Low-depth arithmetic circuit lower bounds via shifted partials

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].

• The paper introduces a novel method using shifted partials to directly establish superpolynomial lower bounds for low-depth arithmetic circuits.
• It demonstrates effective use of homogeneous formulas and unique parse tree structures, bypassing the need for set-multilinear conversions.
• Results validate that polynomial families like IMM and Nisan-Wigderson lead to exponential lower bounds, impacting circuit complexity theory.

Low-depth Arithmetic Circuit Lower Bounds via Shifted Partials

Introduction

The paper explores superpolynomial lower bounds for low-depth arithmetic circuits by employing the shifted partials measure. Unlike prior approaches that require converting circuits into set-multilinear circuits or using random restrictions, this research establishes these bounds directly via homogenous formulas while avoiding set-multilinearity. The work focuses on two main polynomial families: iterated matrix multiplication (IMM) and Nisan-Wigderson design polynomials, providing insights into arithmetic formulas and deriving strong lower bound results for unique parse tree (UPT) formulas.

Shifted Partials and Affine Projections

The incoming approach uses the shifted partials measure $SP_{k,\ell}$, characterized by the complexity of the dimension of spaces formed by taking degree $\ell$ shifts of the $k$-th partial derivatives of a polynomial. Additionally, the affine projects of partials measure $APP_{k,n_0}$, extends this by evaluating dimensions after affine transformations. These measures prove crucial in bounding the complexity of arithmetic circuit evaluations, leading to effective lower bounds, especially for product-depth and unique parse-tree formulas.

Methodology

Lower Bound Strategies

1. Decomposition and Structural Lemmas: By decomposing homogeneous formulas into sums of lower-degree polynomial products, the measures can be bounded effectively.
2. High Residue Effect: Using the residue function of degree sequences, sharp lower bounds are derived.
3. Structural Integrity with UPT: The unique UPT structure, with strictly defined parse trees, facilitates consistent degree sequences across summands, simplifying the lower bound determination.

Polynomial Consideration

IMM and Nisan-Wigderson polynomials serve pivotal roles in demonstrating this paper’s central thesis: for polynomials resident within $VNP$, lower bounds are substantiated without necessitating transformation into set-multilinear equivalents. This assumption holds under rigorous derivations of $SP$ and $APP$ for assorted polynomial configurations.

Results

• Low-Depth Formulas: Establish that for homogeneous formulas computing polynomials with iterated matrix multiplication and nested in low-depth circuits, super-polynomial lower bounds exist.
• Exponential Lower Bounds: A special focus on UPT reveals $n^{\Omega(\log\log(n))}$ size constraints on iterated multiplication polynomials, demonstrating broader implications on circuit complexity.
• Tradeoffs and Assumptions: Curating explicit cases where shifted partials and affine projections assure bounded complexity, challenging existing paradigms established by set-multilinear transformations.

Conclusion

The exploratory and methodological advances consolidated in the research indicate the potency of shifted partials in deriving arithmetic circuit lower bounds. By escaping the confines of set-multilinear transformation requirements, the findings posit a new horizon in proving exponential lower bounds applicable across a spectrum of low-depth circuit configurations. Future pursuits could venture into expanding these bounds across more generalized formulaic and algebraic structures, seeking to further tighten the effective field where these theoretical constructs apply rigorously.