting is an important problem. Recent papers study an approach, unn, which provably minimizes particular convex surrogates under weak assumptions. However, numerical issues make it necessary to experimentally tweak parts of the unn algorithm, at the possible expense of the algorithm’s convergence and performance. In this paper, we propose a lightweight Newton-Raphson alternative optimizing proper scoring rules from a very broad set, and establish formal convergence rates under the boosting framework that compete with those known for unn. To the best of our knowledge, no such boosting-compliant convergence rates were previously known in the popular Gentle Adaboost’s lineage. We provide experiments on a dozen domains, including Caltech and SUN computer vision databases, comparing our approach to major families including support vector machines, (Ada)boosting and stochastic gradient descent. They support three major conclusions: (i) gnnb significantly outperforms unn , in terms of convergence rate and quality of the outputs, (ii) gnnb performs on par with or better than computationally intensive large margin approaches, (iii) on large domains that rule out those latter approaches for computational reasons, gnnb provides a simple and competitive contender to stochastic gradient descent. Experiments include a divide-and-conquer improvement of gnnb exploiting the link with proper scoring rules optimization. [ABSTRACT FROM AUTHOR]

First-order non-convex Riemannian optimization algorithms have gained recent popularity in structured machine learning problems including principal component analysis and low-rank matrix completion. The current paper presents an efficient Riemannian Stochastic Path Integrated Differential EstimatoR (R-SPIDER) algorithm to solve the finite-sum and online Riemannian non-convex minimization problems. At the core of R-SPIDER is a recursive semi-stochastic gradient estimator that can accurately estimate Riemannian gradient under not only exponential mapping and parallel transport, but also general retraction and vector transport operations. Compared with prior Riemannian algorithms, such a recursive gradient estimation mechanism endows R-SPIDER with lower computational cost in first-order oracle complexity. Specifically, for finite-sum problems with $n$ n components, R-SPIDER is proved to converge to an $\epsilon$ ε -approximate stationary point within $\mathcal {O}\big (\min \big (n+\frac{\sqrt{n}}{\epsilon ^2},\frac{1}{\epsilon ^3}\big)\big)$ O min n + n ε 2 , 1 ε 3 stochastic gradient evaluations, beating the best-known complexity $\mathcal {O}\big (n+\frac{1}{\epsilon ^4}\big)$ O n + 1 ε 4 ; for online optimization, R-SPIDER is shown to converge with $\mathcal {O}\big (\frac{1}{\epsilon ^3}\big)$ O 1 ε 3 complexity which is, to the best of our knowledge, the first non-asymptotic result for online Riemannian optimization. For the special case of gradient dominated functions, we further develop a variant of R-SPIDER with improved linear rate of convergence. Extensive experimental results demonstrate the advantage of the proposed algorithms over the state-of-the-art Riemannian non-convex optimization methods. [ABSTRACT FROM AUTHOR]

Numerous tasks at the core of statistics, learning and vision areas are specific cases of ill-posed inverse problems. Recently, learning-based (e.g., deep) iterative methods have been empirically shown to be useful for these problems. Nevertheless, integrating learnable structures into iterations is still a laborious process, which can only be guided by intuitions or empirical insights. Moreover, there is a lack of rigorous analysis about the convergence behaviors of these reimplemented iterations, and thus the significance of such methods is a little bit vague. This paper moves beyond these limits and proposes Flexible Iterative Modularization Algorithm (FIMA), a generic and provable paradigm for nonconvex inverse problems. Our theoretical analysis reveals that FIMA allows us to generate globally convergent trajectories for learning-based iterative methods. Meanwhile, the devised scheduling policies on flexible modules should also be beneficial for classical numerical methods in the nonconvex scenario. Extensive experiments on real applications verify the superiority of FIMA. [ABSTRACT FROM AUTHOR]

