Pursuits in Structured Non-Convex Matrix Factorizations
This work addresses the need for efficient data representation in fields like machine learning and signal processing, offering a flexible framework for structured matrix factorization, though it is incremental as it builds on existing pursuit methods.
The authors tackled the problem of efficiently representing real-world data through structured matrix factorization by introducing a generalized greedy pursuit framework that approximates matrices with few rank-1 structured components, achieving linear convergence and demonstrating efficiency and broad applicability in experiments on real datasets.
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.