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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.
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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.
The recently proposed SPARse Factor Analysis (SPARFA) framework for personalized learning performs factor analysis on ordinal or binary-valued (e.g., correct/incorrect) graded learner responses to questions. The underlying factors are termed concepts (or knowledge components) and are used for learning analytics (LA), the estimation of learner concept-knowledge profiles, and for content analytics (CA), the estimation of question-concept associations and question difficulties. While SPARFA is a powerful tool for LA and CA, it requires a number of algorithm parameters (including the number of concepts), which are difficult to determine in practice. In this paper, we propose SPARFA-Lite, a convex optimization-based method for LA that builds on matrix completion, which only requires a single algorithm parameter and enables us to automatically identify the required number of concepts. Using a variety of educational datasets, we demonstrate that SPARFALite (i) achieves comparable performance in predicting unobserved learner responses to existing methods, including item response theory (IRT) and SPARFA, and (ii) is computationally more efficient.
In the low-rank matrix completion (LRMC) problem, the low-rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. This paper extends this thinking to cases where the columns are points on a low-dimensional nonlinear algebraic variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC). Matrices whose columns belong to a union of subspaces are an important special case. We propose a LADMC algorithm that leverages existing LRMC methods on a tensorized representation of the data. For example, a second-order tensorized representation is formed by taking the Kronecker product of each column with itself, and we consider higher order tensorizations as well. This approach will succeed in many cases where traditional LRMC is guaranteed to fail because the data are low-rank in the tensorized representation but not in the original representation. We provide a formal mathematical justification for the success of our method. In particular, we give bounds of the rank of these data in the tensorized representation, and we prove sampling requirements to guarantee uniqueness of the solution. We also provide experimental results showing that the new approach outperforms existing state-of-the-art methods for matrix completion under a union of subspaces model.
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.
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.
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