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# Average-case linear matrix factorization and reconstruction of low width algebraic branching programs

Kayal, Neeraj and Nair, Vineet and Saha, Chandan (2019) Average-case linear matrix factorization and reconstruction of low width algebraic branching programs. In: COMPUTATIONAL COMPLEXITY, 28 (4). pp. 749-828. PDF com_com_28-4_749_2019.pdf - Published Version Restricted to Registered users only Download (840kB) | Request a copy
Official URL: https://dx.doi.org/10.1007/s00037-019-00189-0

## Abstract

A matrix X is called a linear matrix if its entries are affine forms, i.e., degree one polynomials in n variables. What is a minimalsized representation of a given matrix F as a product of linearmatrices? Finding such a minimal representation is closely related to finding an optimal way to compute a given polynomial via an algebraic branching program. Herewe devise an efficient algorithm for an average-case version of this problem. Specifically, given w, d, n. N and blackbox access to the w2 entries of a matrix product F = X1 Xd, where each Xi is a w x w linear matrix over a given finite field Fq, we wish to recover a factorization F = Y1 Yd , where every Yi is also a linear matrix over Fq (or a small extension of Fq). We show that when the input F is sampled from a distribution defined by choosing random linear matrices X1,..., Xd over Fq independently and taking their product and n = 4w2 and char(Fq) = (dn)O(1), then an equivalent factorization F = Y1 Yd can be recovered in (randomized) time (dn log q)O(1). In fact, we give a (worst-case) polynomial time randomized algorithm to factor any non-degenerate or pure matrix product (a notion we define in the paper) into linear matrices; a matrix product F = X1 Xd is pure with high probability when the Xi's are chosen independently at random. We also show that in this situation, ifwe are instead given a single entry of F rather than its w2 correlated entries, then the recovery can be done in (randomized) time (d(w3) n log q)(O(1)).

Item Type: Journal Article COMPUTATIONAL COMPLEXITY SPRINGER BASEL AG Copyright of this article belongs to SPRINGER BASEL AG Algebraic branching programs; equivalence testing; matrix factorization; pseudorandom polynomial families Division of Electrical Sciences > Computer Science & Automation 21 Nov 2019 09:59 21 Nov 2019 09:59 http://eprints.iisc.ac.in/id/eprint/63801 View Item