Document Open Access Logo

Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory

Authors Peter Bürgisser, Ankit Garg, Rafael Oliveira, Michael Walter, Avi Wigderson



PDF
Thumbnail PDF

File

LIPIcs.ITCS.2018.24.pdf
  • Filesize: 0.61 MB
  • 20 pages

Document Identifiers

Author Details

Peter Bürgisser
Ankit Garg
Rafael Oliveira
Michael Walter
Avi Wigderson

Cite AsGet BibTex

Peter Bürgisser, Ankit Garg, Rafael Oliveira, Michael Walter, and Avi Wigderson. Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 24:1-24:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ITCS.2018.24

Abstract

Alternating minimization heuristics seek to solve a (difficult) global optimization task through iteratively solving a sequence of (much easier) local optimization tasks on different parts (or blocks) of the input parameters. While popular and widely applicable, very few examples of this heuristic are rigorously shown to converge to optimality, and even fewer to do so efficiently. In this paper we present a general framework which is amenable to rigorous analysis, and expose its applicability. Its main feature is that the local optimization domains are each a group of invertible matrices, together naturally acting on tensors, and the optimization problem is minimizing the norm of an input tensor under this joint action. The solution of this optimization problem captures a basic problem in Invariant Theory, called the null-cone problem. This algebraic framework turns out to encompass natural computational problems in combinatorial optimization, algebra, analysis, quantum information theory, and geometric complexity theory. It includes and extends to high dimensions the recent advances on (2-dimensional) operator scaling. Our main result is a fully polynomial time approximation scheme for this general problem, which may be viewed as a multi-dimensional scaling algorithm. This directly leads to progress on some of the problems in the areas above, and a unified view of others. We explain how faster convergence of an algorithm for the same problem will allow resolving central open problems. Our main techniques come from Invariant Theory, and include its rich non-commutative duality theory, and new bounds on the bitsizes of coefficients of invariant polynomials. They enrich the algorithmic toolbox of this very computational field of mathematics, and are directly related to some challenges in geometric complexity theory (GCT).
Keywords
  • alternating minimization
  • tensors
  • scaling
  • quantum marginal problem
  • null cone
  • invariant theory
  • geometric complexity theory

