In this paper we focus on the problem of completion of multidimensional arrays (also referred to as tensors) from limited sampling. Our approach is based on a recently proposed tensor-Singular Value Decomposition (t-SVD) [1]. Using this factorization one can derive notion of tensor rank, referred to as the tensor tubal rank, which has optimality properties similar to that of matrix rank derived from SVD. As shown in [2] some multidimensional data, such as panning video sequences exhibit low tensor tubal rank and we look at the problem of completing such data under random sampling of the data cube. We show that by solving a convex optimization problem, which minimizes the tensor nuclear norm obtained as the convex relaxation of tensor tubal rank, one can guarantee recovery with overwhelming probability as long as samples in proportion to the degrees of freedom in t-SVD are observed. In this sense our results are order-wise optimal. The conditions under which this result holds are very similar to the incoherency conditions for the matrix completion, albeit we define incoherency under the algebraic set-up of t-SVD. We show the performance of the algorithm on some real data sets and compare it with other existing approaches based on tensor flattening and Tucker decomposition.
Nuclear-norm regularization plays a vital role in many learning tasks, such as low-rank matrix recovery (MR), and low-rank representation (LRR). Solving this problem directly can be computationally expensive due to the unknown rank of variables or large-rank singular value decompositions (SVDs). To address this, we propose a proximal Riemannian gradient (PRG) scheme which can efficiently solve trace-norm regularized problems defined on real-algebraic variety $\mMLr$ of real matrices of rank at most $r$. Based on PRG, we further present a simple and novel subspace pursuit (SP) paradigm for general trace-norm regularized problems without the explicit rank constraint $\mMLr$. The proposed paradigm is very scalable by avoiding large-rank SVDs. Empirical studies on several tasks, such as matrix completion and LRR based subspace clustering, demonstrate the superiority of the proposed paradigms over existing methods.
We extend the theory of matrix completion to the case where we make Poisson observations for a subset of entries of a low-rank matrix. We consider the (now) usual matrix recovery formulation through maximum likelihood with proper constraints on the matrix $M$, and establish theoretical upper and lower bounds on the recovery error. Our bounds are nearly optimal up to a factor on the order of $\mathcal{O}(\log(d_1 d_2))$. These bounds are obtained by adapting the arguments used for one-bit matrix completion \cite{davenport20121} (although these two problems are different in nature) and the adaptation requires new techniques exploiting properties of the Poisson likelihood function and tackling the difficulties posed by the locally sub-Gaussian characteristic of the Poisson distribution. Our results highlight a few important distinctions of Poisson matrix completion compared to the prior work in matrix completion including having to impose a minimum signal-to-noise requirement on each observed entry. We also develop an efficient iterative algorithm and demonstrate its good performance in recovering solar flare images.
Recently theoretical guarantees have been obtained for matrix completion in the non-uniform sampling regime. In particular, if the sampling distribution aligns with the underlying matrix's leverage scores, then with high probability nuclear norm minimization will exactly recover the low rank matrix. In this article, we analyze the scenario in which the non-uniform sampling distribution may or may not not align with the underlying matrix's leverage scores. Here we explore learning the parameters for weighted nuclear norm minimization in terms of the empirical sampling distribution. We provide a sufficiency condition for these learned weights which provide an exact recovery guarantee for weighted nuclear norm minimization. It has been established that a specific choice of weights in terms of the true sampling distribution not only allows for weighted nuclear norm minimization to exactly recover the low rank matrix, but also allows for a quantifiable relaxation in the exact recovery conditions. In this article we extend this quantifiable relaxation in exact recovery conditions for a specific choice of weights defined analogously in terms of the empirical distribution as opposed to the true sampling distribution. To accomplish this we employ a concentration of measure bound and a large deviation bound. We also present numerical evidence for the healthy robustness of the weighted nuclear norm minimization algorithm to the choice of empirically learned weights. These numerical experiments show that for a variety of easily computable empirical weights, weighted nuclear norm minimization outperforms unweighted nuclear norm minimization in the non-uniform sampling regime.
