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.
KeywordsConstrained matrix factorization Relaxation methods Multiplicative updates Phase-mapping
We thank Ronan Le Bras and Rich Bernstein for fruitful discussion. This material is supported by NSF awards CCF-1522054, CNS-0832782, CNS-1059284, IIS-1344201 and W911-NF-14-1-0498. Experiments were supported through the Office of Science of the U.S. Department of Energy under Award No. DE-SC0004993. Use of the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515.
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