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Separation Between Read-once Oblivious Algebraic Branching Programs (ROABPs) and Multilinear Depth-three Circuits

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Published:11 February 2020Publication History
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Abstract

We show an exponential separation between two well-studied models of algebraic computation, namely, read-once oblivious algebraic branching programs (ROABPs) and multilinear depth-three circuits. In particular, we show the following:

(1) There exists an explicit n-variate polynomial computable by linear sized multilinear depth-three circuits (with only two product gates) such that every ROABP computing it requires 2Ω(n) size.

(2) Any multilinear depth-three circuit computing IMMn,d (the iterated matrix multiplication polynomial formed by multiplying d, n × n symbolic matrices) has nΩ(d) size. IMMn,d can be easily computed by a poly(n,d) sized ROABP.

(3) Further, the proof of (2) yields an exponential separation between multilinear depth-four and multilinear depth-three circuits: There is an explicit n-variate, degree d polynomial computable by a poly(n) sized multilinear depth-four circuit such that any multilinear depth-three circuit computing it has size nΩ(d). This improves upon the quasi-polynomial separation of Reference [36] between these two models.

The hard polynomial in (1) is constructed using a novel application of expander graphs in conjunction with the evaluation dimension measure [15, 33, 34, 36], while (2) is proved via a new adaptation of the dimension of the partial derivatives measure of Reference [32]. Our lower bounds hold over any field.

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  1. Separation Between Read-once Oblivious Algebraic Branching Programs (ROABPs) and Multilinear Depth-three Circuits

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