Document

# On Multilinear Forms: Bias, Correlation, and Tensor Rank

## File

LIPIcs.APPROX-RANDOM.2020.29.pdf
• Filesize: 0.54 MB
• 23 pages

## Acknowledgements

We would like to thank Suryateja Gavva for helpful discussions. We would like to thank Shubhangi Saraf for suggesting the idea for the proof of Lemma 10.

## Cite As

Abhishek Bhrushundi, Prahladh Harsha, Pooya Hatami, Swastik Kopparty, and Mrinal Kumar. On Multilinear Forms: Bias, Correlation, and Tensor Rank. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 29:1-29:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.29

## Abstract

In this work, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over F₂. We show the following results for multilinear forms and tensors. Correlation bounds. We show that a random d-linear form has exponentially low correlation with low-degree polynomials. More precisely, for d = 2^{o(k)}, we show that a random d-linear form f(X₁,X₂, … , X_d) : (F₂^{k}) ^d → F₂ has correlation 2^{-k(1-o(1))} with any polynomial of degree at most d/2 with high probability. This result is proved by giving near-optimal bounds on the bias of a random d-linear form, which is in turn proved by giving near-optimal bounds on the probability that a sum of t random d-dimensional rank-1 tensors is identically zero. Tensor rank vs Bias. We show that if a 3-dimensional tensor has small rank then its bias, when viewed as a 3-linear form, is large. More precisely, given any 3-dimensional tensor T: [k]³ → F₂ of rank at most t, the bias of the 3-linear form f_T(X₁, X₂, X₃) : = ∑_{(i₁, i₂, i₃) ∈ [k]³} T(i₁, i₂, i₃)⋅ X_{1,i₁}⋅ X_{2,i₂}⋅ X_{3,i₃} is at least (3/4)^t. This bias vs tensor-rank connection suggests a natural approach to proving nontrivial tensor-rank lower bounds. In particular, we use this approach to give a new proof that the finite field multiplication tensor has tensor rank at least 3.52 k, which is the best known rank lower bound for any explicit tensor in three dimensions over F₂. Moreover, this relation between bias and tensor rank holds for d-dimensional tensors for any fixed d.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Randomness, geometry and discrete structures
##### Keywords
• polynomials
• Boolean functions
• tensor rank
• bias
• correlation