Often, tasks are collected for multi-task learning (MTL) because they share similar feature structures. Based on this observation, in this paper, we present novel algorithm-dependent generalization bounds for MTL by exploiting the notion of algorithmic stability. We focus on the performance of one particular task and the average performance over multiple tasks by analyzing the generalization ability of a common parameter that is shared in MTL. When focusing on one particular task, with the help of a mild assumption on the feature structures, we interpret the function of the other tasks as a regularizer that produces a specific inductive bias. The algorithm for learning the common parameter, as well as the predictor, is thereby uniformly stable with respect to the domain of the particular task and has a generalization bound with a fast convergence rate of order \mathcal O(1/n), where n is the sample size of the particular task. When focusing on the average performance over multiple tasks, we prove that a similar inductive bias exists under certain conditions on the feature structures. Thus, the corresponding algorithm for learning the common parameter is also uniformly stable with respect to the domains of the multiple tasks, and its generalization bound is of the order \mathcal {O}(1/T) , where T$ is the number of tasks. These theoretical analyses naturally show that the similarity of feature structures in MTL will lead to specific regularizations for predicting, which enables the learning algorithms to generalize fast and correctly from a few examples. [ABSTRACT FROM PUBLISHER]

In this paper, we propose an algorithm to estimate missing values in tensors of visual data. The values can be missing due to problems in the acquisition process or because the user manually identified unwanted outliers. Our algorithm works even with a small amount of samples and it can propagate structure to fill larger missing regions. Our methodology is built on recent studies about matrix completion using the matrix trace norm. The contribution of our paper is to extend the matrix case to the tensor case by proposing the first definition of the trace norm for tensors and then by building a working algorithm. First, we propose a definition for the tensor trace norm that generalizes the established definition of the matrix trace norm. Second, similarly to matrix completion, the tensor completion is formulated as a convex optimization problem. Unfortunately, the straightforward problem extension is significantly harder to solve than the matrix case because of the dependency among multiple constraints. To tackle this problem, we developed three algorithms: simple low rank tensor completion (SiLRTC), fast low rank tensor completion (FaLRTC), and high accuracy low rank tensor completion (HaLRTC). The SiLRTC algorithm is simple to implement and employs a relaxation technique to separate the dependant relationships and uses the block coordinate descent (BCD) method to achieve a globally optimal solution; the FaLRTC algorithm utilizes a smoothing scheme to transform the original nonsmooth problem into a smooth one and can be used to solve a general tensor trace norm minimization problem; the HaLRTC algorithm applies the alternating direction method of multipliers (ADMMs) to our problem. Our experiments show potential applications of our algorithms and the quantitative evaluation indicates that our methods are more accurate and robust than heuristic approaches. The efficiency comparison indicates that FaLTRC and HaLRTC are more efficient than SiLRTC and between FaLRTC and HaLRTC the former is more efficient to obtain a low accuracy solution and the latter is preferred if a high-accuracy solution is desired. [ABSTRACT FROM AUTHOR]

We consider the estimation and inference of graphical models that characterize the dependency structure of high-dimensional tensor-valued data. To facilitate the estimation of the precision matrix corresponding to each way of the tensor, we assume the data follow a tensor normal distribution whose covariance has a Kronecker product structure. A critical challenge in the estimation and inference of this model is the fact that its penalized maximum likelihood estimation involves minimizing a non-convex objective function. To address it, this paper makes two contributions: (i) In spite of the non-convexity of this estimation problem, we prove that an alternating minimization algorithm, which iteratively estimates each sparse precision matrix while fixing the others, attains an estimator with an optimal statistical rate of convergence. (ii) We propose a de-biased statistical inference procedure for testing hypotheses on the true support of the sparse precision matrices, and employ it for testing a growing number of hypothesis with false discovery rate (FDR) control. The asymptotic normality of our test statistic and the consistency of FDR control procedure are established. Our theoretical results are backed up by thorough numerical studies and our real applications on neuroimaging studies of Autism spectrum disorder and users’ advertising click analysis bring new scientific findings and business insights. The proposed methods are encoded into a publicly available R package Tlasso. [ABSTRACT FROM AUTHOR]

