Skip to main content
SpringerLink
Log in

Matrix Factorization Ranks Via Polynomial Optimization

  • Chapter
  • First Online:
Polynomial Optimization, Moments, and Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 206))

  • 177 Accesses

Abstract

In light of recent data science trends, new interest has fallen in alternative matrix factorizations. By this, we mean various ways of factorizing particular data matrices so that the factors have special properties and reveal insights into the original data. We are interested in the specialized ranks associated with these factorizations, but they are usually difficult to compute. In particular, we consider the nonnegative-, completely positive-, and separable ranks. We focus on a general tool for approximating factorization ranks, the moment hierarchy, a classical technique from polynomial optimization, further augmented by exploiting ideal-sparsity. Contrary to other examples of sparsity, the resulting sparse hierarchy yields equally strong, if not superior, bounds while potentially delivering a speed-up in computation.

This work is supported by the European Union’s Framework Programme for Research and Innovation Horizon 2020 under the Marie Skłodowska-Curie Actions Grant Agreement No. 813211 (POEMA).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    See the code repository: https://github.com/JAndriesJ/ju-cp-rank.

References

  1. Andersen, E.D., Andersen, K.D.: The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm. In: Frenk, H., Roos, K., Terlaky, T., Zhang, S. (eds.) High Performance Optimization, pp. 197–232. Springer US, Boston (2000)

    Chapter  Google Scholar 

  2. Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific, Singapore (2003)

    Book  Google Scholar 

  3. Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59(1), 65–98 (2017)

    Article  MathSciNet  Google Scholar 

  4. Bomze, I.M., Schachinger, W., Ullrich, R.: From seven to eleven: completely positive matrices with high cp-rank. Linear Algebra Appl. 459, 208–221 (2014)

    Article  MathSciNet  Google Scholar 

  5. Bomze, I.M., Schachinger, W., Ullrich, R.: New lower bounds and asymptotics for the cp-rank. SIAM J. Matrix Anal. Appl. 36(1), 20–37 (2015)

    Article  MathSciNet  Google Scholar 

  6. Bron, C., Kerbosch, J.: Algorithm 457: finding all cliques of an undirected graph. Commun. ACM 16(9), 575–577 (1973)

    Article  Google Scholar 

  7. Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120(2), 479–495 (2009)

    Article  MathSciNet  Google Scholar 

  8. Choi, M.-D.: Positive linear maps. In: Operator Algebras and Applications, vol. 38. Proc. Sympos. Pure Math., pp. 583–590 (1982)

    Google Scholar 

  9. Cichocki, A., Zdunek, R., Phan, A.H., Amari, S.-i.: Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-Way Data Analysis and Blind Source Separation. Wiley Publishing, Hoboken (2009)

    Google Scholar 

  10. Curto, R.E., Fialkow, L.A.: Solution of the truncated complex moment problem for flat data. Am. Math. Soc. 119, 568 (1996)

    MathSciNet  Google Scholar 

  11. Curto, R.E., Fialkow, L.A.: The truncated complex -moment problem. Trans. Am. Math. Soc. 352, 2825–2855 (2000)

    Article  MathSciNet  Google Scholar 

  12. de Klerk, E., Laurent, M.: A survey of semidefinite programming approaches to the generalized problem of moments and their error analysis. In: Araujo, C., Benkart, G., Praeger, C.E., Tanbay, B. (eds.) World Women in Mathematics 2018: Proceedings of the First World Meeting for Women in Mathematics (WM), pp. 17–56. Springer International Publishing, Cham (2019)

    Chapter  Google Scholar 

  13. de las Cuevas, G., Netzer, T.: Mixed states in one spatial dimension: decompositions and correspondence with nonnegative matrices. J. Math. Phys. 61(4), 041901 (2020). https://doi.org/10.48550/arXiv.1907.03664

  14. De las Cuevas, G., Drescher, T., Netzer, T.: Separability for mixed states with operator Schmidt rank two. Quantum 3, 203 (2019)

    Google Scholar 

  15. Dickinson, P.J.C., Gijben, L.: On the computational complexity of membership problems for the completely positive cone and its dual. Comput. Optim. Appl. 57, 403–415 (2014)

    Article  MathSciNet  Google Scholar 

  16. Divincenzo, D.P., Terhal, B.M., Thapliyal, A.V.: Optimal decompositions of barely separable states. J. Mod. Opt. 47(2–3), 377–385 (2000)

    Article  MathSciNet  Google Scholar 

  17. Drew, J.H., Johnson, C.R., Loewy, R.: Completely positive matrices associated with M-matrices. Linear Multilinear Algebra 37, 303–310 (1994)

    Article  MathSciNet  Google Scholar 

  18. Dunning, I., Huchette, J., Lubin, M.: JuMP: a modeling language for mathematical optimization. SIAM Rev. 59, 295–320 (2017)

    Article  MathSciNet  Google Scholar 

  19. Fawzi, H., Parrilo, P.A.: Self-scaled bounds for atomic cone ranks: applications to nonnegative rank and cp-rank. Math. Program. 158(1), 417–465 (2016)

    Article  MathSciNet  Google Scholar 

  20. Fawzi, H., Gouveia, J., Parrilo, P.A., Robinson, R.Z., Thomas, R.R.: Positive semidefinite rank. Math. Program. 153(1), 133–177 (2015)