## Metrics

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

## References

1. Boris Alexeev, Michael A. Forbes, and Jacob Tsimerman. Tensor rank: Some lower and upper bounds. In Proc. 26th IEEE Conf. on Comput. Complexity, pages 283-291, 2011. URL: https://doi.org/10.1109/CCC.2011.28.
2. Ido Ben-Eliezer, Rani Hod, and Shachar Lovett. Random low-degree polynomials are hard to approximate. Comput. Complexity, 21(1):63-81, 2012. (Preliminary version in 13th RANDOM, 2009). URL: https://doi.org/10.1007/s00037-011-0020-6.
3. Eli Ben-Sasson and Swastik Kopparty. Affine dispersers from subspace polynomials. SIAM J. Comput., 41(4):880-914, 2012. (Preliminary version in 41st STOC, 2009). URL: https://doi.org/10.1137/110826254.
4. Markus Bläser. A 5/2 n²-lower bound for the rank of n× n matrix multiplication over arbitrary fields. In Proc. 40th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 45-50, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814576.
5. Mark R. Brown and David P. Dobkin. An improved lower bound on polynomial multiplication. IEEE Trans. Computers, C-29(5):337-340, 1980. URL: https://doi.org/10.1109/TC.1980.1675583.
6. Suryajith Chillara, Mrinal Kumar, Ramprasad Saptharishi, and V. Vinay. The chasm at depth four, and tensor rank : Old results, new insights, 2016. URL: http://arxiv.org/abs/1606.04200.
7. David V. Chudnovsky and Gregory V. Chudnovsky. Algebraic complexities and algebraic curves over finite fields. J. Complexity, 4(4):285-316, 1988. URL: https://doi.org/10.1016/0885-064X(88)90012-X.
8. Klim Efremenko, Ankit Garg, Rafael Mendes de Oliveira, and Avi Wigderson. Barriers for rank methods in arithmetic complexity. In Anna Karlin, editor, Proc. 9th Innovations in Theor. Comput. Sci. (ITCS), volume 94 of LIPIcs, pages 1:1-1:19. Schloss Dagstuhl, 2018. URL: http://arxiv.org/abs/1710.09502.
9. William Tomothy Gowers and Julia Wolf. Linear forms and higher-degree uniformity for functions on 𝔽_pⁿ. Geom. Funct. Anal., 21(1):36-69, 2011. URL: http://arxiv.org/abs/1002.2208.
10. Johan Håstad. Tensor rank is NP-complete. J. Algorithms, 11(4):644-654, 1990. (Preliminary version in 16th ICALP, 1989). URL: https://doi.org/10.1016/0196-6774(90)90014-6.
11. Michael Kaminski. A lower bound on the complexity of polynomial multiplication over finite fields. SIAM J. Comput., 34(4):960-992, 2005. (Preliminary version in 22nd STACS, 2005). URL: https://doi.org/10.1137/S0097539704442118.
12. Valentin K. Kolchin. Random Graphs. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1998. URL: https://doi.org/10.1017/CBO9780511721342.
13. Shachar Lovett. The analytic rank of tensors is subadditive, and its applications. Discrete Analysis, 2019(7), 2019. URL: http://arxiv.org/abs/1806.09179.
14. Robert J. Mceliece, Eugene R. Rodemich, Howard Rumsey, Lloyd, and R. Welch. New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities. IEEE Trans. Inform. Theory, 23(2):157-166, 1977. URL: https://doi.org/10.1109/TIT.1977.1055688.
15. Noam Nisan and Avi Wigderson. Hardness vs. randomness. J. Comput. Syst. Sci., 49(2):149-167, October 1994. (Preliminary version in 29th FOCS, 1988). URL: https://doi.org/10.1016/S0022-0000(05)80043-1.
16. Ran Raz. Tensor-rank and lower bounds for arithmetic formulas. J. ACM, 60(6):40:1-40:15, 2013. (Preliminary version in 42nd STOC, 2010). URL: https://doi.org/10.1145/2535928.
17. Alexander A. Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mathematicheskie Zametki, 41(4):598-607, 1987. (English translation in Mathematical Notes of the Academy of Sciences of the USSR, 41(4):333-338, 1987). URL: https://doi.org/10.1007/BF01137685.
18. Igor E. Shparlinski, Michael A. Tsfasman, and Serge G. Vladut. Curves with many points and multiplication in finite fields. In Henning Stichtenoth and Michael A. Tsfasman, editors, Proc. Int. Workshop on Coding Theory and Algebraic Geometry, volume 1518 of LNM, pages 145-169. Springer, 1992. URL: https://doi.org/10.1007/BFb0087999.
19. Amir Shpilka. Lower bounds for matrix product. SIAM J. Comput., 32(5):1185-1200, 2003. (Preliminary version in 42nd FOCS, 2001). URL: http://arxiv.org/abs/cs/0201001.
20. Roman Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proc. 19th ACM Symp. on Theory of Computing (STOC), pages 77-82, 1987. URL: https://doi.org/10.1145/28395.28404.
21. Volker Strassen. Die berechnungskomplexität von elementarsymmetrischen funktionen und von interpolationskoeffizienten (German) [The computational complexity of elementary symmetric functions and interpolation coefficients]. Numerische Mathematik, 20(3):238-251, June 1973. URL: https://doi.org/10.1007/BF01436566.
22. Emanuele Viola. On the power of small-depth computation. Foundations and Trends in Theoretical Computer Science, 5(1):1-72, 2009. URL: https://doi.org/10.1561/0400000033.
23. Emanuele Viola. Matching Smolensky’s correlation bound with majority. (manuscript), 2019.
24. Emanuele Viola and Avi Wigderson. Norms, XOR lemmas, and lower bounds for polynomials and protocols. Theory Comput., 4(1):137-168, 2008. (Preliminary version in 22nd CCC, 2007). URL: https://doi.org/10.4086/toc.2008.v004a007.
X

Feedback for Dagstuhl Publishing