The increasing scale of Structure-from-Motion is fundamentally limited by the conventional optimization framework for the all-in-one global bundle adjustment. In this paper, we propose a distributed approach to coping with this global bundle adjustment for very large scale Structure-from-Motion computation. First, we derive the distributed formulation from the classical optimization algorithm ADMM, Alternating Direction Method of Multipliers, based on the global camera consensus. Then, we analyze the conditions under which the convergence of this distributed optimization would be guaranteed. In particular, we adopt over-relaxation and self-adaption schemes to improve the convergence rate. After that, we propose to split the large scale camera-point visibility graph in order to reduce the communication overheads of the distributed computing. The experiments on both public large scale SfM data-sets and our very large scale aerial photo sets demonstrate that the proposed distributed method clearly outperforms the state-of-the-art method in efficiency and accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel methodology for characterizing the performance of deep learning networks (ResNets and DenseNet) with respect to training convergence and generalization as a function of mini-batch size and learning rate for image classification. This methodology is based on novel measurements derived from the eigenvalues of the approximate Fisher information matrix, which can be efficiently computed even for high capacity deep models. Our proposed measurements can help practitioners to monitor and control the training process (by actively tuning the mini-batch size and learning rate) to allow for good training convergence and generalization. Furthermore, the proposed measurements also allow us to show that it is possible to optimize the training process with a new dynamic sampling training approach that continuously and automatically change the mini-batch size and learning rate during the training process. Finally, we show that the proposed dynamic sampling training approach has a faster training time and a competitive classification accuracy compared to the current state of the art. [ABSTRACT FROM AUTHOR]

Low-rank modeling has many important applications in computer vision and machine learning. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better empirical performance. However, the resulting optimization problem is much more challenging. Recent state-of-the-art requires an expensive full SVD in each iteration. In this paper, we show that for many commonly-used nonconvex low-rank regularizers, the singular values obtained from the proximal operator can be automatically threshold. This allows the proximal operator to be efficiently approximated by the power method. We then develop a fast proximal algorithm and its accelerated variant with inexact proximal step. It can be guaranteed that the squared distance between consecutive iterates converges at a rate of $O(1/T)$O(1/T), where $T$T is the number of iterations. Furthermore, we show the proposed algorithm can be parallelized, and the resultant algorithm achieves nearly linear speedup w.r.t. the number of threads. Extensive experiments are performed on matrix completion and robust principal component analysis. Significant speedup over the state-of-the-art is observed. [ABSTRACT FROM AUTHOR]

This paper revisits the integer programming (IP) problem, which plays a fundamental role in many computer vision and machine learning applications. The literature abounds with many seminal works that address this problem, some focusing on continuous approaches (e.g., linear program relaxation), while others on discrete ones (e.g., min-cut). However, since many of these methods are designed to solve specific IP forms, they cannot adequately satisfy the simultaneous requirements of accuracy, feasibility, and scalability. To this end, we propose a novel and versatile framework called $\ell _p$ℓp-box ADMM, which is based on two main ideas. (1) The discrete constraint is equivalently replaced by the intersection of a box and an $\ell _p$ℓp-norm sphere. (2) We infuse this equivalence into the Alternating Direction Method of Multipliers (ADMM) framework to handle the continuous constraints separately and to harness its attractive properties. More importantly, the ADMM update steps can lead to manageable sub-problems in the continuous domain. To demonstrate its efficacy, we apply it to an optimization form that occurs often in computer vision and machine learning, namely binary quadratic programming (BQP). In this case, the ADMM steps are simple, computationally efficient. Moreover, we present the theoretic analysis about the global convergence of the $\ell _p$ℓp-box ADMM through adding a perturbation with the sufficiently small factor $\epsilon$ε to the original IP problem. Specifically, the globally converged solution generated by $\ell _p$ℓp-box ADMM for the perturbed IP problem will be close to the stationary and feasible point of the original IP problem within $O(\epsilon)$O(ε). We demonstrate the applicability of $\ell _p$ℓp-box ADMM on three important applications: MRF energy minimization, graph matching, and clustering. Results clearly show that it significantly outperforms existing generic IP solvers both in runtime and objective. It also achieves very competitive performance to state-of-the-art methods designed specifically for these applications. [ABSTRACT FROM AUTHOR]

This paper studies the subspace clustering problem. Given some data points approximately drawn from a union of subspaces, the goal is to group these data points into their underlying subspaces. Many subspace clustering methods have been proposed and among which sparse subspace clustering and low-rank representation are two representative ones. Despite the different motivations, we observe that many existing methods own the common block diagonal property, which possibly leads to correct clustering, yet with their proofs given case by case. In this work, we consider a general formulation and provide a unified theoretical guarantee of the block diagonal property. The block diagonal property of many existing methods falls into our special case. Second, we observe that many existing methods approximate the block diagonal representation matrix by using different structure priors, e.g., sparsity and low-rankness, which are indirect. We propose the first block diagonal matrix induced regularizer for directly pursuing the block diagonal matrix. With this regularizer, we solve the subspace clustering problem by Block Diagonal Representation (BDR), which uses the block diagonal structure prior. The BDR model is nonconvex and we propose an alternating minimization solver and prove its convergence. Experiments on real datasets demonstrate the effectiveness of BDR. [ABSTRACT FROM AUTHOR]

The goal of cross-domain matching (CDM) is to find correspondences between two sets of objects in different domains in an unsupervised way. CDM has various interesting applications, including photo album summarization where photos are automatically aligned into a designed frame expressed in the Cartesian coordinate system, and temporal alignment which aligns sequences such as videos that are potentially expressed using different features. In this paper, we propose an information-theoretic CDM framework based on squared-loss mutual information (SMI). The proposed approach can directly handle non-linearly related objects/sequences with different dimensions, with the ability that hyper-parameters can be objectively optimized by cross-validation. We apply the proposed method to several real-world problems including image matching, unpaired voice conversion, photo album summarization, cross-feature video and cross-domain video-to-mocap alignment, and Kinect-based action recognition, and experimentally demonstrate that the proposed method is a promising alternative to state-of-the-art CDM methods. [ABSTRACT FROM AUTHOR]

We propose a new high dimensional semiparametric principal component analysis (PCA) method, named Copula Component Analysis (COCA). The semiparametric model assumes that, after unspecified marginally monotone transformations, the distributions are multivariate Gaussian. COCA improves upon PCA and sparse PCA in three aspects: (i) It is robust to modeling assumptions; (ii) It is robust to outliers and data contamination; (iii) It is scale-invariant and yields more interpretable results. We prove that the COCA estimators obtain fast estimation rates and are feature selection consistent when the dimension is nearly exponentially large relative to the sample size. Careful experiments confirm that COCA outperforms sparse PCA on both synthetic and real-world data sets. [ABSTRACT FROM AUTHOR]

A mixture of shifted asymmetric Laplace distributions is introduced and used for clustering and classification. A variant of the EM algorithm is developed for parameter estimation by exploiting the relationship with the generalized inverse Gaussian distribution. This approach is mathematically elegant and relatively computationally straightforward. Our novel mixture modelling approach is demonstrated on both simulated and real data to illustrate clustering and classification applications. In these analyses, our mixture of shifted asymmetric Laplace distributions performs favourably when compared to the popular Gaussian approach. This work, which marks an important step in the non-Gaussian model-based clustering and classification direction, concludes with discussion as well as suggestions for future work. [ABSTRACT FROM PUBLISHER]

$L_1$ -norm based low rank matrix factorization in the presence of missing data and outliers remains a hot topic in computer vision. Due to non-convexity and non-smoothness, all the existing methods either lack scalability or robustness, or have no theoretical guarantee on convergence. In this paper, we apply the Majorization Minimization technique to solve this problem. At each iteration, we upper bound the original function with a strongly convex surrogate. By minimizing the surrogate and updating the iterates accordingly, the objective function has sufficient decrease, which is stronger than just being non-increasing that other methods could offer. As a consequence, without extra assumptions, we prove that any limit point of the iterates is a stationary point of the objective function. In comparison, other methods either do not have such a convergence guarantee or require extra critical assumptions. Extensive experiments on both synthetic and real data sets testify to the effectiveness of our algorithm. The speed of our method is also highly competitive. [ABSTRACT FROM AUTHOR]