Matrix completion has attracted significant recent attention in many fields including statistics, applied mathematics and electrical engineering. Current literature on matrix completion focuses primarily on independent sampling models under which the individual observed entries are sampled independently. Motivated by applications in genomic data integration, we propose a new framework of structured matrix completion (SMC) to treat structured missingness by design. Specifically, our proposed method aims at efficient matrix recovery when a subset of the rows and columns of an approximately low-rank matrix are observed. We provide theoretical justification for the proposed SMC method and derive lower bound for the estimation errors, which together establish the optimal rate of recovery over certain classes of approximately low-rank matrices. Simulation studies show that the method performs well in finite sample under a variety of configurations. The method is applied to integrate several ovarian cancer genomic studies with different extent of genomic measurements, which enables us to construct more accurate prediction rules for ovarian cancer survival.
We address the collective matrix completion problem of jointly recovering a collection of matrices with shared structure from partial (and potentially noisy) observations. To ensure well--posedness of the problem, we impose a joint low rank structure, wherein each component matrix is low rank and the latent space of the low rank factors corresponding to each entity is shared across the entire collection. We first develop a rigorous algebra for representing and manipulating collective--matrix structure, and identify sufficient conditions for consistent estimation of collective matrices. We then propose a tractable convex estimator for solving the collective matrix completion problem, and provide the first non--trivial theoretical guarantees for consistency of collective matrix completion. We show that under reasonable assumptions stated in Section 3.1, with high probability, the proposed estimator exactly recovers the true matrices whenever sample complexity requirements dictated by Theorem 1 are met. The sample complexity requirement derived in the paper are optimum up to logarithmic factors, and significantly improve upon the requirements obtained by trivial extensions of standard matrix completion. Finally, we propose a scalable approximate algorithm to solve the proposed convex program, and corroborate our results through simulated experiments.
In this paper, we consider the streaming memory-limited matrix completion problem when the observed entries are noisy versions of a small random fraction of the original entries. We are interested in scenarios where the matrix size is very large so the matrix is very hard to store and manipulate. Here, columns of the observed matrix are presented sequentially and the goal is to complete the missing entries after one pass on the data with limited memory space and limited computational complexity. We propose a streaming algorithm which produces an estimate of the original matrix with a vanishing mean square error, uses memory space scaling linearly with the ambient dimension of the matrix, i.e. the memory required to store the output alone, and spends computations as much as the number of non-zero entries of the input matrix.
The matrix completion problem consists in reconstructing a matrix from a sample of entries, possibly observed with noise. A popular class of estimator, known as nuclear norm penalized estimators, are based on minimizing the sum of a data fitting term and a nuclear norm penalization. Here, we investigate the case where the noise distribution belongs to the exponential family and is sub-exponential. Our framework alllows for a general sampling scheme. We first consider an estimator defined as the minimizer of the sum of a log-likelihood term and a nuclear norm penalization and prove an upper bound on the Frobenius prediction risk. The rate obtained improves on previous works on matrix completion for exponential family. When the sampling distribution is known, we propose another estimator and prove an oracle inequality w.r.t. the Kullback-Leibler prediction risk, which translates immediatly into an upper bound on the Frobenius prediction risk. Finally, we show that all the rates obtained are minimax optimal up to a logarithmic factor.
In this paper, we develop a relative error bound for nuclear norm regularized matrix completion, with the focus on the completion of full-rank matrices. Under the assumption that the top eigenspaces of the target matrix are incoherent, we derive a relative upper bound for recovering the best low-rank approximation of the unknown matrix. Although multiple works have been devoted to analyzing the recovery error of full-rank matrix completion, their error bounds are usually additive, making it impossible to obtain the perfect recovery case and more generally difficult to leverage the skewed distribution of eigenvalues. Our analysis is built upon the optimality condition of the regularized formulation and existing guarantees for low-rank matrix completion. To the best of our knowledge, this is the first relative bound that has been proved for the regularized formulation of matrix completion.
