[Paper Review] Quasi-polynomial Hitting-set for Set-depth-Delta Formulas

Manindra Agrawal, Chandan Saha+1 more|arXiv (Cornell University)|Sep 11, 2012

Complexity and Algorithms in Graphs3 citations

TL;DR

This paper presents a quasi-polynomial time hitting-set generator for set-depth-Δ formulas over any field, achieving blackbox identity testing in time polynomial in exp((Δ² log s)^{Δ−1}), where s is the formula size. The approach uses variable shifting and low-support rank concentration in Hadamard algebras, proving a key algebraic conjecture for this class and resolving an open problem for depth-3 formulas and up to depth ε log s / log log s for ε < 1.

ABSTRACT

We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition (X_1,...,X_d) of the variable indices [n] that the top product layer respects, i.e. C(x) = \sum_{i=1}^k \prod_{j=1}^{d} f_{i,j}(x_{X_j}), where f_{i,j} is a sparse polynomial in F[x_{X_j}]. Extending this definition to any depth - we call a depth-Delta formula C (consisting of alternating layers of Sigma and Pi gates, with a Sigma-gate on top) a set-depth-Delta formula if every Pi-layer in C respects a (unknown) partition on the variables; if Delta is even then the product gates of the bottom-most Pi-layer are allowed to compute arbitrary monomials. In this work, we give a hitting-set generator for set-depth-Delta formulas (over any field) with running time polynomial in exp(({Delta}^2 log s)^{Delta - 1}), where s is the size bound on the input set-depth-Delta formula. In other words, we give a quasi-polynomial time blackbox polynomial identity test for such constant-depth formulas. Previously, the very special case of Delta=3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator. This was declared as an open problem by Shpilka &amp; Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan &amp; Wigderson (FOCS 1995). Our work settles this question, not only for depth-3 but, up to depth epsilon.log s / loglog s, for a fixed constant epsilon &lt; 1. The technique is to investigate depth-Delta formulas via depth-(Delta-1) formulas over a Hadamard algebra, after applying a `shift' on the variables. We propose a new algebraic conjecture about the low-support rank-concentration in the latter formulas, and manage to prove it in the case of set-depth-Delta formulas.

Motivation & Objective

• To develop a blackbox polynomial identity test (PIT) for set-depth-Δ formulas, a natural class of constant-depth arithmetic circuits.
• To resolve the open problem of constructing a sub-exponential hitting-set generator for depth-3 set-multilinear formulas, previously unresolved since Nisan & Wigderson (1995).
• To extend the reach of hitting-set generation to formulas of depth up to ε log s / log log s for a fixed ε < 1.
• To prove a new algebraic conjecture on low-support rank concentration in depth-(Δ−1) formulas over Hadamard algebras, which enables the construction of efficient hitting sets.

Proposed method

• Apply a variable shift to transform the original formula into a form where low-support monomials become detectable.
• Use a recursive construction in a Hadamard algebra to reduce the depth-Δ formula to a depth-(Δ−1) formula with structured coefficients.
• Prove that after shifting, the resulting formula exhibits low-support rank concentration, meaning a small number of low-degree monomials span the space of low-support terms.
• Leverage this concentration to construct a hitting set by testing the formula on restricted variable sets of size ℓ, where ℓ is small and polynomial in the depth parameter.
• Use sparse polynomial identity testing on the restricted formulas, which is efficient due to their low sparsity.
• Combine the shift and sparse PIT steps into a hitting-set generator with quasi-polynomial running time in terms of the formula size s and depth Δ.

Experimental results

Research questions

• RQ1 Can a quasi-polynomial time hitting-set generator be constructed for set-depth-Δ formulas over any field, extending beyond depth-3 circuits?
• RQ2 Does the phenomenon of low-support rank concentration hold in depth-(Δ−1) formulas over Hadamard algebras after a variable shift, enabling efficient identity testing?
• RQ3 Can the hitting-set generator be extended to formulas of depth up to ε log s / log log s for a fixed ε < 1, improving on previous exponential-time bounds?
• RQ4 Is it possible to derandomize blackbox PIT for set-depth-Δ formulas using partial derivatives and variable shifting techniques?
• RQ5 Can the low-support rank concentration conjecture be proven for a broader class of formulas beyond set-depth-Δ, potentially leading to polynomial-time PIT?

Key findings

• A quasi-polynomial time hitting-set generator is constructed for set-depth-Δ formulas, with running time polynomial in exp((Δ² log s)^{Δ−1}), where s is the formula size.
• The construction resolves the open problem of sub-exponential hitting-set generation for depth-3 set-multilinear formulas, which was posed by Shpilka & Yehudayoff (2010).
• The method achieves blackbox PIT for formulas of depth up to ε log s / log log s for any fixed ε < 1, extending the reach of known derandomization results.
• The key technical contribution is a proof of low-support rank concentration in depth-(Δ−1) formulas over Hadamard algebras after a variable shift, which enables the construction.
• For semi-diagonal formulas and set-depth-6 formulas, improved hitting-set times of s^{O(log s)} and s^{O(log² s)} are achieved, respectively, via duality and refined analysis.
• The results demonstrate that low-support rank concentration is a viable and powerful structural property for blackbox identity testing in constant-depth arithmetic circuits.