The group Lasso is an extension of the Lasso for feature selection on (predefined) nonoverlapping groups of features. The nonoverlapping group structure limits its applicability in practice. There have been several recent attempts to study a more general formulation where groups of features are given, potentially with overlaps between the groups. The resulting optimization is, however, much more challenging to solve due to the group overlaps. In this paper, we consider the efficient optimization of the overlapping group Lasso penalized problem. We reveal several key properties of the proximal operator associated with the overlapping group Lasso, and compute the proximal operator by solving the smooth and convex dual problem, which allows the use of the gradient descent type of algorithms for the optimization. Our methods and theoretical results are then generalized to tackle the general overlapping group Lasso formulation based on the $(\ell_q)$ norm. We further extend our algorithm to solve a nonconvex overlapping group Lasso formulation based on the capped norm regularization, which reduces the estimation bias introduced by the convex penalty. We have performed empirical evaluations using both a synthetic and the breast cancer gene expression dataset, which consists of 8,141 genes organized into (overlapping) gene sets. Experimental results show that the proposed algorithm is more efficient than existing state-of-the-art algorithms. Results also demonstrate the effectiveness of the nonconvex formulation for overlapping group Lasso. [ABSTRACT FROM AUTHOR]

This paper proposes a novel temporal knowledge representation and learning framework to perform large-scale temporal signature mining of longitudinal heterogeneous event data. The framework enables the representation, extraction, and mining of high-order latent event structure and relationships within single and multiple event sequences. The proposed knowledge representation maps the heterogeneous event sequences to a geometric image by encoding events as a structured spatial-temporal shape process. We present a doubly constrained convolutional sparse coding framework that learns interpretable and shift-invariant latent temporal event signatures. We show how to cope with the sparsity in the data as well as in the latent factor model by inducing a double sparsity constraint on the \beta-divergence to learn an overcomplete sparse latent factor model. A novel stochastic optimization scheme performs large-scale incremental learning of group-specific temporal event signatures. We validate the framework on synthetic data and on an electronic health record dataset. [ABSTRACT FROM AUTHOR]

Multi-view learning aims to integrate multiple data information from different views to improve the learning performance. The key problem is to handle the unconformities or distortions among view-specific samples or measurements of similarity or dissimilarity. This paper models the view-specific samples as a nonlinear mapping of uniform but latent intact samples for all the views, and the view-specific dissimilarity matrices or similarity matrices are estimated in terms of the uniform latent one. Two methods are then developed for multi-view clustering. One makes use of uniform multidimensional scaling (UMDS) on multi-view dissimilarities or kernels. The other one uses a uniform class assignment (UCA) procedure that optimally extracts the cluster components contained in the view-specific similarity matrices. These two methods result in the same optimization model, subjected to some slightly different constraints. A first-order condition of solutions is given as a nonlinear eigenvalue problem, and a second order condition guarantees local optimality. The nonlinear eigenvalue problem is solved by an iterative algorithm via eigen-space updating, and its convergence is proven. Furthermore, a fast implementation of the algorithm is discussed, which adopts the strategy of restarting subspace extension. Numerical experiments on some real-world data sets provide good support to the proposed methods. [ABSTRACT FROM PUBLISHER]

In this paper, we study Stochastic Composite Optimization (SCO) for sparse learning that aims to learn a sparse solution from a composite function. Most of the recent SCO algorithms have already reached the optimal expected convergence rate \mathcal O(1/\lambda T) , but they often fail to deliver sparse solutions at the end either due to the limited sparsity regularization during stochastic optimization (SO) or due to the limitation in online-to-batch conversion. Even when the objective function is strongly convex, their high probability bounds can only attain \mathcal O(\sqrt\log (1/\delta)/T) with $\delta$ is the failure probability, which is much worse than the expected convergence rate. To address these limitations, we propose a simple yet effective two-phase Stochastic Composite Optimization scheme by adding a novel powerful sparse online-to-batch conversion to the general Stochastic Optimization algorithms. We further develop three concrete algorithms, OptimalSL, LastSL and AverageSL, directly under our scheme to prove the effectiveness of the proposed scheme. Both the theoretical analysis and the experiment results show that our methods can really outperform the existing methods at the ability of sparse learning and at the meantime we can improve the high probability bound to approximately \mathcal {O}(\log (\log (T)/\delta)/\lambda T) . [ABSTRACT FROM AUTHOR]

