The image nonlocal self-similarity (NSS) prior refers to the fact that a local patch often has many nonlocal similar patches to it across the image and has been widely applied in many recently proposed machining learning algorithms for image processing. However, there is no theoretical analysis on its working principle in the literature. In this paper, we discover a potential causality between NSS and low-rank property of color images, which is also available to grey images. A new patch group based NSS prior scheme is proposed to learn explicit NSS models of natural color images. The numerical low-rank property of patched matrices is also rigorously proved. The NSS-based QMC algorithm computes an optimal low-rank approximation to the high-rank color image, resulting in high PSNR and SSIM measures and particularly the better visual quality. A new tensor NSS-based QMC method is also presented to solve the color video inpainting problem based on quaternion tensor representation. The numerical experiments on color images and videos indicate the advantages of NSS-based QMC over the state-of-the-art methods.
Value function based reinforcement learning (RL) algorithms, for example, $Q$-learning, learn optimal policies from datasets of actions, rewards, and state transitions. However, when the underlying state transition dynamics are stochastic and evolve on a high-dimensional space, generating independent and identically distributed (IID) data samples for creating these datasets poses a significant challenge due to the intractability of the associated normalizing integral. In these scenarios, Hamiltonian Monte Carlo (HMC) sampling offers a computationally tractable way to generate data for training RL algorithms. In this paper, we introduce a framework, called \textit{Hamiltonian $Q$-Learning}, that demonstrates, both theoretically and empirically, that $Q$ values can be learned from a dataset generated by HMC samples of actions, rewards, and state transitions. Furthermore, to exploit the underlying low-rank structure of the $Q$ function, Hamiltonian $Q$-Learning uses a matrix completion algorithm for reconstructing the updated $Q$ function from $Q$ value updates over a much smaller subset of state-action pairs. Thus, by providing an efficient way to apply $Q$-learning in stochastic, high-dimensional settings, the proposed approach broadens the scope of RL algorithms for real-world applications.
Completing low-rank matrices from subsampled measurements has received much attention in the past decade. Existing works indicate that $\mathcal{O}(nr\log^2(n))$ datums are required to theoretically secure the completion of an $n \times n$ noisy matrix of rank $r$ with high probability, under some quite restrictive assumptions: (1) the underlying matrix must be incoherent; (2) observations follow the uniform distribution. The restrictiveness is partially due to ignoring the roles of the leverage score and the oracle information of each element. In this paper, we employ the leverage scores to characterize the importance of each element and significantly relax assumptions to: (1) not any other structure assumptions are imposed on the underlying low-rank matrix; (2) elements being observed are appropriately dependent on their importance via the leverage score. Under these assumptions, instead of uniform sampling, we devise an ununiform/biased sampling procedure that can reveal the ``importance'' of each observed element. Our proofs are supported by a novel approach that phrases sufficient optimality conditions based on the Golfing Scheme, which would be of independent interest to the wider areas. Theoretical findings show that we can provably recover an unknown $n\times n$ matrix of rank $r$ from just about $\mathcal{O}(nr\log^2 (n))$ entries, even when the observed entries are corrupted with a small amount of noisy information. The empirical results align precisely with our theories.
We consider robust low rank matrix estimation as a trace regression when outputs are contaminated by adversaries. The adversaries are allowed to add arbitrary values to arbitrary outputs. Such values can depend on any samples. We deal with matrix compressed sensing, including lasso as a partial problem, and matrix completion, and then we obtain sharp estimation error bounds. To obtain the error bounds for different models such as matrix compressed sensing and matrix completion, we propose a simple unified approach based on a combination of the Huber loss function and the nuclear norm penalization, which is a different approach from the conventional ones. Some error bounds obtained in the present paper are sharper than the past ones.
In this paper, we introduce TITAN, a novel inerTIal block majorizaTion minimizAtioN framework for non-smooth non-convex optimization problems. To the best of our knowledge, TITAN is the first framework of block-coordinate update method that relies on the majorization-minimization framework while embedding inertial force to each step of the block updates. The inertial force is obtained via an extrapolation operator that subsumes heavy-ball and Nesterov-type accelerations for block proximal gradient methods as special cases. By choosing various surrogate functions, such as proximal, Lipschitz gradient, Bregman, quadratic, and composite surrogate functions, and by varying the extrapolation operator, TITAN produces a rich set of inertial block-coordinate update methods. We study sub-sequential convergence as well as global convergence for the generated sequence of TITAN. We illustrate the effectiveness of TITAN on two important machine learning problems, namely sparse non-negative matrix factorization and matrix completion.
