Relaxation Methods for Constrained Matrix Factorization Problems: Solving the Phase Mapping Problem in Materials Discovery

Matrix factorization is a robust and widely adopted technique in data science, in which a given matrix is decomposed as the product of low rank matrices. We study a challenging constrained matrix factorization problem in materials discovery, the so-called phase mapping problem. We introduce a novel lazy Iterative Agile Factor Decomposition (IAFD) approach that relaxes and postpones non-convex constraint sets (the lazy constraints), iteratively enforcing them when violations are detected. IAFD interleaves multiplicative gradient-based updates with efficient modular algorithms that detect and repair constraint violations, while still ensuring fast run times. Experimental results show that IAFD is several orders of magnitude faster and its solutions are also in general considerably better than previous approaches. IAFD solves a key problem in materials discovery while also paving the way towards tackling constrained matrix factorization problems in general, with broader implications for data science.