The Iterative Closest Point (ICP) algorithm is one of the most widely used methods for point-set registration. However, being based on local iterative optimization, ICP is known to be susceptible to local minima. Its performance critically relies on the quality of the initialization and only local optimality is guaranteed. This paper presents the first globally optimal algorithm, named Go-ICP, for Euclidean (rigid) registration of two 3D point-sets under the $L_2$ error metric defined in ICP. The Go-ICP method is based on a branch-and-bound scheme that searches the entire 3D motion space $SE(3)$ . By exploiting the special structure of $SE(3)$ geometry, we derive novel upper and lower bounds for the registration error function. Local ICP is integrated into the BnB scheme, which speeds up the new method while guaranteeing global optimality. We also discuss extensions, addressing the issue of outlier robustness. The evaluation demonstrates that the proposed method is able to produce reliable registration results regardless of the initialization. Go-ICP can be applied in scenarios where an optimal solution is desirable or where a good initialization is not always available. [ABSTRACT FROM PUBLISHER]

In recent years, sparse coding has been widely used in many applications ranging from image processing to pattern recognition. Most existing sparse coding based applications require solving a class of challenging non-smooth and non-convex optimization problems. Despite the fact that many numerical methods have been developed for solving these problems, it remains an open problem to find a numerical method which is not only empirically fast, but also has mathematically guaranteed strong convergence. In this paper, we propose an alternating iteration scheme for solving such problems. A rigorous convergence analysis shows that the proposed method satisfies the global convergence property: the whole sequence of iterates is convergent and converges to a critical point. Besides the theoretical soundness, the practical benefit of the proposed method is validated in applications including image restoration and recognition. Experiments show that the proposed method achieves similar results with less computation when compared to widely used methods such as K-SVD. [ABSTRACT FROM PUBLISHER]

In this paper, we study a classification problem in which sample labels are randomly corrupted. In this scenario, there is an unobservable sample with noise-free labels. However, before being observed, the true labels are independently flipped with a probability $\rho \in [0,0.5)$ , and the random label noise can be class-conditional. Here, we address two fundamental problems raised by this scenario. The first is how to best use the abundant surrogate loss functions designed for the traditional classification problem when there is label noise. We prove that any surrogate loss function can be used for classification with noisy labels by using importance reweighting, with consistency assurance that the label noise does not ultimately hinder the search for the optimal classifier of the noise-free sample. The other is the open problem of how to obtain the noise rate \rho$ . We show that the rate is upper bounded by the conditional probability P(\hat{Y}|X) of the noisy sample. Consequently, the rate can be estimated, because the upper bound can be easily reached in classification problems. Experimental results on synthetic and real datasets confirm the efficiency of our methods. [ABSTRACT FROM PUBLISHER]

Hidden conditional random fields (HCRFs) are discriminative latent variable models which have been shown to successfully learn the hidden structure of a given classification problem. An Infinite hidden conditional random field is a hidden conditional random field with a countably infinite number of hidden states, which rids us not only of the necessity to specify a priori a fixed number of hidden states available but also of the problem of overfitting. Markov chain Monte Carlo (MCMC) sampling algorithms are often employed for inference in such models. However, convergence of such algorithms is rather difficult to verify, and as the complexity of the task at hand increases the computational cost of such algorithms often becomes prohibitive. These limitations can be overcome by variational techniques. In this paper, we present a generalized framework for infinite HCRF models, and a novel variational inference approach on a model based on coupled Dirichlet Process Mixtures, the HCRF-DPM. We show that the variational HCRF-DPM is able to converge to a correct number of represented hidden states, and performs as well as the best parametric HCRFs—chosen via cross-validation—for the difficult tasks of recognizing instances of agreement, disagreement, and pain in audiovisual sequences. [ABSTRACT FROM AUTHOR]

