We consider the problem of noisy matrix completion, in which the goal is to reconstruct a structured matrix whose entries are partially observed in noise. Standard approaches to this underdetermined inverse problem are based on assuming that the underlying matrix has low rank, or is well-approximated by a low rank matrix. In this paper, we propose a richer model based on what we term the "permutation-rank" of a matrix. We first describe how the classical non-negative rank model enforces restrictions that may be undesirable in practice, and how and these restrictions can be avoided by using the richer permutation-rank model. Second, we establish the minimax rates of estimation under the new permutation-based model, and prove that surprisingly, the minimax rates are equivalent up to logarithmic factors to those for estimation under the typical low rank model. Third, we analyze a computationally efficient singular-value-thresholding algorithm, known to be optimal for the low-rank setting, and show that it also simultaneously yields a consistent estimator for the low-permutation rank setting. Finally, we present various structural results characterizing the uniqueness of the permutation-rank decomposition, and characterizing convex approximations of the permutation-rank polytope.
Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite number of loss functions. The present paper proposes a Riemannian stochastic quasi-Newton algorithm with variance reduction (R-SQN-VR). The key challenges of averaging, adding, and subtracting multiple gradients are addressed with notions of retraction and vector transport. We present convergence analyses of R-SQN-VR on both non-convex and retraction-convex functions under retraction and vector transport operators. The proposed algorithm is evaluated on the Karcher mean computation on the symmetric positive-definite manifold and the low-rank matrix completion on the Grassmann manifold. In all cases, the proposed algorithm outperforms the state-of-the-art Riemannian batch and stochastic gradient algorithms.
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, a cutoff can be derived to automatically threshold the singular values obtained from the proximal operator. This allows such operator being efficiently approximated by power method. Based on it, we develop a proximal gradient algorithm (and its accelerated variant) with inexact proximal splitting and prove that a convergence rate of O(1/T) where T is the number of iterations is guaranteed. Furthermore, we show the proposed algorithm can be well parallelized, which achieves nearly linear speedup w.r.t the number of threads. Extensive experiments are performed on matrix completion and robust principal component analysis, which shows a significant speedup over the state-of-the-art. Moreover, the matrix solution obtained is more accurate and has a lower rank than that of the nuclear norm regularizer.
Nowadays, the availability of large-scale data in disparate application domains urges the deployment of sophisticated tools for extracting valuable knowledge out of this huge bulk of information. In that vein, low-rank representations (LRRs) which seek low-dimensional embeddings of data have naturally appeared. In an effort to reduce computational complexity and improve estimation performance, LRR has been viewed via a matrix factorization (MF) perspective. Recently, low-rank MF (LRMF) approaches have been proposed for tackling the inherent weakness of MF i.e., the unawareness of the dimension of the low-dimensional space where data reside. Herein, inspired by the merits of iterative reweighted schemes for rank minimization, we come up with a generic low-rank promoting regularization function. Then, focusing on a specific instance of it, we propose a regularizer that imposes column-sparsity jointly on the two matrix factors that result from MF, thus promoting low-rankness on the optimization problem. The problems of denoising, matrix completion and non-negative matrix factorization (NMF) are redefined according to the new LRMF formulation and solved via efficient Newton-type algorithms with proven theoretical guarantees as to their convergence and rates of convergence to stationary points. The effectiveness of the proposed algorithms is verified in diverse simulated and real data experiments.
We propose a novel algorithm for sequential matrix completion in a recommender system setting, where the $(i,j)$th entry of the matrix corresponds to a user $i$'s rating of product $j$. The objective of the algorithm is to provide a sequential policy for user-product pair recommendation which will yield the highest possible ratings after a finite time horizon. The algorithm uses a Gamma process factor model with two posterior-focused bandit policies, Thompson Sampling and Information-Directed Sampling. While Thompson Sampling shows competitive performance in simulations, state-of-the-art performance is obtained from Information-Directed Sampling, which makes its recommendations based off a ratio between the expected reward and a measure of information gain. To our knowledge, this is the first implementation of Information Directed Sampling on large real datasets. This approach contributes to a recent line of research on bandit approaches to collaborative filtering including Kawale et al. (2015), Li et al. (2010), Bresler et al. (2014), Li et al. (2016), Deshpande & Montanari (2012), and Zhao et al. (2013). The setting of this paper, as has been noted in Kawale et al. (2015) and Zhao et al. (2013), presents significant challenges to bounding regret after finite horizons. We discuss these challenges in relation to simpler models for bandits with side information, such as linear or gaussian process bandits, and hope the experiments presented here motivate further research toward theoretical guarantees.
We consider matrix completion for recommender systems from the point of view of link prediction on graphs. Interaction data such as movie ratings can be represented by a bipartite user-item graph with labeled edges denoting observed ratings. Building on recent progress in deep learning on graph-structured data, we propose a graph auto-encoder framework based on differentiable message passing on the bipartite interaction graph. Our model shows competitive performance on standard collaborative filtering benchmarks. In settings where complimentary feature information or structured data such as a social network is available, our framework outperforms recent state-of-the-art methods.
This paper examines fundamental error characteristics for a general class of matrix completion problems, where the matrix of interest is a product of two a priori unknown matrices, one of which is sparse, and the observations are noisy. Our main contributions come in the form of minimax lower bounds for the expected per-element squared error for this problem under under several common noise models. Specifically, we analyze scenarios where the corruptions are characterized by additive Gaussian noise or additive heavier-tailed (Laplace) noise, Poisson-distributed observations, and highly-quantized (e.g., one-bit) observations, as instances of our general result. Our results establish that the error bounds derived in (Soni et al., 2016) for complexity-regularized maximum likelihood estimators achieve, up to multiplicative constants and logarithmic factors, the minimax error rates in each of these noise scenarios, provided that the nominal number of observations is large enough, and the sparse factor has (on an average) at least one non-zero per column.
In an era when big data are becoming the norm, there is less concern with the quantity but more with the quality and completeness of the data. In many disciplines, data are collected from heterogeneous sources, resulting in multi-view or multi-modal datasets. The missing data problem has been challenging to address in multi-view data analysis. Especially, when certain samples miss an entire view of data, it creates the missing view problem. Classic multiple imputations or matrix completion methods are hardly effective here when no information can be based on in the specific view to impute data for such samples. The commonly-used simple method of removing samples with a missing view can dramatically reduce sample size, thus diminishing the statistical power of a subsequent analysis. In this paper, we propose a novel approach for view imputation via generative adversarial networks (GANs), which we name by VIGAN. This approach first treats each view as a separate domain and identifies domain-to-domain mappings via a GAN using randomly-sampled data from each view, and then employs a multi-modal denoising autoencoder (DAE) to reconstruct the missing view from the GAN outputs based on paired data across the views. Then, by optimizing the GAN and DAE jointly, our model enables the knowledge integration for domain mappings and view correspondences to effectively recover the missing view. Empirical results on benchmark datasets validate the VIGAN approach by comparing against the state of the art. The evaluation of VIGAN in a genetic study of substance use disorders further proves the effectiveness and usability of this approach in life science.
Background subtraction is the primary task of the majority of video inspection systems. The most important part of the background subtraction which is common among different algorithms is background modeling. In this regard, our paper addresses the problem of background modeling in a computationally efficient way, which is important for current eruption of "big data" processing coming from high resolution multi-channel videos. Our model is based on the assumption that background in natural images lies on a low-dimensional subspace. We formulated and solved this problem in a low-rank matrix completion framework. In modeling the background, we benefited from the in-face extended Frank-Wolfe algorithm for solving a defined convex optimization problem. We evaluated our fast robust matrix completion (fRMC) method on both background models challenge (BMC) and Stuttgart artificial background subtraction (SABS) datasets. The results were compared with the robust principle component analysis (RPCA) and low-rank robust matrix completion (RMC) methods, both solved by inexact augmented Lagrangian multiplier (IALM). The results showed faster computation, at least twice as when IALM solver is used, while having a comparable accuracy even better in some challenges, in subtracting the backgrounds in order to detect moving objects in the scene.
