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

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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 minimal-sized representation of a given matrix F as a product of linear matrices? Finding such a minimal representation is closely related to finding an optimal way to compute a given polynomial via an algebraic branching program. Here we devise an efficient algorithm for an average-case version of this problem. Specifically, given \(w,d,n \in \mathbb{N}\) and blackbox access to the w2 entries of a matrix product \(F = X_1 \cdots X_d\), where each \(X_i\) is a \(w \times w\) linear matrix over a given finite field \(\mathbb{F}_q\), we wish to recover a factorization \(F = Y_1 \cdots Y_{d'}\), where every \(Y_i\) is also a linear matrix over \(\mathbb{F}_q\) (or a small extension of \(\mathbb{F}_q\)). We show that when the input F is sampled from a distribution defined by choosing random linear matrices \(X_1, \ldots, X_d\) over \(\mathbb{F}_q\) independently and taking their product and \(n \geq 4w^2\) and \({\rm char}(\mathbb{F}_q) = (dn)^{\varOmega(1)}\), then an equivalent factorization \(F = Y_1 \cdots Y_d\) 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 \(F = X_1 \cdots X_d\) is pure with high probability when the \(X_i\)'s are chosen independently at random. We also show that in this situation, if we are instead given a single entry of F rather than its w2 correlated entries, then the recovery can be done in (randomized) time \((d^{w^3} n \log q)^{O(1)}\).

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Acknowledgements

We thank Sébastien Tavenas for a few initial discussions on this work. Thanks also to the anonymous reviewers for their helpful comments to improve the presentation of the paper.

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Correspondence to Vineet Nair.

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Kayal, N., Nair, V. & Saha, C. Average-case linear matrix factorization and reconstruction of low width algebraic branching programs. comput. complex. 28, 749–828 (2019). https://doi.org/10.1007/s00037-019-00189-0

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