For most problems in science and engineering we can obtain data sets that describe the observed system from variousperspectives and record the behavior of its individual components. Heterogeneous data sets can be collectively mined by data fusion. Fusion can focus on a specific target relation and exploit directly associated data together with contextual data and data about system’s constraints. In the paper we describe a data fusion approach with penalized matrix tri-factorization (DFMF) that simultaneouslyfactorizes data matrices to reveal hidden associations. The approach can directly consider any data that can be expressed in a matrix, including those from feature-based representations, ontologies, associations and networks. We demonstrate the utility of DFMF for gene function prediction task with eleven different data sources and for prediction of pharmacologic actions by fusing six data sources. Our data fusion algorithm compares favorably to alternative data integration approaches and achieves higher accuracy than can be obtained from any single data source alone. [ABSTRACT FROM AUTHOR]

We study properties of the mean shift (MS)-type algorithms for estimating modes of probability density functions (PDFs), via regarding these algorithms as gradient ascent on estimated PDFs with adaptive step sizes. We rigorously prove convergence of mode estimate sequences generated by the MS-type algorithms, under the assumption that an analytic kernel function is used. Moreover, our analysis on the MS function finds several new properties of mode estimate sequences and corresponding density estimate sequences, including the result that in the MS-type algorithm using a Gaussian kernel the density estimate monotonically increases between two consecutive mode estimates. This implies that, in the one-dimensional case, the mode estimate sequence monotonically converges to the stationary point nearest to an initial point without jumping over any stationary point. [ABSTRACT FROM AUTHOR]

We propose a new learning rule for sparse multinomial logistic regression (SMLR). The new rule is the generalization of the one proposed in the pioneering work by Krishnapuram et al. In our proposed method, the parameters of SMLR are iteratively estimated from log-posterior by using some approximations. The proposed update rule provides a faster convergence compared to the state-of the-art methods used for SMLR parameter estimation. The estimated parameters are tested on the pixel-based classification of hyperspectral images. The experimental results on real hyperspectral images show that the classification accuracy of proposed method is also better than those of the state-of-the-art methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the heterogeneous domain adaptation (HDA) problem, in which the data from the source domain and the target domain are represented by heterogeneous features with different dimensions. By introducing two different projection matrices, we first transform the data from two domains into a common subspace such that the similarity between samples across different domains can be measured. We then propose a new feature mapping function for each domain, which augments the transformed samples with their original features and zeros. Existing supervised learning methods (e.g., SVM and SVR) can be readily employed by incorporating our newly proposed augmented feature representations for supervised HDA. As a showcase, we propose a novel method called Heterogeneous Feature Augmentation (HFA) based on SVM. We show that the proposed formulation can be equivalently derived as a standard Multiple Kernel Learning (MKL) problem, which is convex and thus the global solution can be guaranteed. To additionally utilize the unlabeled data in the target domain, we further propose the semi-supervised HFA (SHFA) which can simultaneously learn the target classifier as well as infer the labels of unlabeled target samples. Comprehensive experiments on three different applications clearly demonstrate that our SHFA and HFA outperform the existing HDA methods. [ABSTRACT FROM PUBLISHER]

Recovering a large matrix from a small subset of its entries is a challenging problem arising in many real applications, such as image inpainting and recommender systems. Many existing approaches formulate this problem as a general low-rank matrix approximation problem. Since the rank operator is nonconvex and discontinuous, most of the recent theoretical studies use the nuclear norm as a convex relaxation. One major limitation of the existing approaches based on nuclear norm minimization is that all the singular values are simultaneously minimized, and thus the rank may not be well approximated in practice. In this paper, we propose to achieve a better approximation to the rank of matrix by truncated nuclear norm, which is given by the nuclear norm subtracted by the sum of the largest few singular values. In addition, we develop a novel matrix completion algorithm by minimizing the Truncated Nuclear Norm. We further develop three efficient iterative procedures, TNNR-ADMM, TNNR-APGL, and TNNR-ADMMAP, to solve the optimization problem. TNNR-ADMM utilizes the alternating direction method of multipliers (ADMM), while TNNR-AGPL applies the accelerated proximal gradient line search method (APGL) for the final optimization. For TNNR-ADMMAP, we make use of an adaptive penalty according to a novel update rule for ADMM to achieve a faster convergence rate. Our empirical study shows encouraging results of the proposed algorithms in comparison to the state-of-the-art matrix completion algorithms on both synthetic and real visual datasets. [ABSTRACT FROM AUTHOR]

