Proceedings ArticleDOI
Non-commutative circuits and the sum-of-squares problem
Pavel Hrubeš,Avi Wigderson,Amir Yehudayoff +2 more
- pp 667-676
Reads0
Chats0
TLDR
It is shown that any non-commutative circuit computing an <i>ordered</i> non-Commutative polynomial can be efficiently transformed to a syntactically multilinear circuit computing that polynomials that is ordered.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
Citations
More filters
Book
Arithmetic Circuits: A Survey of Recent Results and Open Questions
Amir Shpilka,Amir Yehudayoff +1 more
TL;DR: The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what it finds to be the most interesting and accessible research directions, with an emphasis on works from the last two decades.
Journal Article
An exponential lower bound for the sum of powers of bounded degree polynomials.
TL;DR: The main theorem is a lower bound on the number of summands in any representation of the form (1) for an explicit polynomial f, which is an easy corollary of Fischer [Fis94].
Proceedings ArticleDOI
Quasipolynomial-Time Identity Testing of Non-commutative and Read-Once Oblivious Algebraic Branching Programs
Michael A. Forbes,Amir Shpilka +1 more
TL;DR: This work gives the first quasi-polynomial sized hitting sets for size S circuits from this class, when the order of the variables is known, and strengthens a result of Mulmuley, and shows that derandomizing a particular case of the No ether Normalization Lemma is reducible to black-box PIT of read-once oblivious ABPs.
Posted Content
A deterministic polynomial time algorithm for non-commutative rational identity testing
TL;DR: In this article, a deterministic polynomial time algorithm for testing if a symbolic matrix in non-commuting variables is invertible or not is presented, and the algorithm efficiently solves the word problem for the free skew field and the identity testing problem for arithmetic formulae with division, two problems which had only exponential-time algorithms.
References
More filters
Journal ArticleDOI
A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries
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.
Proceedings ArticleDOI
Completeness classes in algebra
TL;DR: The aim of this paper is to demonstrate that for both algebraic and combinatorial problems this phenomenon exists in a form that is purely algebraic in both of the respects (A) and (B).
Algebraic Complexity Theory.
TL;DR: Algebraic complexity theory as mentioned in this paper is a project of lower bounds and optimality, which unifies two quite different traditions: mathematical logic and the theory of recursive functions, and numerical algebra.
Journal ArticleDOI
The complexity of partial derivatives
Walter Baur,Volker Strassen +1 more
TL;DR: Using the nonscalar complexity in k, the complexity of single power sums, single elementary symmetric functions, the resultant and the discriminant as root functions are determined up to order of magnitude.
Journal ArticleDOI
Vermeidung von Divisionen.
K. Ramachandra,Volker Strassen +1 more
TL;DR: In this article, it was shown that the use of divisions does not decrease the number of (*,/)-operations for multiplication of general matrices, and that multiplication of orthogonal matrices does not increase the computational complexity.