We show that given an estimate $\widehat{A}$ that is close to a general high-rank positive semi-definite (PSD) matrix $A$ in spectral norm (i.e., $\|\widehat{A}-A\|_2 \leq \delta$), the simple truncated SVD of $\widehat{A}$ produces a multiplicative approximation of $A$ in Frobenius norm. This observation leads to many interesting results on general high-rank matrix estimation problems, which we briefly summarize below ($A$ is an $n\times n$ high-rank PSD matrix and $A_k$ is the best rank-$k$ approximation of $A$): (1) High-rank matrix completion: By observing $\Omega(\frac{n\max\{\epsilon^{-4},k^2\}\mu_0^2\|A\|_F^2\log n}{\sigma_{k+1}(A)^2})$ elements of $A$ where $\sigma_{k+1}\left(A\right)$ is the $\left(k+1\right)$-th singular value of $A$ and $\mu_0$ is the incoherence, the truncated SVD on a zero-filled matrix satisfies $\|\widehat{A}_k-A\|_F \leq (1+O(\epsilon))\|A-A_k\|_F$ with high probability. (2)High-rank matrix de-noising: Let $\widehat{A}=A+E$ where $E$ is a Gaussian random noise matrix with zero mean and $\nu^2/n$ variance on each entry. Then the truncated SVD of $\widehat{A}$ satisfies $\|\widehat{A}_k-A\|_F \leq (1+O(\sqrt{\nu/\sigma_{k+1}(A)}))\|A-A_k\|_F + O(\sqrt{k}\nu)$. (3) Low-rank Estimation of high-dimensional covariance: Given $N$ i.i.d.~samples $X_1,\cdots,X_N\sim\mathcal N_n(0,A)$, can we estimate $A$ with a relative-error Frobenius norm bound? We show that if $N = \Omega\left(n\max\{\epsilon^{-4},k^2\}\gamma_k(A)^2\log N\right)$ for $\gamma_k(A)=\sigma_1(A)/\sigma_{k+1}(A)$, then $\|\widehat{A}_k-A\|_F \leq (1+O(\epsilon))\|A-A_k\|_F$ with high probability, where $\widehat{A}=\frac{1}{N}\sum_{i=1}^N{X_iX_i^\top}$ is the sample covariance.
In this paper, we consider the Graphical Lasso (GL), a popular optimization problem for learning the sparse representations of high-dimensional datasets, which is well-known to be computationally expensive for large-scale problems. Recently, we have shown that the sparsity pattern of the optimal solution of GL is equivalent to the one obtained from simply thresholding the sample covariance matrix, for sparse graphs under different conditions. We have also derived a closed-form solution that is optimal when the thresholded sample covariance matrix has an acyclic structure. As a major generalization of the previous result, in this paper we derive a closed-form solution for the GL for graphs with chordal structures. We show that the GL and thresholding equivalence conditions can significantly be simplified and are expected to hold for high-dimensional problems if the thresholded sample covariance matrix has a chordal structure. We then show that the GL and thresholding equivalence is enough to reduce the GL to a maximum determinant matrix completion problem and drive a recursive closed-form solution for the GL when the thresholded sample covariance matrix has a chordal structure. For large-scale problems with up to 450 million variables, the proposed method can solve the GL problem in less than 2 minutes, while the state-of-the-art methods converge in more than 2 hours.
Missing data is a ubiquitous problem. It is especially challenging in medical settings because many streams of measurements are collected at different - and often irregular - times. Accurate estimation of those missing measurements is critical for many reasons, including diagnosis, prognosis and treatment. Existing methods address this estimation problem by interpolating within data streams or imputing across data streams (both of which ignore important information) or ignoring the temporal aspect of the data and imposing strong assumptions about the nature of the data-generating process and/or the pattern of missing data (both of which are especially problematic for medical data). We propose a new approach, based on a novel deep learning architecture that we call a Multi-directional Recurrent Neural Network (M-RNN) that interpolates within data streams and imputes across data streams. We demonstrate the power of our approach by applying it to five real-world medical datasets. We show that it provides dramatically improved estimation of missing measurements in comparison to 11 state-of-the-art benchmarks (including Spline and Cubic Interpolations, MICE, MissForest, matrix completion and several RNN methods); typical improvements in Root Mean Square Error are between 35% - 50%. Additional experiments based on the same five datasets demonstrate that the improvements provided by our method are extremely robust.
In this letter, we study the deterministic sampling patterns for the completion of low rank matrix, when corrupted with a sparse noise, also known as robust matrix completion. We extend the recent results on the deterministic sampling patterns in the absence of noise based on the geometric analysis on the Grassmannian manifold. A special case where each column has a certain number of noisy entries is considered, where our probabilistic analysis performs very efficiently. Furthermore, assuming that the rank of the original matrix is not given, we provide an analysis to determine if the rank of a valid completion is indeed the actual rank of the data corrupted with sparse noise by verifying some conditions.
Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often require proper regularization (e.g. trimming, regularized cost, projection) in order to guarantee fast convergence. For vanilla procedures such as gradient descent, however, prior theory either recommends highly conservative learning rates to avoid overshooting, or completely lacks performance guarantees. This paper uncovers a striking phenomenon in nonconvex optimization: even in the absence of explicit regularization, gradient descent enforces proper regularization implicitly under various statistical models. In fact, gradient descent follows a trajectory staying within a basin that enjoys nice geometry, consisting of points incoherent with the sampling mechanism. This "implicit regularization" feature allows gradient descent to proceed in a far more aggressive fashion without overshooting, which in turn results in substantial computational savings. Focusing on three fundamental statistical estimation problems, i.e. phase retrieval, low-rank matrix completion, and blind deconvolution, we establish that gradient descent achieves near-optimal statistical and computational guarantees without explicit regularization. In particular, by marrying statistical modeling with generic optimization theory, we develop a general recipe for analyzing the trajectories of iterative algorithms via a leave-one-out perturbation argument. As a byproduct, for noisy matrix completion, we demonstrate that gradient descent achieves near-optimal error control --- measured entrywise and by the spectral norm --- which might be of independent interest.
