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

  • Junwen Bai
  • Johan BjorckEmail author
  • Yexiang Xue
  • Santosh K. Suram
  • John Gregoire
  • Carla Gomes
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10335)


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.


Constrained 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.


  1. 1.
    Atkins, P., De Paula, J.: Atkins’ Physical Chemistry, p. 77. Oxford University Press, New York (2006)Google Scholar
  2. 2.
    Ermon, S., Bras, R.L., Suram, S.K., Gregoire, J.M., Gomes, C., Selman, B., Van Dover, R.B.: Pattern decomposition with complex combinatorial constraints: application to materials discovery. arXiv preprint arXiv:1411.7441 (2014)
  3. 3.
    Ermon, S., Bras, R., Gomes, C.P., Selman, B., Dover, R.B.: SMT-aided combinatorial materials discovery. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 172–185. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-31612-8_14 CrossRefGoogle Scholar
  4. 4.
    Le Bras, R., Bernstein, R., Suram, S.K., Gregoire, J.M., Selman, B., Gomes, C.P., van Dover, R.B.: A computational challenge problem in materials discovery: synthetic problem generator and real-world datasets (2014)Google Scholar
  5. 5.
    LeBras, R., Damoulas, T., Gregoire, J.M., Sabharwal, A., Gomes, C.P., Dover, R.B.: Constraint reasoning and kernel clustering for pattern decomposition with scaling. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 508–522. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-23786-7_39 CrossRefGoogle Scholar
  6. 6.
    Lee, D.D., Seung, H.S.: Learning the parts of objects by non-negative matrix factorization. Nature 401(6755), 788–791 (1999)CrossRefGoogle Scholar
  7. 7.
    Lee, D.T., Schachter, B.J.: Two algorithms for constructing a delaunay triangulation. Int. J. Comput. Inf. Sci. 9(3), 219–242 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Long, C., Bunker, D., Li, X., Karen, V., Takeuchi, I.: Rapid identification of structural phases in combinatorial thin-film libraries using X-ray diffraction and non-negative matrix factorization. Rev. Sci. Instrum. 80(10), 103902 (2009)CrossRefGoogle Scholar
  9. 9.
    Long, C., Hattrick-Simpers, J., Murakami, M., Srivastava, R., Takeuchi, I., Karen, V.L., Li, X.: Rapid structural mapping of ternary metallic alloy systems using the combinatorial approach and cluster analysis. Rev. Sci. Instrum. 78(7), 072217 (2007)CrossRefGoogle Scholar
  10. 10.
    Shashua, A., Hazan, T.: Non-negative tensor factorization with applications to statistics and computer vision. In: Proceedings of the 22nd International Conference on Machine Learning, pp. 792–799. ACM (2005)Google Scholar
  11. 11.
    Smaragdis, P.: Non-negative matrix factor deconvolution; extraction of multiple sound sources from monophonic inputs. In: Puntonet, C.G., Prieto, A. (eds.) ICA 2004. LNCS, vol. 3195, pp. 494–499. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-30110-3_63 CrossRefGoogle Scholar
  12. 12.
    Suram, S.K., Xue, Y., Bai, J., Bras, R.L., Rappazzo, B., Bernstein, R., Bjorck, J., Zhou, L., van Dover, R.B., Gomes, C.P., et al.: Automated phase mapping with agilefd and its application to light absorber discovery in the V-Mn-Nb oxide system. arXiv preprint arXiv:1610.02005 (2016)
  13. 13.
    Vavasis, S.A.: On the complexity of nonnegative matrix factorization. SIAM J. Optim. 20(3), 1364–1377 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Xue, Y., Bai, J., Le Bras, R., Rappazzo, B., Bernstein, R., Bjorck, J., Longpre, L., Suram, S., van Dover, B., Gregoire, J., Gomes, C.: Phase mapper: an AI platform to accelerate high throughput materials discovery. In: Twenty-Ninth International Conference on Innovative Applications of Artificial Intelligence (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Junwen Bai
    • 1
  • Johan Bjorck
    • 1
    Email author
  • Yexiang Xue
    • 1
  • Santosh K. Suram
    • 2
  • John Gregoire
    • 2
  • Carla Gomes
    • 1
  1. 1.Department of Computer ScienceCornell UniversityIthacaUSA
  2. 2.Joint Center for Artificial Photosynthesis, California Institute of TechnologyPasadenaUSA

Personalised recommendations