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Velleda Baldoni, Michèle Vergne, and Michael Walter. Computation of dilated Kronecker coefficients. Journal of Symbolic Computation, 84:113-146, 2018. Google Scholar
  2. Charles H Bennett, Sandu Popescu, Daniel Rohrlich, John A Smolin, and Ashish V Thapliyal. Exact and asymptotic measures of multipartite pure-state entanglement. Physical Review A, 63(1):012307, 2000. Google Scholar
  3. Jonah Blasiak, Thomas Church, Henry Cohn, Joshua A. Grochow, Eric Naslund, William F Sawin, and Chris Umans. On cap sets and the group-theoretic approach to matrix multiplication. Discrete Analysis, 2017. URL: http://dx.doi.org/10.19086/da.1245.
  4. Herm Brascamp and Elliot Lieb. Best constants in Young’s inequality, its converse and its generalization to more than three functions. Advances in Mathematics, 20:151-172, 1976. Google Scholar
  5. Peter Bürgisser, Matthias Christandl, Ketan D Mulmuley, and Michael Walter. Membership in moment polytopes is in NP and coNP. SIAM Journal on Computing, 46(3):972-991, 2017. Google Scholar
  6. Peter Bürgisser, Ankit Garg, Rafael Mendes de Oliveira, Michael Walter, and Avi Wigderson. Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory. CoRR, abs/1711.08039, 2017. URL: http://arxiv.org/abs/1711.08039.
  7. Arthur Cayley. On linear transformations. Cambridge and Dublin Mathematical Journal, 1:104-122, 1846. Google Scholar
  8. Imre Csiszár and Gábor Tusnády. Information geometry and alternating minimization procedures. Stat. Decis. Suppl., 1:205-237, 1984. Google Scholar
  9. H. Derksen and G. Kemper. Computational Invariant Theory, volume 130. Springer-Verlag, Berlin, 2002. Google Scholar
  10. Harm Derksen. Polynomial bounds for rings of invariants. Proceedings of the American Mathematical Society, 129(4):955-963, 2001. Google Scholar
  11. Ankit Garg, Leonid Gurvits, Rafael Mendes de Oliveira, and Avi Wigderson. A deterministic polynomial time algorithm for non-commutative rational identity testing. CoRR, abs/1511.03730, 2015. URL: http://arxiv.org/abs/1511.03730.
  12. Ankit Garg, Leonid Gurvits, Rafael Mendes de Oliveira, and Avi Wigderson. Algorithmic aspects of brascamp-lieb inequalities. CoRR, abs/1607.06711, 2016. URL: http://arxiv.org/abs/1607.06711.
  13. Valentina Georgoulas, Joel W Robbin, and Dietmar A Salamon. The moment-weight inequality and the Hilbert-Mumford criterion. math, abs/1311.0410, 2013. URL: http://arxiv.org/abs/1311.0410.
  14. Leonid Gurvits. Classical complexity and quantum entanglement. Journal of Computer and System Sciences, 69(3):448-484, 2004. Google Scholar
  15. Leonid Gurvits. Hyperbolic polynomials approach to van der waerden/schrijver-valiant like conjectures: sharper bounds, simpler proofs and algorithmic applications. STOC, pages 417-426, 2006. Google Scholar
  16. Leonid Gurvits and Peter N. Yianilos. The deflation-inflation method for certain semidefinite programming and maximum determinant completion problems. Technical Report, NECI, 1998. Google Scholar
  17. Moritz Hardt. Understanding alternating minimization for matrix completion. FOCS, pages 651-660, 2014. Google Scholar
  18. David Hilbert. Ueber die Theorie der algebraischen Formen. Mathematische Annalen, 36(4):473-534, 1890. Google Scholar
  19. David Hilbert. Uber die vollen Invariantensysteme. Math. Ann., 42:313-370, 1893. Google Scholar
  20. Christian Ikenmeyer, Ketan D. Mulmuley, and Michael Walter. On vanishing of Kronecker coefficients. Computational Complexity, 2017. Google Scholar
  21. Gabor Ivanyos, Youming Qiao, and K. V. Subrahmanyam. Non-commutative Edmonds' problem and matrix semi-invariants. Computational Complexity, 2016. Google Scholar
  22. Gábor Ivanyos, Youming Qiao, and K. V. Subrahmanyam. Constructive noncommutative rank computation in deterministic polynomial time over fields of arbitrary characteristics. ITCS, 2017. Google Scholar
  23. Prateek Jain, Praneeth Netrapalli, and Sujay Sanghavi. Low-rank matrix completion using alternating minimization. STOC, pages 665-674, 2013. Google Scholar
  24. George Kempf and Linda Ness. The length of vectors in representation spaces. In Algebraic Geometry, volume 732 of Lecture Notes in Math., pages 233-243. Springer, 1979. URL: http://dx.doi.org/10.1007/BFb0066647.
  25. Frances Kirwan. Cohomology of quotients in symplectic and algebraic geometry, volume 31 of Mathematical Notes. Princeton University Press, 1984. Google Scholar
  26. Alexander Klyachko. Coherent states, entanglement, and geometric invariant theory. Quantum Physics, abs/quant-ph/0206012, 2002. URL: http://arxiv.org/abs/quant-ph/0206012.
  27. Alexander A. Klyachko. Quantum marginal problem and n-representability. Journal of Physics: Conference Series, 36(1):72, 2006. URL: http://dx.doi.org/10.1088/1742-6596/36/1/014.
  28. Carlton Lemke and J. T. Howson. Equilibrium points of bimatrix games. SIAM Journal on Applied Mathematics, 12(2):413-423, 1964. Google Scholar
  29. Elliot Lieb. Gaussian kernels have only Gaussian maximizers. Inventiones Mathematicae, 102:179-208, 1990. Google Scholar
  30. Nati Linial, Alex Samorodnitsky, and Avi Wigderson. A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents. STOC, pages 644-652, 1998. Google Scholar
  31. Ketan Mulmuley. Geometric Complexity Theory V: Equivalence between Blackbox Derandomization of Polynomial Identity Testing and Derandomization of Noether’s Normalization Lemma. FOCS, pages 629-638, 2012. Google Scholar
  32. Ketan D. Mulmuley. Geometric Complexity Theory V: Efficient algorithms for Noether Normalization. J. Amer. Math. Soc., 30(1):225-309, 2017. URL: http://dx.doi.org/10.1090/jams/864.
  33. Ketan D. Mulmuley and Milind Sohoni. Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems. SIAM J. Comput, 31(2):496-526, 2001. Google Scholar
  34. David Mumford. Geometric invariant theory. Springer-Verlag, Berlin-New York, 1965. Google Scholar
  35. Linda Ness and David Mumford. A stratification of the null cone via the moment map. American Journal of Mathematics, 106(6):1281-1329, 1984. URL: http://www.jstor.org/stable/2374395.
  36. John von Neumann. Functional Operators, Vol. II: The Geometry of Orthogonal Spaces, volume 22 of Annals of Math. Studies. Princeton University Press, 1950. Google Scholar
  37. Meisam Razaviyayn, Mingyi Hong, and Zhi-Quan Luo. A unified convergence analysis of block successive minimization methods for nonsmooth optimization. SIAM Journal on Optimization, 23(2):1126-1153, 2013. Google Scholar
  38. R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. The Annals of Mathematical Statistics, 35:876-879, 1964. Google Scholar
  39. Bernd Sturmfels. Algorithms in Invariant Theory. Texts &Monographs in Symbolic Computation. Springer, 2nd edition, 2008. Google Scholar
  40. B. Sury. An elementary proof of the Hilbert-Mumford criterion. Electron. J. Linear Algebra, 7:174-177, 2000. URL: http://dx.doi.org/10.13001/1081-3810.1053.
  41. Frank Verstraete, Jeroen Dehaene, and De Moor Bart. Normal forms and entanglement measures for multipartite quantum states. Physical Review A, 68(1), 2003. Google Scholar
  42. Michael Walter. Multipartite Quantum States and their Marginals. PhD thesis, ETH Zurich, 2014. Google Scholar
  43. Michael Walter, Brent Doran, David Gross, and Matthias Christandl. Entanglement polytopes: multiparticle entanglement from single-particle information. Science, 340(6137):1205-1208, 2013. Google Scholar
  44. Yu Wang, Wotao Yin, and Jinshan Zeng. Global convergence of ADMM in nonconvex nonsmooth optimization. math, abs/1511.06324, 2015. URL: http://arxiv.org/abs/1511.06324.
  45. Chris Woodward. Moment maps and geometric invariant theory. math, abs/0912.1132, 2011. URL: http://arxiv.org/abs/0912.1132.
  46. Yangyang Xu and Wotao Yin. A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion. SIAM Journal on Imaging Sciences, 6(3):1758-1789, 2013. Google Scholar
  47. Hongyi Zhang and Suvrit Sra. First-order methods for geodesically convex optimization. math, abs/1602.06053, 2016. URL: http://arxiv.org/abs/1602.06053.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail