Document Open Access Logo

Simple Norm Bounds for Polynomial Random Matrices via Decoupling

Authors Madhur Tulsiani, June Wu

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.ITCS.2025.91.pdf
  • Filesize: 1.63 MB
  • 22 pages
HTML5 Logo HTML (experimental)
LIPIcs.ITCS.2025.91.html

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probability and statistics
  • Theory of computation → Theory and algorithms for application domains
Keywords
  • Matrix Concentration
  • Decoupling
  • Graph Matrices

Metrics

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

Abstract

We present a new method for obtaining norm bounds for random matrices, where each entry is a low-degree polynomial in an underlying set of independent real-valued random variables. Such matrices arise in a variety of settings in the analysis of spectral and optimization algorithms, which require understanding the spectrum of a random matrix depending on data obtained as independent samples.
Using ideas of decoupling and linearization from analysis, we show a simple way of expressing norm bounds for such matrices, in terms of matrices of lower-degree polynomials corresponding to derivatives. Iterating this method gives a simple bound with an elementary proof, which can recover many bounds previously required more involved techniques.

Cite As Get BibTex

Madhur Tulsiani and June Wu. Simple Norm Bounds for Polynomial Random Matrices via Decoupling. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 91:1-91:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.91

Author Details

Madhur Tulsiani
  • Toyota Technological Institute at Chicago, IL, USA
June Wu
  • University of Chicago, IL, USA

Funding

  • Tulsiani, Madhur: NSF grants CCF-1816372 and CCF-2326685.

References

  1. Radosław Adamczak and Paweł Wolff. Concentration inequalities for non-lipschitz functions with bounded derivatives of higher order. Probability Theory and Related Fields, 162(3):531-586, 2015. Google Scholar
  2. Kwangjun Ahn, Dhruv Medarametla, and Aaron Potechin. Graph matrices: Norm bounds and applications. arXiv preprint, 2021. URL: https://arxiv.org/abs/1604.03423.
  3. Richard Aoun, Marwa Banna, and Pierre Youssef. Matrix poincaré inequalities and concentration. Advances in Mathematics, 371:107251, 2020. Google Scholar
  4. Mitali Bafna, Jun-Ting Hsieh, Pravesh K. Kothari, and Jeff Xu. Polynomial-time power-sum decomposition of polynomials. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 956-967, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00094.
  5. Afonso S Bandeira, March T Boedihardjo, and Ramon van Handel. Matrix concentration inequalities and free probability. Inventiones Mathematicae, pages 1-69, 2023. Google Scholar
  6. Nikhil Bansal, Haotian Jiang, and Raghu Meka. Resolving matrix Spencer conjecture up to poly-logarithmic rank. In Proceedings of the 55th ACM Symposium on Theory of Computing, pages 1814-1819, 2023. URL: https://doi.org/10.1145/3564246.3585103.
  7. Boaz Barak, Samuel Hopkins, Jonathan Kelner, Pravesh K Kothari, Ankur Moitra, and Aaron Potechin. A nearly tight sum-of-squares lower bound for the planted clique problem. SIAM Journal on Computing, 48(2):687-735, 2019. URL: https://doi.org/10.1137/17M1138236.
  8. Sergey G Bobkov, Friedrich Götze, and Holger Sambale. Higher order concentration of measure. Communications in Contemporary Mathematics, 21(03):1850043, 2019. Google Scholar
  9. Tatiana Brailovskaya and Ramon van Handel. Universality and sharp matrix concentration inequalities. arXiv preprint, 2022. URL: https://arxiv.org/abs/2201.05142.
  10. Jingqiu Ding, Tommaso d’Orsi, Chih-Hung Liu, David Steurer, and Stefan Tiegel. Fast algorithm for overcomplete order-3 tensor decomposition. In Conference on Learning Theory, pages 3741-3799. PMLR, 2022. URL: https://proceedings.mlr.press/v178/ding22a.html.
  11. Rong Ge, Qingqing Huang, and Sham M. Kakade. Learning mixtures of gaussians in high dimensions. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, STOC '15, pages 761-770, New York, NY, USA, 2015. Association for Computing Machinery. URL: https://doi.org/10.1145/2746539.2746616.
  12. Samuel B Hopkins, Tselil Schramm, and Jonathan Shi. A robust spectral algorithm for overcomplete tensor decomposition. In Conference on Learning Theory, pages 1683-1722. PMLR, 2019. URL: http://proceedings.mlr.press/v99/hopkins19b.html.
  13. Samuel B Hopkins, Tselil Schramm, Jonathan Shi, and David Steurer. Fast spectral algorithms from sum-of-squares proofs: tensor decomposition and planted sparse vectors. In Proceedings of the 48th ACM Symposium on Theory of Computing, pages 178-191, 2016. URL: https://doi.org/10.1145/2897518.2897529.
  14. De Huang and Joel A. Tropp. From poincaré inequalities to nonlinear matrix concentration. Bernoulli, 2020. Google Scholar
  15. De Huang and Joel A. Tropp. Nonlinear matrix concentration via semigroup methods. Electronic Journal of Probability, 26:Art. No. 8, January 2021. Google Scholar
  16. Chris Jones, Aaron Potechin, Goutham Rajendran, Madhur Tulsiani, and Jeff Xu. Sum-of-squares lower bounds for sparse independent set. In Proceedings of the 62nd IEEE Symposium on Foundations of Computer Science, 2021. Google Scholar
  17. Jeong Han Kim and Van H Vu. Concentration of multivariate polynomials and its applications. Combinatorica, 20(3):417-434, 2000. URL: https://doi.org/10.1007/S004930070014.
  18. Bohdan Kivva and Aaron Potechin. Exact nuclear norm, completion and decomposition for random overcomplete tensors via degree-4 sos. arXiv preprint arXiv:2011.09416, 2020. URL: https://arxiv.org/abs/2011.09416.
  19. Stanislaw Kwapien. Decoupling Inequalities for Polynomial Chaos. The Annals of Probability, 15(3):1062-1071, 1987. Google Scholar
  20. Cécilia Lancien and Pierre Youssef. A note on quantum expanders, 2023. URL: https://arxiv.org/abs/2302.07772.
  21. Rafał Latała. Estimates of moments and tails of gaussian chaoses. The Annals of Probability, 34(6):2315-2331, 2006. Google Scholar
  22. Lester Mackey, Michael Jordan, Richard Chen, Brendan Farrell, and Joel Tropp. Matrix concentration inequalities via the method of exchangeable pairs. The Annals of Probability, 42, 2012. Google Scholar
  23. Terry R. McConnell and Murad S. Taqqu. Double integration with respect to symmetric stable processes. Technical report, Technical Report 618, Cornell Univ., 1984. Google Scholar
  24. Ankur Moitra and Alexander S Wein. Spectral methods from tensor networks. In Proceedings of the 51st ACM Symposium on Theory of Computing, pages 926-937, 2019. URL: https://doi.org/10.1145/3313276.3316357.
  25. Ryan O'Donnell and Yu Zhao. Polynomial bounds for decoupling, with applications. In Proceedings of the 31st Conference on Computational Complexity, CCC '16, Dagstuhl, DEU, 2016. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.24.
  26. Daniel Paulin, Lester Mackey, and Joel A. Tropp. Efron–Stein inequalities for random matrices. The Annals of Probability, 44(5):3431-3473, 2016. Google Scholar
  27. Víctor H. Peña and Evarist Giné. Decoupling: From dependence to independence. Springer-Verlag, 1999. Google Scholar
  28. Goutham Rajendran. Nonlinear Random Matrices and Applications to the Sum of Squares Hierarchy. PhD thesis, University of Chicago, 2022. Google Scholar
  29. Goutham Rajendran and Madhur Tulsiani. Concentration of polynomial random matrices via efron-stein inequalities. Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2023. Google Scholar
  30. Holger Rauhut. Compressive sensing and structured random matrices. In Theoretical Foundations and Numerical Methods for Sparse Recovery, pages 1-92. De Gruyter, Berlin, New York, 2010. Google Scholar
  31. Warren Schudy and Maxim Sviridenko. Bernstein-like concentration and moment inequalities for polynomials of independent random variables: multilinear case. arXiv preprint, 2011. URL: https://arxiv.org/abs/1109.5193.
  32. Warren Schudy and Maxim Sviridenko. Concentration and moment inequalities for polynomials of independent random variables. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 437-446. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.37.
  33. Khalid Shebrawi and Hussien Albadawi. Trace inequalities for matrices. Bulletin of the Australian Mathematical Society, 87(1):139-148, 2013. Google Scholar
  34. Joel A. Tropp. An introduction to matrix concentration inequalities. Foundations and Trends in Machine Learning, 8(1-2):1-230, 2015. Google Scholar
  35. Roman Vershynin. High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge University Press, 2018. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact