Pavel Hrubeš
ABSTRACT
We initiate a direction for proving lower bounds on the size of non-commutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of non-commutative arithmetic circuits and a problem about commutative degree four polynomials, the classical sum-of-squares problem: find the smallest n such that there exists an identity (x12+x22+•• + xk2)• (y1^2+y22+•• + yk2)= f12+f22+ ... +fn2, where each fi = fi(X,Y) is bilinear in X={x1,... ,xk} and Y={y1,..., yk}. Over the complex numbers, we show that a sufficiently strong super-linear lower bound on n in, namely, n ≥ k1+ε with ε >0, implies an exponential lower bound on the size of arithmetic circuits computing the non-commutative permanent.
More generally, we consider such sum-of-squares identities for any M polynomial h(X,Y), namely: h(X,Y) = f12+f22+...+fn2.
Again, proving n ≥ k1+ε in for any explicit h over the complex numbers gives an exponential lower bound for the non-commutative permanent. Our proofs relies on several new structure theorems for non-commutative circuits, as well as a non-commutative analog of Valiant's completeness of the permanent.
We proceed to prove such super-linear bounds in some restricted cases. We prove that n ≥ Ω(k6/5) in (1), if f1,..., fn are required to have integer coefficients. Over the real numbers, we construct an explicit M polynomial h such that n in (2) must be at least Ω(k2). Unfortunately, these results do not imply circuit lower bounds. We also present other structural results about non-commutative arithmetic circuits. We show that any non-commutative circuit computing an ordered non-commutative polynomial can be efficiently transformed to a syntactically multilinear circuit computing that polynomial. The permanent, for example, is ordered. Hence, lower bounds on the size of syntactically multilinear circuits computing the permanent imply unrestricted non-commutative lower bounds. We also prove an exponential lower bound on the size of non-commutative syntactically multilinear circuit computing an explicit polynomial. This polynomial is, however, not ordered and an unrestricted circuit lower bound does not follow.
- A. Barvinok. A simple polynomial time algorithm to approximatethe permanent within a simply exponential factor. Random Structures and Algorithms 14(1), pages 29--61, 1999. Google Scholar
Digital Library
- W. Baur and V. Strassen. The complexity of partial derivatives. Theoretical computer science (22), pages 317--330, 1983.Google Scholar
- P. Burgisser. Completeness and reduction in algebraic complexity theory. Springer-Verlag Berlin Heidelberg 2000.Google Scholar
- S. Chien and A. Sinclair. Algebras with polynomial identities and computing the determinant. SIAM Journal on Computing 37, pages 252--266, 2007. Google ScholarDigital Library
- S. Chien, L. Rasmussen and A. Sinclair. Clifford algebras an approximating the permanent. STOC '02, pages 222--231, 2002. Google ScholarDigital Library
- J. von zur Gathen. Algebraic complexity theory. Ann. Rev. Comp. Sci. (3), pages 317--347, 1988. Google ScholarDigital Library
- C. Godsil and I. Gutman. On the matching polynomial of a graph. Algebraic Methods in Graph Theory, pages 241--249, 1981.Google Scholar
- L. Hyafil. On the parallel evaluation of multivariate polynomials. SIAM J. Comput. 8(2), pages 120--123, 1979.Google Scholar
- P. Hrubes and A. Yehudayoff. Homogeneous formulas and symmetric polynomials. arXiv:0907.2621 Google ScholarDigital Library
- P. Hrubes, A. Wigderson and A. Yehudayoff. Relationless completeness and separations. To appear in CCC. Google ScholarDigital Library
- A. Hurwitz. Über die Komposition der quadratischen Formen von beliebigvielen Variabeln. Nach. Ges. der Wiss. Göttingen, pages 309--316, 1898.Google Scholar