    Article  MathSciNet  Google Scholar 

  21. Fawzi, H., Gouveia, J., Parrilo, P.A., Saunderson, J., Thomas, R.R.: Lifting for simplicity: concise descriptions of convex sets. SIAM Rev. 64(4), 866–918 (2022)

    Article  MathSciNet  Google Scholar 

  22. Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., de Wolf, R.: Exponential lower bounds for polytopes in combinatorial optimization. J. ACM 62(2), 1–23 (2015)

    Article  MathSciNet  Google Scholar 

  23. Gharibian, S.: Strong NP-hardness of the quantum separability problem. Quantum Inf. Comput. 10, 343–360 (2010)

    MathSciNet  Google Scholar 

  24. Gillis, N.: Nonnegative Matrix Factorization. Society for Industrial and Applied Mathematics, Philadelphia (2020)

    Book  Google Scholar 

  25. Gillis, N., Glineur, F.: On the geometric interpretation of the nonnegative rank. Linear Algebra Appl. 437(11), 2685–2712 (2012)

    Article  MathSciNet  Google Scholar 

  26. Gribling, S., de Laat, D., Laurent, M.: Lower bounds on matrix factorization ranks via noncommutative polynomial optimization. Found. Comput. Math. 19, 1013–1070 (2019)

    Article  MathSciNet  Google Scholar 

  27. Gribling, S., Laurent, M., Steenkamp, A.: Bounding the separable rank via polynomial optimization. Linear Algebra Appl. 648, 1–55 (2022)

    Article  MathSciNet  Google Scholar 

  28. Gurvits, L.: Classical deterministic complexity of Edmonds’ problem and quantum entanglement. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, STOC ’03, pp. 10–19. Association for Computing Machinery, New York (2003)

    Google Scholar 

  29. Hall, M.: Combinatorial Theory. Wiley, Hoboken (1988)

    Book  Google Scholar 

  30. Henrion, D., Lasserre, J.-B.: Detecting global optimality and extracting solutions in GloptiPoly. In: Henrion, D., Garulli, A. (eds.) Positive Polynomials in Control, pp. 293–310. Springer, Berlin (2005)

    Chapter  Google Scholar 

  31. Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)

    Article  MathSciNet  Google Scholar 

  32. Korda, M., Laurent, M., Magron, V., Steenkamp, A.: Exploiting ideal-sparsity in the generalized moment problem with application to matrix factorization ranks. Math. Program. (2023)

    Google Scholar 

  33. Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, Singapore (2009)

    Google Scholar 

  34. Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry, pp. 157–270. Springer, New York (2009)

    Chapter  Google Scholar 

  35. Lee, D.D., Seung, H.S.: Learning the parts of objects by non-negative matrix factorization. Nature 401, 788–791 (1999)

    Article  Google Scholar 

  36. Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics, Philadelphia (1994)

    Book  Google Scholar 

  37. Nie, J.: The A-truncated K-moment problem. Found. Comput. Math. 14(6), 1243–1276 (2014)

    Article  MathSciNet  Google Scholar 

  38. Shaked-Monderer, N., Berman, A.: Copositive and Completely Positive Matrices. World Scientific Publishing, Singapore (2021)

    Book  Google Scholar 

  39. Shaked-Monderer, N., Bomze, I.M., Jarre, F., Schachinger, W.: On the cp-rank and minimal cp factorizations of a completely positive matrix. SIAM J. Matrix Anal. Appl. 34, 355–368 (2013)

    Article  MathSciNet  Google Scholar 

  40. Sidiropoulos, n.d., Bro, R.: On the uniqueness of multilinear decomposition of N-way arrays. J. Chemometrics 14(3), 229–239 (2000)

    Google Scholar 

  41. Uhlmann, A.: Entropy and optimal decompositions of states relative to a maximal commutative subalgebra. Open. Syst. Inf. Dyn. 5(3), 209–228 (1998)

    Article  Google Scholar 

  42. Vavasis, S.A.: On the complexity of nonnegative matrix factorization. SIAM J. Optim. 20, 1364–1377 (2009)

    Article  MathSciNet  Google Scholar 

  43. Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. In: STOC ’88 (1988)

    Google Scholar 

Download references

Acknowledgements

We want to thank Prof. Dr. Monique Laurent for proofreading several drafts of this chapter and providing key insights when the author’s knowledge was lacking. We also thank the editors for the opportunity to consolidate and share our expertise on this fascinating topic.

Author information

Authors and Affiliations

  1. Centrum Wiskunde & Informatica (CWI), Amsterdam, the Netherlands

    Andries Steenkamp

Authors
  1. Andries Steenkamp
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andries Steenkamp .

Editor information

Editors and Affiliations

  1. University of Birmingham, Birmingham, UK

    Michal Kočvara

  2. Aromath, Inria at Université Côte d’Azur, Sophia Antipolis, France

    Bernard Mourrain

  3. UiT The Arctic University of Norway, Tromsø, Norway

    Cordian Riener

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Steenkamp, A. (2023). Matrix Factorization Ranks Via Polynomial Optimization. In: Kočvara, M., Mourrain, B., Riener, C. (eds) Polynomial Optimization, Moments, and Applications. Springer Optimization and Its Applications, vol 206. Springer, Cham. https://doi.org/10.1007/978-3-031-38659-6_5

Download citation

Publish with us

Policies and ethics

Search

Navigation