Querying a Matrix Through Matrix-Vector Products

Authors Xiaoming Sun, David P. Woodruff, Guang Yang, Jialin Zhang



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2019.94.pdf
  • Filesize: 0.59 MB
  • 16 pages

Document Identifiers

Author Details

Xiaoming Sun
  • CAS Key Lab of Network Data Science and Technology, Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China
David P. Woodruff
  • Carnegie Mellon University, Pittsburgh, PA, US
Guang Yang
  • Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
  • Conflux, Beijing, China
Jialin Zhang
  • CAS Key Lab of Network Data Science and Technology, Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China

Acknowledgements

We want to thank Roman Vershynin and Yan Shuo Tan for the helpful comments.

Cite AsGet BibTex

Xiaoming Sun, David P. Woodruff, Guang Yang, and Jialin Zhang. Querying a Matrix Through Matrix-Vector Products. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 94:1-94:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.94

Abstract

We consider algorithms with access to an unknown matrix M in F^{n x d} via matrix-vector products, namely, the algorithm chooses vectors v^1, ..., v^q, and observes Mv^1, ..., Mv^q. Here the v^i can be randomized as well as chosen adaptively as a function of Mv^1, ..., Mv^{i-1}. Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, this fundamental model does not appear to have been studied on its own, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Lower bounds and information complexity
Keywords
  • Communication complexity
  • linear algebra
  • sketching

Metrics

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

References

  1. Yuqing Ai, Wei Hu, Yi Li, and David P. Woodruff. New Characterizations in Turnstile Streams with Applications. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pages 20:1-20:22, 2016. Google Scholar
  2. Noga Alon, Yossi Matias, and Mario Szegedy. The Space Complexity of Approximating the Frequency Moments. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, pages 20-29, 1996. Google Scholar
  3. Sepehr Assadi, Sanjeev Khanna, and Yang Li. On Estimating Maximum Matching Size in Graph Streams. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1723-1742, 2017. Google Scholar
  4. Sepehr Assadi, Sanjeev Khanna, Yang Li, and Grigory Yaroslavtsev. Maximum Matchings in Dynamic Graph Streams and the Simultaneous Communication Model. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1345-1364, 2016. Google Scholar
  5. Khanh Do Ba, Piotr Indyk, Eric Price, and David P. Woodruff. Lower Bounds for Sparse Recovery. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 1190-1197, 2010. Google Scholar
  6. Maria-Florina Balcan, Yi Li, David P. Woodruff, and Hongyang Zhang. Testing Matrix Rank, Optimally. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 727-746, 2019. Google Scholar
  7. Anders Björner, László Lovász, and Andrew CC Yao. Linear decision trees: volume estimates and topological bounds. In Proceedings of the twenty-fourth annual ACM symposium on Theory of computing, pages 170-177. ACM, 1992. Google Scholar
  8. Eric Blais, Joshua Brody, and Kevin Matulef. Property Testing Lower Bounds via Communication Complexity. Computational Complexity, 21(2):311-358, 2012. Google Scholar
  9. Vladimir Braverman, Stephen R. Chestnut, Robert Krauthgamer, Yi Li, David P. Woodruff, and Lin Yang. Matrix Norms in Data Streams: Faster, Multi-Pass and Row-Order. In Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholmsmässan, Stockholm, Sweden, July 10-15, 2018, pages 648-657, 2018. Google Scholar
  10. Emmanuel J Candes, Justin K Romberg, and Terence Tao. Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 59(8):1207-1223, 2006. Google Scholar
  11. Talya Eden, Amit Levi, Dana Ron, and C Seshadhri. Approximately counting triangles in sublinear time. SIAM Journal on Computing, 46(5):1603-1646, 2017. Google Scholar
  12. Piotr Indyk, Eric Price, and David P. Woodruff. On the Power of Adaptivity in Sparse Recovery. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 285-294, 2011. Google Scholar
  13. Akshay Kamath and Eric Price. Adaptive Sparse Recovery with Limited Adaptivity. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 2729-2744, 2019. Google Scholar
  14. Daniel M. Kane, Shachar Lovett, and Shay Moran. Near-optimal linear decision trees for k-SUM and related problems. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 554-563, 2018. Google Scholar
  15. Sampath Kannan, Elchanan Mossel, Swagato Sanyal, and Grigory Yaroslavtsev. Linear Sketching over F_2. In 33rd Computational Complexity Conference, CCC 2018, June 22-24, 2018, San Diego, CA, USA, pages 8:1-8:37, 2018. Google Scholar
  16. Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, and Aaron Sidford. Single Pass Spectral Sparsification in Dynamic Streams. SIAM J. Comput., 46(1):456-477, 2017. Google Scholar
  17. John T Kent and R J Muirhead. Aspects of Multivariate Statistical Theory. The Statistician, 33(2):251, 1984. Google Scholar
  18. Christian Konrad. Maximum Matching in Turnstile Streams. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, pages 840-852, 2015. Google Scholar
  19. Yi Li, Huy L. Nguyen, and David P. Woodruff. On Sketching Matrix Norms and the Top Singular Vector. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1562-1581, 2014. Google Scholar
  20. Yi Li, Huy L. Nguyen, and David P. Woodruff. Turnstile streaming algorithms might as well be linear sketches. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 174-183, 2014. Google Scholar
  21. Yi Li and David P. Woodruff. On approximating functions of the singular values in a stream. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 726-739, 2016. Google Scholar
  22. Yi Li and David P. Woodruff. Embeddings of Schatten Norms with Applications to Data Streams. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, pages 60:1-60:14, 2017. Google Scholar
  23. Vasileios Nakos, Xiaofei Shi, David P. Woodruff, and Hongyang Zhang. Improved Algorithms for Adaptive Compressed Sensing. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, pages 90:1-90:14, 2018. Google Scholar
  24. Eric Price and David P. Woodruff. Lower Bounds for Adaptive Sparse Recovery. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 652-663, 2013. Google Scholar
  25. Jianhong Shen. On the singular values of Gaussian random matrices. Linear Algebra and its Applications, 326:1-14, 2001. Google Scholar
  26. Roman Vershynin. Introduction to the non-asymptotic analysis of random matrices. Compressed Sensing: Theory and Applications, November 2010. URL: http://dx.doi.org/10.1017/CBO9780511794308.006.
  27. Martin Wainwright. High-dimensional statistics: A non-asymptotic viewpoint. URL: https://www.stat.berkeley.edu/~mjwain/stat210b/Chap2_TailBounds_Jan22_2015.pdf.
  28. Hermann Weyl. Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Mathematische Annalen, 71(4):441-479, 1912. Google Scholar
  29. David P. Woodruff. Sketching as a Tool for Numerical Linear Algebra. Foundations and Trends in Theoretical Computer Science, 10(1-2):1-157, 2014. Google Scholar
  30. Andrew Chi-Chin Yao. Probabilistic computations: Toward a unified measure of complexity. In 18th Annual Symposium on Foundations of Computer Science, FOCS 1977, pages 222-227. IEEE, 1977. Google Scholar
  31. Qiaochu Yuan. Singular value decomposition, 2017. URL: https://qchu.wordpress.com/2017/03/13/singular-value-decomposition/.