Recommendation system is a common demand in daily life and matrix completion is a widely adopted technique for this task. However, most matrix completion methods lack semantic interpretation and usually result in weak-semantic recommendations. To this end, this paper proposes a $S$emantic $A$nalysis approach for $R$ecommendation systems $(SAR)$, which applies a two-level hierarchical generative process that assigns semantic properties and categories for user and item. $SAR$ learns semantic representations of users/items merely from user ratings on items, which offers a new path to recommendation by semantic matching with the learned representations. Extensive experiments demonstrate $SAR$ outperforms other state-of-the-art baselines substantially.
This work studies low-rank approximation of a positive semidefinite matrix from partial entries via nonconvex optimization. We characterized how well local-minimum based low-rank factorization approximates a fixed positive semidefinite matrix without any assumptions on the rank-matching, the condition number or eigenspace incoherence parameter. Furthermore, under certain assumptions on rank-matching and well-boundedness of condition numbers and eigenspace incoherence parameters, a corollary of our main theorem improves the state-of-the-art sampling rate results for nonconvex matrix completion with no spurious local minima in Ge et al. [2016, 2017]. In addition, we investigated when the proposed nonconvex optimization results in accurate low-rank approximations even in presence of large condition numbers, large incoherence parameters, or rank mismatching. We also propose to apply the nonconvex optimization to memory-efficient Kernel PCA. Compared to the well-known Nystr\"{o}m methods, numerical experiments indicate that the proposed nonconvex optimization approach yields more stable results in both low-rank approximation and clustering.
A standard model for Recommender Systems is the Matrix Completion setting: given partially known matrix of ratings given by users (rows) to items (columns), infer the unknown ratings. In the last decades, few attempts where done to handle that objective with Neural Networks, but recently an architecture based on Autoencoders proved to be a promising approach. In current paper, we enhanced that architecture (i) by using a loss function adapted to input data with missing values, and (ii) by incorporating side information. The experiments demonstrate that while side information only slightly improve the test error averaged on all users/items, it has more impact on cold users/items.
This paper considers the problem of matrix completion when the observed entries are noisy and contain outliers. It begins with introducing a new optimization criterion for which the recovered matrix is defined as its solution. This criterion uses the celebrated Huber function from the robust statistics literature to downweigh the effects of outliers. A practical algorithm is developed to solve the optimization involved. This algorithm is fast, straightforward to implement, and monotonic convergent. Furthermore, the proposed methodology is theoretically shown to be stable in a well defined sense. Its promising empirical performance is demonstrated via a sequence of simulation experiments, including image inpainting.
Indian regional movie dataset is the first database of regional Indian movies, users and their ratings. It consists of movies belonging to 18 different Indian regional languages and metadata of users with varying demographics. Through this dataset, the diversity of Indian regional cinema and its huge viewership is captured. We analyze the dataset that contains roughly 10K ratings of 919 users and 2,851 movies using some supervised and unsupervised collaborative filtering techniques like Probabilistic Matrix Factorization, Matrix Completion, Blind Compressed Sensing etc. The dataset consists of metadata information of users like age, occupation, home state and known languages. It also consists of metadata of movies like genre, language, release year and cast. India has a wide base of viewers which is evident by the large number of movies released every year and the huge box-office revenue. This dataset can be used for designing recommendation systems for Indian users and regional movies, which do not, yet, exist. The dataset can be downloaded from \href{https://goo.gl/EmTPv6}{https://goo.gl/EmTPv6}.
Factoring a matrix into two low rank matrices is at the heart of many problems. The problem of matrix completion especially uses it to decompose a sparse matrix into two non sparse, low rank matrices which can then be used to predict unknown entries of the original matrix. We present a scalable and decentralized approach in which instead of learning two factors for the original input matrix, we decompose the original matrix into a grid blocks, each of whose factors can be individually learned just by communicating (gossiping) with neighboring blocks. This eliminates any need for a central server. We show that our algorithm performs well on both synthetic and real datasets.