We consider the problem of reconstructing a rank-one matrix from a revealed subset of its entries when some of the revealed entries are corrupted with perturbations that are unknown and can be arbitrarily large. It is not known which revealed entries are corrupted. We propose a new algorithm combining alternating minimization with extreme-value filtering and provide sufficient and necessary conditions to recover the original rank-one matrix. In particular, we show that our proposed algorithm is optimal when the set of revealed entries is given by an Erd\H{o}s-R\'enyi random graph. These results are then applied to the problem of classification from crowdsourced data under the assumption that while the majority of the workers are governed by the standard single-coin David-Skene model (i.e., they output the correct answer with a certain probability), some of the workers can deviate arbitrarily from this model. In particular, the "adversarial" workers could even make decisions designed to make the algorithm output an incorrect answer. Extensive experimental results show our algorithm for this problem, based on rank-one matrix completion with perturbations, outperforms all other state-of-the-art methods in such an adversarial scenario.
Sampling is a fundamental and arguably very important task with numerous applications in Machine Learning. One approach to sample from a high dimensional distribution $e^{-f}$ for some function $f$ is the Langevin Algorithm (LA). Recently, there has been a lot of progress in showing fast convergence of LA even in cases where $f$ is non-convex, notably [53], [39] in which the former paper focuses on functions $f$ defined in $\mathbb{R}^n$ and the latter paper focuses on functions with symmetries (like matrix completion type objectives) with manifold structure. Our work generalizes the results of [53] where $f$ is defined on a manifold $M$ rather than $\mathbb{R}^n$. From technical point of view, we show that KL decreases in a geometric rate whenever the distribution $e^{-f}$ satisfies a log-Sobolev inequality on $M$.
We propose a totally functional view of geometric matrix completion problem. Differently from existing work, we propose a novel regularization inspired from the functional map literature that is more interpretable and theoretically sound. On synthetic tasks with strong underlying geometric structure, our framework outperforms state of the art by a huge margin (two order of magnitude) demonstrating the potential of our approach. On real datasets, we achieve state-of-the-art results at a fraction of the computational effort of previous methods. Our code is publicly available at https://github.com/Not-IITian/functional-matrix-completion
We present Wikipedia-based Polyglot Dirichlet Allocation (WikiPDA), a crosslingual topic model that learns to represent Wikipedia articles written in any language as distributions over a common set of language-independent topics. It leverages the fact that Wikipedia articles link to each other and are mapped to concepts in the Wikidata knowledge base, such that, when represented as bags of links, articles are inherently language-independent. WikiPDA works in two steps, by first densifying bags of links using matrix completion and then training a standard monolingual topic model. A human evaluation shows that WikiPDA produces more coherent topics than monolingual text-based LDA, thus offering crosslinguality at no cost. We demonstrate WikiPDA's utility in two applications: a study of topical biases in 28 Wikipedia editions, and crosslingual supervised classification. Finally, we highlight WikiPDA's capacity for zero-shot language transfer, where a model is reused for new languages without any fine-tuning. Researchers can benefit from WikiPDA as a practical tool for studying Wikipedia's content across its 299 language editions in interpretable ways, via an easy-to-use library publicly available at https://github.com/epfl-dlab/WikiPDA.
This work formulates antiviral repositioning as a matrix completion problem where the antiviral drugs are along the rows and the viruses along the columns. The input matrix is partially filled, with ones in positions where the antiviral has been known to be effective against a virus. The curated metadata for antivirals (chemical structure and pathways) and viruses (genomic structure and symptoms) is encoded into our matrix completion framework as graph Laplacian regularization. We then frame the resulting multiple graph regularized matrix completion problem as deep matrix factorization. This is solved by using a novel optimization method called HyPALM (Hybrid Proximal Alternating Linearized Minimization). Results on our curated RNA drug virus association (DVA) dataset shows that the proposed approach excels over state-of-the-art graph regularized matrix completion techniques. When applied to "in silico" prediction of antivirals for COVID-19, our approach returns antivirals that are either used for treating patients or are under for trials for the same.
This paper makes a selective survey on the recent development of the factor model and its application on statistical learnings. We focus on the perspective of the low-rank structure of factor models, and particularly draws attentions to estimating the model from the low-rank recovery point of view. The survey mainly consists of three parts: the first part is a review on new factor estimations based on modern techniques on recovering low-rank structures of high-dimensional models. The second part discusses statistical inferences of several factor-augmented models and applications in econometric learning models. The final part summarizes new developments dealing with unbalanced panels from the matrix completion perspective.
In this paper, we propose a novel solution for non-convex problems of multiple variables, especially for those typically solved by an alternating minimization (AM) strategy that splits the original optimization problem into a set of sub-problems corresponding to each variable, and then iteratively optimize each sub-problem using a fixed updating rule. However, due to the intrinsic non-convexity of the original optimization problem, the optimization can usually be trapped into spurious local minimum even when each sub-problem can be optimally solved at each iteration. Meanwhile, learning-based approaches, such as deep unfolding algorithms, are highly limited by the lack of labelled data and restricted explainability. To tackle these issues, we propose a meta-learning based alternating minimization (MLAM) method, which aims to minimize a partial of the global losses over iterations instead of carrying minimization on each sub-problem, and it tends to learn an adaptive strategy to replace the handcrafted counterpart resulting in advance on superior performance. Meanwhile, the proposed MLAM still maintains the original algorithmic principle, which contributes to a better interpretability. We evaluate the proposed method on two representative problems, namely, bi-linear inverse problem: matrix completion, and non-linear problem: Gaussian mixture models. The experimental results validate that our proposed approach outperforms AM-based methods in standard settings, and is able to achieve effective optimization in challenging cases while other comparing methods would typically fail.
We propose an iterative algorithm for low-rank matrix completion that can be interpreted as both an iteratively reweighted least squares (IRLS) algorithm and a saddle-escaping smoothing Newton method applied to a non-convex rank surrogate objective. It combines the favorable data efficiency of previous IRLS approaches with an improved scalability by several orders of magnitude. Our method attains a local quadratic convergence rate already for a number of samples that is close to the information theoretical limit. We show in numerical experiments that unlike many state-of-the-art approaches, our approach is able to complete very ill-conditioned matrices with a condition number of up to $10^{10}$ from few samples.
A model-based collaborative filtering (CF) approach utilizing fast adaptive randomized singular value decomposition (SVD) is proposed for the matrix completion problem in recommender system. Firstly, a fast adaptive PCA frameworkis presented which combines the fixed-precision randomized matrix factorization algorithm [1] and accelerating skills for handling large sparse data. Then, a novel termination mechanism for the adaptive PCA is proposed to automatically determine a number of latent factors for achieving the near optimal prediction accuracy during the subsequent model-based CF. The resulted CF approach has good accuracy while inheriting high runtime efficiency. Experiments on real data show that, the proposed adaptive PCA is up to 2.7X and 6.7X faster than the original fixed-precision SVD approach [1] and svds in Matlab repsectively, while preserving accuracy. The proposed model-based CF approach is able to efficiently process the MovieLens data with 20M ratings and exhibits more than 10X speedup over the regularized matrix factorization based approach [2] and the fast singular value thresholding approach [3] with comparable or better accuracy. It also owns the advantage of parameter free. Compared with the deep-learning-based CF approach, the proposed approach is much more computationally efficient, with just marginal performance loss.
In this paper, we propose a simple algorithm to cluster nonnegative data lying in disjoint subspaces. We analyze its performance in relation to a certain measure of correlation between said subspaces. We use our clustering algorithm to develop a matrix completion algorithm which can outperform standard matrix completion algorithms on data matrices satisfying certain natural conditions.
Existing results for low-rank matrix recovery largely focus on quadratic loss, which enjoys favorable properties such as restricted strong convexity/smoothness (RSC/RSM) and well conditioning over all low rank matrices. However, many interesting problems involve more general, non-quadratic losses, which do not satisfy such properties. For these problems, standard nonconvex approaches such as rank-constrained projected gradient descent (a.k.a. iterative hard thresholding) and Burer-Monteiro factorization could have poor empirical performance, and there is no satisfactory theory guaranteeing global and fast convergence for these algorithms. In this paper, we show that a critical component in provable low-rank recovery with non-quadratic loss is a regularity projection oracle. This oracle restricts iterates to low-rank matrices within an appropriate bounded set, over which the loss function is well behaved and satisfies a set of approximate RSC/RSM conditions. Accordingly, we analyze an (averaged) projected gradient method equipped with such an oracle, and prove that it converges globally and linearly. Our results apply to a wide range of non-quadratic low-rank estimation problems including one bit matrix sensing/completion, individualized rank aggregation, and more broadly generalized linear models with rank constraints.