Social trust prediction addresses the significant problem of exploring interactions among users in social networks. Naturally, this problem can be formulated in the matrix completion framework, with each entry indicating the trustness or distrustness. However, there are two challenges for the social trust problem: 1) the observed data are with sign (1-bit) measurements; 2) they are typically sampled non-uniformly. Most of the previous matrix completion methods do not well handle the two issues. Motivated by the recent progress of max-norm, we propose to solve the problem with a 1-bit max-norm constrained formulation. Since max-norm is not easy to optimize, we utilize a reformulation of max-norm which facilitates an efficient projected gradient decent algorithm. We demonstrate the superiority of our formulation on two benchmark datasets.
Over the past few years, trace regression models have received considerable attention in the context of matrix completion, quantum state tomography, and compressed sensing. Estimation of the underlying matrix from regularization-based approaches promoting low-rankedness, notably nuclear norm regularization, have enjoyed great popularity. In the present paper, we argue that such regularization may no longer be necessary if the underlying matrix is symmetric positive semidefinite (\textsf{spd}) and the design satisfies certain conditions. In this situation, simple least squares estimation subject to an \textsf{spd} constraint may perform as well as regularization-based approaches with a proper choice of the regularization parameter, which entails knowledge of the noise level and/or tuning. By contrast, constrained least squares estimation comes without any tuning parameter and may hence be preferred due to its simplicity.
Principal components analysis (PCA) is a well-known technique for approximating a tabular data set by a low rank matrix. Here, we extend the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types. This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix completion, sparse and robust PCA, $k$-means, $k$-SVD, and maximum margin matrix factorization. The method handles heterogeneous data sets, and leads to coherent schemes for compressing, denoising, and imputing missing entries across all data types simultaneously. It also admits a number of interesting interpretations of the low rank factors, which allow clustering of examples or of features. We propose several parallel algorithms for fitting generalized low rank models, and describe implementations and numerical results.
We provide scientific foundations for athletic performance prediction on an individual level, exposing the phenomenology of individual athletic running performance in the form of a low-rank model dominated by an individual power law. We present, evaluate, and compare a selection of methods for prediction of individual running performance, including our own, \emph{local matrix completion} (LMC), which we show to perform best. We also show that many documented phenomena in quantitative sports science, such as the form of scoring tables, the success of existing prediction methods including Riegel's formula, the Purdy points scheme, the power law for world records performances and the broken power law for world record speeds may be explained on the basis of our findings in a unified way.
Tensors play a central role in many modern machine learning and signal processing applications. In such applications, the target tensor is usually of low rank, i.e., can be expressed as a sum of a small number of rank one tensors. This motivates us to consider the problem of low rank tensor recovery from a class of linear measurements called separable measurements. As specific examples, we focus on two distinct types of separable measurement mechanisms (a) Random projections, where each measurement corresponds to an inner product of the tensor with a suitable random tensor, and (b) the completion problem where measurements constitute revelation of a random set of entries. We present a computationally efficient algorithm, with rigorous and order-optimal sample complexity results (upto logarithmic factors) for tensor recovery. Our method is based on reduction to matrix completion sub-problems and adaptation of Leurgans' method for tensor decomposition. We extend the methodology and sample complexity results to higher order tensors, and experimentally validate our theoretical results.
We consider the problem of signal recovery on graphs as graphs model data with complex structure as signals on a graph. Graph signal recovery implies recovery of one or multiple smooth graph signals from noisy, corrupted, or incomplete measurements. We propose a graph signal model and formulate signal recovery as a corresponding optimization problem. We provide a general solution by using the alternating direction methods of multipliers. We next show how signal inpainting, matrix completion, robust principal component analysis, and anomaly detection all relate to graph signal recovery, and provide corresponding specific solutions and theoretical analysis. Finally, we validate the proposed methods on real-world recovery problems, including online blog classification, bridge condition identification, temperature estimation, recommender system, and expert opinion combination of online blog classification.
