Document

# Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

## File

LIPIcs.CCC.2015.304.pdf
• Filesize: 0.58 MB
• 19 pages

## Cite As

Rafael Oliveira, Amir Shpilka, and Ben Lee Volk. Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 304-322, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.CCC.2015.304

## Abstract

In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds we obtain lower bounds for these models. For depth-3 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 2*delta/3))). This implies a lower bound of exp(~Omega(n^(1/2))) for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 4*delta/3)). This implies a lower bound of exp(~Omega(n^(1/4))) for depth-4 multilinear formulas, for some explicit polynomial. A regular formula consists of alternating layers of +,* gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) exp(n^(1-delta)), for regular depth-d multilinear formulas of size exp(n^delta), where delta = O(1/sqrt(5)^d)). This result implies a lower bound of roughly exp(~Omega(n^(1/sqrt(5)^d))) for such formulas. We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known. Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a read-once algebraic branching program (from depth-3 formulas we go straight to read-once algebraic branching programs).
##### Keywords
• Arithmetic Circuits
• Derandomization
• Polynomial Identity Testing

## Metrics

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

## References

1. M. Agrawal. Proving lower bounds via pseudo-random generators. In Proceedings of the 25th FSTTCS, volume 3821 of LNCS, pages 92-105, 2005.
2. M. Agrawal, R. Gurjar, A. Korwar, and N. Saxena. Hitting-sets for ROABP and sum of set-multilinear circuits. Electronic Colloquium on Computational Complexity (ECCC), 21:85, 2014.
3. M. Agrawal, N. Kayal, and N. Saxena. Primes is in P. Annals of Mathematics, 160(2):781-793, 2004.
4. M. Agrawal, C. Saha, R. Saptharishi, and N. Saxena. Jacobian hits circuits: hitting-sets, lower bounds for depth-d occur-k formulas & depth-3 transcendence degree-k circuits. In STOC, pages 599-614, 2012.
5. M. Agrawal, C. Saha, and N. Saxena. Quasi-polynomial hitting-set for set-depth-Δ formulas. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 321-330, 2013.
6. M. Agrawal and V. Vinay. Arithmetic circuits: A chasm at depth four. In Proceedings of the 49th Annual FOCS, pages 67-75, 2008.
7. N. Alon and J. H. Spencer. The probabilistic method. J. Wiley, 3 edition, 2008.
8. M. Anderson, D. van Melkebeek, and I. Volkovich. Derandomizing polynomial identity testing for multilinear constant-read formulae. In Proceedings of the 26th Annual CCC, pages 273-282, 2011.
9. R. A. DeMillo and R. J. Lipton. A probabilistic remark on algebraic program testing. Inf. Process. Lett., 7(4):193-195, 1978.
10. Z. Dvir and A. Shpilka. Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits. SIAM J. on Computing, 36(5):1404-1434, 2006.
11. Z. Dvir, A. Shpilka, and A. Yehudayoff. Hardness-randomness tradeoffs for bounded depth arithmetic circuits. SIAM J. on Computing, 39(4):1279-1293, 2009.
12. M. A. Forbes, R. Saptharishi, and A. Shpilka. Hitting sets for multilinear read-once algebraic branching programs, in any order. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 867-875, 2014.
13. M. A. Forbes and A. Shpilka. On identity testing of tensors, low-rank recovery and compressed sensing. In Proceedings of the 44th annual STOC, pages 163-172, 2012.
14. M. A. Forbes and A. Shpilka. Quasipolynomial-time identity testing of non-commutative and read-once oblivious algebraic branching programs. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 243-252, 2013.
15. A. Gupta, P. Kamath, N. Kayal, and R. Saptharishi. Arithmetic circuits: A chasm at depth three. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, pages 578-587, 2013.
16. J. Heintz and C. P. Schnorr. Testing polynomials which are easy to compute (extended abstract). In Proceedings of the 12th annual STOC, pages 262-272, 1980.
17. V. Kabanets and R. Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. In Proceedings of the 35th Annual STOC, pages 355-364, 2003.
18. V. Kabanets and R. Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1-2):1-46, 2004.
19. Z. S. Karnin, P. Mukhopadhyay, A. Shpilka, and I. Volkovich. Deterministic identity testing of depth-4 multilinear circuits with bounded top fan-in. SIAM J. Comput., 42(6):2114-2131, 2013.
20. Z. S. Karnin and A. Shpilka. Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in. Combinatorica, 31(3):333-364, 2011.
21. N. Kayal, C. Saha, and R. Saptharishi. A super-polynomial lower bound for regular arithmetic formulas. In Symposium on Theory of Computing, STOC 2014, pages 146-153, 2014.
22. N. Kayal and S. Saraf. Blackbox polynomial identity testing for depth 3 circuits. In Proceedings of the 50th Annual FOCS, pages 198-207, 2009.
23. N. Kayal and N. Saxena. Polynomial identity testing for depth 3 circuits. Computational Complexity, 16(2):115-138, 2007.
24. P. Koiran. Arithmetic circuits: the chasm at depth four gets wider. CoRR, abs/1006.4700, 2010.
25. N. Nisan. Lower bounds for non-commutative computation. In Proceedings of the 23rd Annual STOC, pages 410-418, 1991.
26. N. Nisan and A. Wigderson. Lower bound on arithmetic circuits via partial derivatives. Computational Complexity, 6:217-234, 1996.
27. R. Oliveira, A. Shpilka, and B. L. Volk. Subexponential size hitting sets for bounded depth multilinear formulas. Electronic Colloquium on Computational Complexity (ECCC), 21:157, 2014.
28. R. Raz and A. Shpilka. Deterministic polynomial identity testing in non-commutative models. Computational Complexity, 14(1):1-19, 2005.
29. R. Raz and A. Yehudayoff. Lower bounds and separations for constant depth multilinear circuits. Computational Complexity, 18(2):171-207, 2009.
30. S. Saraf and I. Volkovich. Black-box identity testing of depth-4 multilinear circuits. In Proceedings of the 43rd annual STOC, pages 421-430, 2011.
31. N. Saxena and C. Seshadhri. Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter. In Proceedings of the 43rd Annual STOC, pages 431-440, 2011.
32. J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, 1980.
33. A. Shpilka and I. Volkovich. Read-once polynomial identity testing. In Proceedings of the 40th Annual STOC, pages 507-516, 2008.
34. A. Shpilka and I. Volkovich. Improved polynomial identity testing for read-once formulas. In APPROX-RANDOM, pages 700-713, 2009.
35. A. Shpilka and A. Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5(3-4):207-388, 2010.
36. S. Tavenas. Improved bounds for reduction to depth 4 and depth 3. In Mathematical Foundations of Computer Science 2013 - 38th International Symposium, MFCS 2013, pages 813-824, 2013.
37. L. G. Valiant, S. Skyum, S. Berkowitz, and C. Rackoff. Fast parallel computation of polynomials using few processors. SIAM J. on Computing, 12(4):641-644, November 1983.
38. R. Zippel. Probabilistic algorithms for sparse polynomials. In EUROSAM, pages 216-226, 1979.