- A. Hurwitz. Über die Komposition der quadratischen Formen. Math. Ann., 88, pages 1--25, 1923.Google ScholarCross Ref
- I. M. James. On the immersion problem for real projective spaces. Bull. Am. Math. Soc., 69, pages 231--238, 1967.Google ScholarCross Ref
- M. Jerrum, A. Sinclair and E. Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM 51(4), pages 671--697, 2004. Google ScholarDigital Library
- S. Jukna Boolean function complexity: advances and frontiers. Book in preparation. Google ScholarDigital Library
- N. Karmarkar, R. Karp, R. Lipton, L. Lovasz and M. Luby. A Monte-Carlo algorithm for estimating the permanent. SIAM Journal on Computing 22(2), pages 284--293, 1993. Google ScholarDigital Library
- T. Kirkman. On pluquatemions, and horaoid products of sums of n squares. Philos.Mag. (ser. 3), 33, pages 447--459; 494--509, 1848.Google Scholar
- K. Y. Lam. Some new results on composition of quadratic forms. Inventiones Mathematicae., 1985.Google ScholarCross Ref
- T. Y. Lam and T. Smith. On Yuzvinsky's monomial pairings. Quart. J. Math. Oxford. (2), 44, pages 215--237, 1993.Google ScholarCross Ref
- K. Mulmuley. On P vs. NP, Geometric Complexity Theory, and the Riemann Hypothesis. Technical Report, Computer Science department, The University of Chicago, 2009.Google Scholar
- N. Nisan. Lower bounds for non-commutative computation. STOC '91, pages 410--418, 1991. Google ScholarDigital Library
- N. Nisan and A. Wigderson. Lower bounds on arithmetic circuits via partial derivatives. Computational Complexity, vol. 6, pages 217--234, 1996. Google ScholarDigital Library
- A. Pfister. Zur Darstellung definiter Funktionen als Summe von Quadraten. ph Inventiones Mathematicae., 1967.Google Scholar
- J. Radon. Lineare scharen orthogonalen matrizen. Abh. Math. Sem. Univ. Hamburg 1, pages 2--14, 1922.Google Scholar
- R. Raz. Multi-linear formulas for permanent and determinant are of super-polynomial size. Journal of the Association for Computing Machinery 56 (2), 2009. ıgnore Google ScholarDigital Library
- R. Raz. Elusive functions and lower bounds for arithmetic circuits. To appear in Theory of Computing. Google ScholarDigital Library
- R. Raz and A. Yehudayoff. Lower bounds and separation for constant depth multilinear circuits. Proceedings of Computational Complexity, pages 128--139, 2008. Google ScholarDigital Library
- R. Raz, A. Shpilka and A. Yehudayoff. A lower bound for the size of syntactically multilinear arithmetic circuits. SIAM Journal on Computing 38 (4), pages 1624--1647, 2008. Google ScholarDigital Library
- D. B. Shapiro. Composition of quadratic forms. W. de Gruyter Verlag, 2000.Google Scholar
- . Strassen. Die berechnungskomplexitat von elementarsymmetrischen funktionen und von interpolationskoeffizienten. Numerische Mathematik (20), pages 238--251, 1973.Google Scholar
- V. Strassen. Vermeidung von Divisionen. J. Reine Angew. Math. 264, pages 182--202, 1973.Google Scholar
- L. G. Valiant. Completeness classes in algebra. STOC '79, pages 249--261. Google ScholarDigital Library
- S. Winograd. On the number of multiplications needed to compute certain functions. Comm. on Pure and Appl. Math. (23), pages 165--179, 1970.Google Scholar
- P. Yiu. Sums of squares formulae with integer coefficients. Canad. Math. Bull., 30, pages 318--324, 1987.Google ScholarCross Ref
- P. Yiu. On the product of two sums of 16 squares as a sum of squares of integral bilinear forms. Quart. J. Math. Oxford. (2), 41, pages 463--500, 1990.Google ScholarCross Ref
- S. Yuzvinsky. A series of monomial pairings. phLinear and multilinear algebra, 15, pages 19--119, 1984.Google Scholar
Index Terms
- Non-commutative circuits and the sum-of-squares problem
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