This paper is concerned with the column $\ell_{2,0}$-regularized factorization model of low-rank matrix recovery problems and its computation. The column $\ell_{2,0}$-norm of factor matrices is introduced to promote column sparsity of factors and low-rank solutions. For this nonconvex discontinuous optimization problem, we develop an alternating majorization-minimization (AMM) method with extrapolation, and a hybrid AMM in which a majorized alternating proximal method is proposed to seek an initial factor pair with less nonzero columns and the AMM with extrapolation is then employed to minimize of a smooth nonconvex loss. We provide the global convergence analysis for the proposed AMM methods and apply them to the matrix completion problem with non-uniform sampling schemes. Numerical experiments are conducted with synthetic and real data examples, and comparison results with the nuclear-norm regularized factorization model and the max-norm regularized convex model show that the column $\ell_{2,0}$-regularized factorization model has an advantage in offering solutions of lower error and rank within less time.
Robust low-rank matrix completion (RMC), or robust principal component analysis with partially observed data, has been studied extensively for computer vision, signal processing and machine learning applications. This problem aims to decompose a partially observed matrix into the superposition of a low-rank matrix and a sparse matrix, where the sparse matrix captures the grossly corrupted entries of the matrix. A widely used approach to tackle RMC is to consider a convex formulation, which minimizes the nuclear norm of the low-rank matrix (to promote low-rankness) and the l1 norm of the sparse matrix (to promote sparsity). In this paper, motivated by some recent works on low-rank matrix completion and Riemannian optimization, we formulate this problem as a nonsmooth Riemannian optimization problem over Grassmann manifold. This new formulation is scalable because the low-rank matrix is factorized to the multiplication of two much smaller matrices. We then propose an alternating manifold proximal gradient continuation (AManPGC) method to solve the proposed new formulation. The convergence rate of the proposed algorithm is rigorously analyzed. Numerical results on both synthetic data and real data on background extraction from surveillance videos are reported to demonstrate the advantages of the proposed new formulation and algorithm over several popular existing approaches.
Matrix learning is at the core of many machine learning problems. A number of real-world applications such as collaborative filtering and text mining can be formulated as a low-rank matrix completion problem, which recovers incomplete matrix using low-rank assumptions. To ensure that the matrix solution has a low rank, a recent trend is to use nonconvex regularizers that adaptively penalize singular values. They offer good recovery performance and have nice theoretical properties, but are computationally expensive due to repeated access to individual singular values. In this paper, based on the key insight that adaptive shrinkage on singular values improve empirical performance, we propose a new nonconvex low-rank regularizer called "nuclear norm minus Frobenius norm" regularizer, which is scalable, adaptive and sound. We first show it provably holds the adaptive shrinkage property. Further, we discover its factored form which bypasses the computation of singular values and allows fast optimization by general optimization algorithms. Stable recovery and convergence are guaranteed. Extensive low-rank matrix completion experiments on a number of synthetic and real-world data sets show that the proposed method obtains state-of-the-art recovery performance while being the fastest in comparison to existing low-rank matrix learning methods.
This paper considers stochastic convex optimization problems where the objective and constraint functions involve expectations with respect to the data indices or environmental variables, in addition to deterministic convex constraints on the domain of the variables. Although the setting is generic and arises in different machine learning applications, online and efficient approaches for solving such problems have not been widely studied. Since the underlying data distribution is unknown a priori, a closed-form solution is generally not available, and classical deterministic optimization paradigms are not applicable. State-of-the-art approaches, such as those using the saddle point framework, can ensure that the optimality gap as well as the constraint violation decay as $\O\left(T^{-\frac{1}{2}}\right)$ where $T$ is the number of stochastic gradients. The domain constraints are assumed simple and handled via projection at every iteration. In this work, we propose a novel conservative stochastic optimization algorithm (CSOA) that achieves zero constraint violation and $\O\left(T^{-\frac{1}{2}}\right)$ optimality gap. Further, the projection operation (for scenarios when calculating projection is expensive) in the proposed algorithm can be avoided by considering the conditional gradient or Frank-Wolfe (FW) variant of the algorithm. The state-of-the-art stochastic FW variants achieve an optimality gap of $\O\left(T^{-\frac{1}{3}}\right)$ after $T$ iterations, though these algorithms have not been applied to problems with functional expectation constraints. In this work, we propose the FW-CSOA algorithm that is not only projection-free but also achieves zero constraint violation with $\O\left(T^{-\frac{1}{4}}\right)$ decay of the optimality gap. The efficacy of the proposed algorithms is tested on two relevant problems: fair classification and structured matrix completion.