The PAC-Bayesian approach is a powerful set of techniques to derive non- asymptotic risk bounds for random estimators. The corresponding optimal distribution of estimators, usually called the Gibbs posterior, is unfortunately intractable. One may sample from it using Markov chain Monte Carlo, but this is often too slow for big datasets. We consider instead variational approximations of the Gibbs posterior, which are fast to compute. We undertake a general study of the properties of such approximations. Our main finding is that such a variational approximation has often the same rate of convergence as the original PAC-Bayesian procedure it approximates. We specialise our results to several learning tasks (classification, ranking, matrix completion),discuss how to implement a variational approximation in each case, and illustrate the good properties of said approximation on real datasets.
Forward stagewise regression follows a very simple strategy for constructing a sequence of sparse regression estimates: it starts with all coefficients equal to zero, and iteratively updates the coefficient (by a small amount $\epsilon$) of the variable that achieves the maximal absolute inner product with the current residual. This procedure has an interesting connection to the lasso: under some conditions, it is known that the sequence of forward stagewise estimates exactly coincides with the lasso path, as the step size $\epsilon$ goes to zero. Furthermore, essentially the same equivalence holds outside of least squares regression, with the minimization of a differentiable convex loss function subject to an $\ell_1$ norm constraint (the stagewise algorithm now updates the coefficient corresponding to the maximal absolute component of the gradient). Even when they do not match their $\ell_1$-constrained analogues, stagewise estimates provide a useful approximation, and are computationally appealing. Their success in sparse modeling motivates the question: can a simple, effective strategy like forward stagewise be applied more broadly in other regularization settings, beyond the $\ell_1$ norm and sparsity? The current paper is an attempt to do just this. We present a general framework for stagewise estimation, which yields fast algorithms for problems such as group-structured learning, matrix completion, image denoising, and more.
This paper presents a unified geometric framework for the statistical analysis of a general ill-posed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix estimation, and noisy matrix completion. We propose computationally feasible convex programs for statistical inference including estimation, confidence intervals and hypothesis testing. A theoretical framework is developed to characterize the local estimation rate of convergence and to provide statistical inference guarantees. Our results are built based on the local conic geometry and duality. The difficulty of statistical inference is captured by the geometric characterization of the local tangent cone through the Gaussian width and Sudakov minoration estimate.
For the problems of low-rank matrix completion, the efficiency of the widely-used nuclear norm technique may be challenged under many circumstances, especially when certain basis coefficients are fixed, for example, the low-rank correlation matrix completion in various fields such as the financial market and the low-rank density matrix completion from the quantum state tomography. To seek a solution of high recovery quality beyond the reach of the nuclear norm, in this paper, we propose a rank-corrected procedure using a nuclear semi-norm to generate a new estimator. For this new estimator, we establish a non-asymptotic recovery error bound. More importantly, we quantify the reduction of the recovery error bound for this rank-corrected procedure. Compared with the one obtained for the nuclear norm penalized least squares estimator, this reduction can be substantial (around 50%). We also provide necessary and sufficient conditions for rank consistency in the sense of Bach (2008). Very interestingly, these conditions are highly related to the concept of constraint nondegeneracy in matrix optimization. As a byproduct, our results provide a theoretical foundation for the majorized penalty method of Gao and Sun (2010) and Gao (2010) for structured low-rank matrix optimization problems. Extensive numerical experiments demonstrate that our proposed rank-corrected procedure can simultaneously achieve a high recovery accuracy and capture the low-rank structure.
We consider the problem of completing a matrix with categorical-valued entries from partial observations. This is achieved by extending the formulation and theory of one-bit matrix completion. We recover a low-rank matrix $X$ by maximizing the likelihood ratio with a constraint on the nuclear norm of $X$, and the observations are mapped from entries of $X$ through multiple link functions. We establish theoretical upper and lower bounds on the recovery error, which meet up to a constant factor $\mathcal{O}(K^{3/2})$ where $K$ is the fixed number of categories. The upper bound in our case depends on the number of categories implicitly through a maximization of terms that involve the smoothness of the link functions. In contrast to one-bit matrix completion, our bounds for categorical matrix completion are optimal up to a factor on the order of the square root of the number of categories, which is consistent with an intuition that the problem becomes harder when the number of categories increases. By comparing the performance of our method with the conventional matrix completion method on the MovieLens dataset, we demonstrate the advantage of our method.