Pursuits in Structured Non-Convex Matrix Factorizations

Efficiently representing real world data in a succinct and parsimonious manner is of central importance in many fields. We present a generalized greedy pursuit framework, allowing us to efficiently solve structured matrix factorization problems, where the factors are allowed to be from arbitrary sets of structured vectors. Such structure may include sparsity, non-negativeness, order, or a combination thereof. The algorithm approximates a given matrix by a linear combination of few rank-1 matrices, each factorized into an outer product of two vector atoms of the desired structure. For the non-convex subproblems of obtaining good rank-1 structured matrix atoms, we employ and analyze a general atomic power method. In addition to the above applications, we prove linear convergence for generalized pursuit variants in Hilbert spaces - for the task of approximation over the linear span of arbitrary dictionaries - which generalizes OMP and is useful beyond matrix problems. Our experiments on real datasets confirm both the efficiency and also the broad applicability of our framework in practice

On Approximation Guarantees for Greedy Low Rank Optimization

We provide new approximation guarantees for greedy low rank matrix estimation under standard assumptions of restricted strong convexity and smoothness. Our novel analysis also uncovers previously unknown connections between the low rank estimation and combinatorial optimization, so much so that our bounds are reminiscent of corresponding approximation bounds in submodular maximization. Additionally, we also provide statistical recovery guarantees. Finally, we present empirical comparison of greedy estimation with established baselines on two important real-world problems