Non-commutative circuits and the sum-of-squares problem
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In this paper, 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, was made.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+e with e >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+e 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.read more
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Dissertation
Polynomial identity testing of read-once oblivious algebraic branching programs
TL;DR: This work gives the first quasi-polynomial sized hitting set for size S circuits from this class, and strengthens a result of Mulmuley [Mul12a], and shows that derandomizing a particular case of the Noether Normalization Lemma is reducible to black-box PIT of read-once oblivious ABPs.
Proceedings ArticleDOI
Non-commutative arithmetic circuits with division
Pavel Hrubeš,Avi Wigderson +1 more
TL;DR: It is shown how divisions can be eliminated from non-commutative circuits and formulae which compute polynomials, and the non-Commutative version of the "rational function identity testing" problem is addressed.
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Operator Scaling: Theory and Applications
TL;DR: In this paper, the authors present a deterministic polynomial time algorithm for testing whether a symbolic matrix in non-commuting variables over the free skew field is invertible or not.
Proceedings ArticleDOI
Relationless Completeness and Separations
TL;DR: It is proved that even in a completely relationless world which assumes no commutativity nor associativity, permanent remains VNP-complete, and determinant can polynomially simulate any arithmetic formula, just as in the standard commutative, associative world of Valiant.
Posted Content
Operator scaling: theory and applications
TL;DR: A complexity analysis of an existing algorithm due to Gurvits (J Comput Syst Sci 69(3):448–484, 2004 ), who proved it was polynomial time for certain classes of inputs, that is extended to actually approximate capacity to any accuracy in polynometric time.
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TL;DR: A polynomial-time randomized algorithm for estimating the permanent of an arbitrary n × n matrix with nonnegative entries computes an approximation that is within arbitrarily small specified relative error of the true value of the permanent.
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TL;DR: In this article, a comprehensive description of basic lower bound arguments, covering many of the gems of this complexity Waterloo that have been discovered over the past several decades, right up to results from the last year or two, is given.