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Register a new userMatrix Completion under Low-Rank Missing Mechanism
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Matrix completion is a modern missing data problem where both the missing structure and the underlying parameter are high dimensional. Although missing structure is a key component to any missing data problems, existing matrix completion methods often assume a simple uniform missing mechanism. In this work, we study matrix completion from corrupted data under a novel low-rank missing mechanism. The probability matrix of observation is estimated via a high dimensional low-rank matrix estimation procedure, and further used to complete the target matrix via inverse probabilities weighting. Due to both high dimensional and extreme (i.e., very small) nature of the true probability matrix, the effect of inverse probability weighting requires careful study. We derive optimal asymptotic convergence rates of the proposed estimators for both the observation probabilities and the target matrix.
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Inductive Matrix Completion (IMC) is an important class of matrix completion problems that allows direct inclusion of available features to enhance estimation capabilities. These models have found applications in personalized recommendation systems, multilabel learning, dictionary learning, etc. This paper examines a general class of noisy matrix completion tasks where the underlying matrix is following an IMC model i.e., it is formed by a mixing matrix (a priori unknown) sandwiched between two known feature matrices. The mixing matrix here is assumed to be well approximated by the product of two sparse matrices---referred here to as sparse factor models. We leverage the main theorem of Soni:2016:NMC and extend it to provide theoretical error bounds for the sparsity-regularized maximum likelihood estimators for the class of problems discussed in this paper. The main result is general in the sense that it can be used to derive error bounds for various noise models. In this paper, we instantiate our main result for the case of Gaussian noise and provide corresponding error bounds in terms of squared loss.
We consider the matrix completion problem of recovering a structured low rank matrix with partially observed entries with mixed data types. Vast majority of the solutions have proposed computationally feasible estimators with strong statistical guarantees for the case where the underlying distribution of data in the matrix is continuous. A few recent approaches have extended using similar ideas these estimators to the case where the underlying distributions belongs to the exponential family. Most of these approaches assume that there is only one underlying distribution and the low rank constraint is regularized by the matrix Schatten Norm. We propose a computationally feasible statistical approach with strong recovery guarantees along with an algorithmic framework suited for parallelization to recover a low rank matrix with partially observed entries for mixed data types in one step. We also provide extensive simulation evidence that corroborate our theoretical results.
Most recent results in matrix completion assume that the matrix under consideration is low-rank or that the columns are in a union of low-rank subspaces. In real-world settings, however, the linear structure underlying these models is distorted by a (typically unknown) nonlinear transformation. This paper addresses the challenge of matrix completion in the face of such nonlinearities. Given a few observations of a matrix that are obtained by applying a Lipschitz, monotonic function to a low rank matrix, our task is to estimate the remaining unobserved entries. We propose a novel matrix completion method that alternates between low-rank matrix estimation and monotonic function estimation to estimate the missing matrix elements. Mean squared error bounds provide insight into how well the matrix can be estimated based on the size, rank of the matrix and properties of the nonlinear transformation. Empirical results on synthetic and real-world datasets demonstrate the competitiveness of the proposed approach.
This paper addresses the problem of low-rank distance matrix completion. This problem amounts to recover the missing entries of a distance matrix when the dimension of the data embedding space is possibly unknown but small compared to the number of considered data points. The focus is on high-dimensional problems. We recast the considered problem into an optimization problem over the set of low-rank positive semidefinite matrices and propose two efficient algorithms for low-rank distance matrix completion. In addition, we propose a strategy to determine the dimension of the embedding space. The resulting algorithms scale to high-dimensional problems and monotonically converge to a global solution of the problem. Finally, numerical experiments illustrate the good performance of the proposed algorithms on benchmarks.
Low rank tensor ring model is powerful for image completion which recovers missing entries in data acquisition and transformation. The recently proposed tensor ring (TR) based completion algorithms generally solve the low rank optimization problem by alternating least squares method with predefined ranks, which may easily lead to overfitting when the unknown ranks are set too large and only a few measurements are available. In this paper, we present a Bayesian low rank tensor ring model for image completion by automatically learning the low rank structure of data. A multiplicative interaction model is developed for the low-rank tensor ring decomposition, where core factors are enforced to be sparse by assuming their entries obey Student-T distribution. Compared with most of the existing methods, the proposed one is free of parameter-tuning, and the TR ranks can be obtained by Bayesian inference. Numerical Experiments, including synthetic data, color images with different sizes and YaleFace dataset B with respect to one pose, show that the proposed approach outperforms state-of-the-art ones, especially in terms of recovery accuracy.
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