Non-commutative circuits and the sum-of-squares problem

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 <i>non-commutative</i> arithmetic circuits and a problem about <i>commutative</i> degree four polynomials, the classical sum-of-squares problem: find the smallest n such that there exists an identity (x<sub>1</sub><sup>2</sup>+x<sub>2</sub><sup>2</sup>+•• + x<sub>k</sub><sup>2</sup>)• (y<sub>1</sub>^2+y<sub>2</sub><sup>2</sup>+•• + y<sub>k</sub><sup>2</sup>)= f<sub>1</sub><sup>2</sup>+f<sub>2</sub><sup>2</sup>+ ... +f<sub>n</sub><sup>2</sup>, where each f<sub>i</sub> = f<sub>i</sub>(X,Y) is bilinear in X={x<sub>1</sub>,... ,x<sub>k</sub>} and Y={y<sub>1</sub>,..., y<sub>k</sub>}. Over the complex numbers, we show that a sufficiently strong <i>super-linear</i> lower bound on n in, namely, n ≥ k<sup>1+ε</sup> with ε >0, implies an <i>exponential</i> 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) = f<sub>1</sub><sup>2</sup>+f<sub>2</sub><sup>2</sup>+...+f<sub>n</sub><sup>2</sup>. Again, proving n ≥ k<sup>1+ε</sup> in for <i>any</i> explicit h over the complex numbers gives an <i>exponential</i> 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(k<sup>6/5</sup>) in (1), if f<sub>1</sub>,..., f<sub>n</sub> are required to have <i>integer</i> coefficients. Over the <i>real</i> numbers, we construct an explicit M polynomial h such that n in (2) must be at least Ω(k<sup>2</sup>). 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 <i>ordered</i> 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.

[1]  S. Winograd.  On the number of multiplications necessary to compute certain functions , 1970 .

[2]  Martin Tompa,et al.  A Direct Version of Shamir and Snir's Lower Bounds on Monotone Circuit Depth , 1994, Inf. Process. Lett..

[3]  Ran Raz,et al.  Lower Bounds and Separations for Constant Depth Multilinear Circuits , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[4]  K. Ramachandra,et al.  Vermeidung von Divisionen. , 1973 .

[5]  S Winograd,et al.  On the number of multiplications required to compute certain functions. , 1967, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Sergey Yuzvinsky,et al.  A series of monomial pairings , 1984 .

[7]  Amir Yehudayoff,et al.  Homogeneous Formulas and Symmetric Polynomials , 2011, computational complexity.

[8]  Leslie G. Valiant,et al.  Completeness classes in algebra , 1979, STOC.

[9]  Noam Nisan,et al.  Lower bounds for non-commutative computation , 1991, STOC '91.

[10]  Eric Vigoda,et al.  A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries , 2004, JACM.

[11]  Ran Raz,et al.  Multi-linear formulas for permanent and determinant are of super-polynomial size , 2004, STOC '04.

[12]  V. Strassen.  Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten , 1973 .

[13]  Avi Wigderson,et al.  Relationless Completeness and Separations , 2010, 2010 IEEE 25th Annual Conference on Computational Complexity.

[14]  J. Gathen.  Algebraic complexity theory , 1988 .

[15]  Ketan Mulmuley,et al.  On P vs. NP, Geometric Complexity Theory, and the Riemann Hypothesis , 2009, ArXiv.

[16]  Laurent Hyafil,et al.  On the parallel evaluation of multivariate polynomials , 1978, SIAM J. Comput..

[17]  A. Hurwitz,et al.  Über die Komposition der quadratischen Formen , 1922 .

[18]  Ran Raz.  Elusive functions and lower bounds for arithmetic circuits , 2008, STOC '08.

[19]  A. Pfister.  Zur Darstellung definiter Funktionen als Summe von Quadraten , 1967 .

[20]  Ran Raz.  Elusive Functions and Lower Bounds for Arithmetic Circuits , 2010, Theory Comput..

[21]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[22]  K. Y. Lam,et al.  Some new results on composition of quadratic forms , 1985 .

[23]  Raoul Bott,et al.  ON THE IMMERSION PROBLEM FOR REAL PROJECTIVE SPACES , 2007 .

[24]  P. Yiu,et al.  Sums of Squares Formulae With Integer Coefficients , 1987, Canadian Mathematical Bulletin.

[25]  Lin Yu-qing,et al.  Matching polynomial of graph , 2007 .

[26]  A. Hurwitz.  Über die Komposition der quadratischen Formen von beliebig vielen Variablen , 1963 .

[27]  Roy Dubisch.  Composition of Quadratic Forms , 1946 .

[28]  A. Barvinok.  Polynomial time algorithms to approximate permanents and mixed discriminants within a simply exponential factor , 1999 .

[29]  T. Y. Lam,et al.  ON YUZVINSKY'S MONOMIAL PAIRINGS , 1993 .

[30]  Steve Chien,et al.  Clifford algebras and approximating the permanent , 2002, STOC '02.

[31]  Ran Raz,et al.  A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[32]  E. Jenkins,et al.  On the composition of quadratic forms , 1935 .

[33]  Peter Bürgisser,et al.  Completeness and Reduction in Algebraic Complexity Theory , 2000, Algorithms and computation in mathematics.

[34]  Noam Nisan,et al.  Lower bounds on arithmetic circuits via partial derivatives , 2005, computational complexity.

[35]  Paul Yiu,et al.  ON THE PRODUCT OF TWO SUMS OF 16 SQUARES AS A SUM OF SQUARES OF INTEGRAL BILINEAR FORMS , 1990 .

[36]  Steve Chien,et al.  Algebras with polynomial identities and computing the determinant , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[37]  Ran Raz,et al.  Lower Bounds and Separations for Constant Depth Multilinear Circuits , 2008, Computational Complexity Conference.

[38]  Alexander Barvinok.  A simple polynomial time algorithm to approximate the permanent within a simply exponential factor , 1997 .

[39]  Richard J. Lipton,et al.  A Monte-Carlo Algorithm for Estimating the Permanent , 1993, SIAM J. Comput..