Feature selection with specific multivariate performance measures is the key to the success of many applications such as image retrieval and text classification. The existing feature selection methods are usually designed for classification error. In this paper, we propose a generalized sparse regularizer. Based on the proposed regularizer, we present a unified feature selection framework for general loss functions. In particular, we study the novel feature selection paradigm by optimizing multivariate performance measures. The resultant formulation is a challenging problem for high-dimensional data. Hence, a two-layer cutting plane algorithm is proposed to solve this problem, and the convergence is presented. In addition, we adapt the proposed method to optimize multivariate measures for multiple-instance learning problems. The analyses by comparing with the state-of-the-art feature selection methods show that the proposed method is superior to others. Extensive experiments on large-scale and high-dimensional real-world datasets show that the proposed method outperforms (l_1)-SVM and SVM-RFE when choosing a small subset of features, and achieves significantly improved performances over (SVM^perf) in terms of $(F_1)$-score. [ABSTRACT FROM AUTHOR]

As a leading partitional clustering technique, $(k)$-modes is one of the most computationally efficient clustering methods for categorical data. In the $(k)$-modes, a cluster is represented by a “mode,” which is composed of the attribute value that occurs most frequently in each attribute domain of the cluster, whereas, in real applications, using only one attribute value in each attribute to represent a cluster may not be adequate as it could in turn affect the accuracy of data analysis. To get rid of this deficiency, several modified clustering algorithms were developed by assigning appropriate weights to several attribute values in each attribute. Although these modified algorithms are quite effective, their convergence proofs are lacking. In this paper, we analyze their convergence property and prove that they cannot guarantee to converge under their optimization frameworks unless they degrade to the original $(k)$--modes type algorithms. Furthermore, we propose two different modified algorithms with weighted cluster prototypes to overcome the shortcomings of these existing algorithms. We rigorously derive updating formulas for the proposed algorithms and prove the convergence of the proposed algorithms. The experimental studies show that the proposed algorithms are effective and efficient for large categorical datasets. [ABSTRACT FROM AUTHOR]

Spectral Matching (SM) is a computationally efficient approach to approximate the solution of pairwise matching problems that are np-hard. In this paper, we present a probabilistic interpretation of spectral matching schemes and derive a novel Probabilistic Matching (PM) scheme that is shown to outperform previous approaches. We show that spectral matching can be interpreted as a Maximum Likelihood (ML) estimate of the assignment probabilities and that the Graduated Assignment (GA) algorithm can be cast as a Maximum a Posteriori (MAP) estimator. Based on this analysis, we derive a ranking scheme for spectral matchings based on their reliability, and propose a novel iterative probabilistic matching algorithm that relaxes some of the implicit assumptions used in prior works. We experimentally show our approaches to outperform previous schemes when applied to exhaustive synthetic tests as well as the analysis of real image sequences. [ABSTRACT FROM AUTHOR]

Principal component analysis (PCA) based on L1-norm maximization is an emerging technique that has drawn growing interest in the signal processing and machine learning research communities, especially due to its robustness to outliers. The present work proves that L1-norm PCA can perform independent component analysis (ICA) under the whitening assumption. However, when the source probability distributions fulfil certain conditions, the L1-norm criterion needs to be minimized rather than maximized, which can be accomplished by simple modifications on existing optimal algorithms for L1-PCA. If the sources have symmetric distributions, we show in addition that L1-PCA is linked to kurtosis optimization. A number of numerical experiments illustrate the theoretical results and analyze the comparative performance of different algorithms for ICA via L1-PCA. Although our analysis is asymptotic in the sample size, this equivalence opens interesting new perspectives for performing ICA using optimal algorithms for L1-PCA with guaranteed global convergence while inheriting the increased robustness to outliers of the L1-norm criterion. [ABSTRACT FROM AUTHOR]