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Register a new userMatrix Completion Under Monotonic Single Index Models
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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.
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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.
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.
Matrix completion aims to reconstruct a data matrix based on observations of a small number of its entries. Usually in matrix completion a single matrix is considered, which can be, for example, a rating matrix in recommendation system. However, in practical situations, data is often obtained from multiple sources which results in a collection of matrices rather than a single one. In this work, we consider the problem of collective matrix completion with multiple and heterogeneous matrices, which can be count, binary, continuous, etc. We first investigate the setting where, for each source, the matrix entries are sampled from an exponential family distribution. Then, we relax the assumption of exponential family distribution for the noise and we investigate the distribution-free case. In this setting, we do not assume any specific model for the observations. The estimation procedures are based on minimizing the sum of a goodness-of-fit term and the nuclear norm penalization of the whole collective matrix. We prove that the proposed estimators achieve fast rates of convergence under the two considered settings and we corroborate our results with numerical experiments.
Single Index Models (SIMs) are simple yet flexible semi-parametric models for classification and regression. Response variables are modeled as a nonlinear, monotonic function of a linear combination of features. Estimation in this context requires learning both the feature weights, and the nonlinear function. While methods have been described to learn SIMs in the low dimensional regime, a method that can efficiently learn SIMs in high dimensions has not been forthcoming. We propose three variants of a computationally and statistically efficient algorithm for SIM inference in high dimensions. We establish excess risk bounds for the proposed algorithms and experimentally validate the advantages that our SIM learning methods provide relative to Generalized Linear Model (GLM) and low dimensional SIM based learning methods.
Predicting unobserved entries of a partially observed matrix has found wide applicability in several areas, such as recommender systems, computational biology, and computer vision. Many scalable methods with rigorous theoretical guarantees have been developed for algorithms where the matrix is factored into low-rank components, and embeddings are learned for the row and column entities. While there has been recent research on incorporating explicit side information in the low-rank matrix factorization setting, often implicit information can be gleaned from the data, via higher-order interactions among entities. Such implicit information is especially useful in cases where the data is very sparse, as is often the case in real-world datasets. In this paper, we design a method to learn embeddings in the context of recommendation systems, using the observation that higher powers of a graph transition probability matrix encode the probability that a random walker will hit that node in a given number of steps. We develop a coordinate descent algorithm to solve the resulting optimization, that makes explicit computation of the higher order powers of the matrix redundant, preserving sparsity and making computations efficient. Experiments on several datasets show that our method, that can use higher order information, outperforms methods that only use explicitly available side information, those that use only second-order implicit information and in some cases, methods based on deep neural networks as well.
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