Matrix completion with structure
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Often, data organized in matrix form contains missing entries. Further, such data has been observed to exhibit effective low-rank, and has led to interest in the particular problem of low-rank matrix-completion: Given a partially-observed matrix, estimate the missing entries such that the output completion is low-rank. The goal of this thesis is to improve matrix-completion algorithms by explicitly analyzing two sources of information in the observed entries: their locations and their values. First, we provide a categorization of a new approach to matrix-completion, which we call structural. Structural methods quantify the possibility of completion using tests applied only to the locations of known entries. By framing each test as the class of partially-observed matrices that pass the test, we provide the first organizing framework for analyzing the relationship among structural completion methods. Building on the structural approach, we then develop a new algorithm for active matrix-completion that is combinatorial in nature. The algorithm uses just the locations of known entries to suggest a small number of queries to be made on the missing entries that allow it to produce a full and accurate completion. If a budget is placed on the number of queries, the algorithm outputs a partial completion, indicating which entries it can and cannot accurately estimate given the observations at hand. Finally, we propose a local approach to matrix-completion that analyzes the values of the observed entries to discover a structure that is more fine-grained than the traditional low-rank assumption. Motivated by the Singular Value Decomposition, we develop an algorithm that finds low-rank submatrices using only the first few singular vectors of a matrix. By completing low-rank submatrices separately from the rest of the matrix, the local approach to matrix-completion produces more accurate reconstructions than traditional algorithms.