Tasuku Soma

Finding a low-rank basis in a matrix subspace

For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained by the tensor CP decomposition. For the higher rank case, the situation is not as straightforward. In this work we present an algorithm based on a greedy process applicable to higher rank problems. Our algorithm first estimates the minimum rank by applying soft singular value thresholding to a nuclear norm relaxation, and then computes a matrix with that rank using the method of alternating projections. We provide local convergence results, and compare our algorithm with several alternative approaches. Applications include data compression beyond the classical truncated SVD, computing accurate eigenvectors of a near-multiple eigenvalue, image separation and graph Laplacian eigenproblems.

Maximizing Monotone Submodular Functions over the Integer Lattice

The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade. In this paper, we address the problem for functions defined over the integer lattice. Suppose that a non-negative monotone submodular function

Maximizing monotone submodular functions over the integer lattice

f:\mathbb {Z}_+^n \rightarrow \mathbb {R}_+

Fast Deterministic Algorithms for Matrix Completion Problems

is given via an evaluation oracle. Assume further that f satisfies the diminishing return property, which is not an immediate consequence of the submodularity when the domain is the integer lattice. Then, we show polynomial-time

Multicasting in linear deterministic relay network by matrix completion

1-1/e-\epsilon

Optimal Budget Allocation: Theoretical Guarantee and Efficient Algorithm

-approximation algorithm for cardinality constraints, polymatroid constraints, and knapsack constraints. For a cardinality constraint, we also show a