Low-depth arithmetic circuit lower bounds via shifted partials

Prashanth Amireddy; Ankit Garg; Neeraj Kayal; Chandan Saha; Bhargav Thankey

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arXiv · 2211.07691

Low-depth arithmetic circuit lower bounds via shifted partials

Prashanth Amireddy·Ankit Garg·Neeraj Kayal·Chandan Saha·Bhargav Thankey

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Abstract

We prove super-polynomial lower bounds for low-depth arithmetic circuits using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013], [Kayal, ECCC 2012] and the affine projections of partials measure [Garg-Kayal-Saha, FOCS 2020], [Kayal-Nair-Saha, STACS 2016]. The recent breakthrough work of Limaye, Srinivasan and Tavenas [FOCS 2021] proved these lower bounds by proving lower bounds for low-depth set-multilinear circuits. An interesting aspect of our proof is that it does not require conversion of a circuit to a set-multilinear circuit, nor does it involve a random restriction. We are able to upper bound the measures for homogeneous formulas directly, without going via set-multilinearity. Our lower bounds hold for the iterated matrix multiplication as well as the Nisan-Wigderson design polynomials. We also define a subclass of homogeneous formulas which we call unique parse tree (UPT) formulas, and prove superpolynomial lower bounds for these. This generalizes the superpolynomial lower bounds for regular formulas in [Kayal-Saha-Saptharishi, STOC 2014], [Fournier-Limaye-Malod-Srinivasan, STOC 2014].

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Prashanth Amireddy, Ankit Garg, Neeraj Kayal, Chandan Saha, Bhargav Thankey. 2022-11-14. Low-depth arithmetic circuit lower bounds via shifted partials. https://arxiv.org/abs/2211.07691

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