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Polynomial Identity Testing and Reconstruction for Depth-4 Powering Circuits of High Degree

Amir Shpilka, Yann Tal

TL;DR

These algorithms provide the first polynomial-time deterministic solutions for depth-$4$ powering circuits with unbounded top fan-in and a robust version of the Klivans-Spielman hitting set.

Abstract

We study deterministic polynomial identity testing (PIT) and reconstruction algorithms for depth-$4$ arithmetic circuits of the form \[ Σ^{[r]}\!\wedge^{[d]}\!Σ^{[s]}\!Π^{[δ]}. \] This model generalizes Waring decompositions and diagonal circuits, and captures sums of powers of low-degree sparse polynomials. Specifically, each circuit computes a sum of $r$ terms, where each term is a $d$-th power of an $s$-sparse polynomial of degree $δ$. This model also includes algebraic representations that arise in tensor decomposition and moment-based learning tasks such as mixture models and subspace learning. We give deterministic worst-case algorithms for PIT and reconstruction in this model. Our PIT construction applies when $d>r^2$ and yields explicit hitting sets of size $O(r^4 s^4 n^2 d δ^3)$. The reconstruction algorithm runs in time $\textrm{poly}(n,s,d)$ under the condition $d=Ω(r^4δ)$, and in particular it tolerates polynomially large top fan-in $r$ and bottom degree $δ$. Both results hold over fields of characteristic zero and over fields of sufficiently large characteristic. These algorithms provide the first polynomial-time deterministic solutions for depth-$4$ powering circuits with unbounded top fan-in. In particular, the reconstruction result improves upon previous work which required non-degeneracy or average-case assumptions. The PIT construction relies on the ABC theorem for function fields (Mason-Stothers theorem), which ensures linear independence of high-degree powers of sparse polynomials after a suitable projection. The reconstruction algorithm combines this with Wronskian-based differential operators, structural properties of their kernels, and a robust version of the Klivans-Spielman hitting set.

Polynomial Identity Testing and Reconstruction for Depth-4 Powering Circuits of High Degree

TL;DR

These algorithms provide the first polynomial-time deterministic solutions for depth- powering circuits with unbounded top fan-in and a robust version of the Klivans-Spielman hitting set.

Abstract

We study deterministic polynomial identity testing (PIT) and reconstruction algorithms for depth- arithmetic circuits of the form \[ Σ^{[r]}\!\wedge^{[d]}\!Σ^{[s]}\!Π^{[δ]}. \] This model generalizes Waring decompositions and diagonal circuits, and captures sums of powers of low-degree sparse polynomials. Specifically, each circuit computes a sum of terms, where each term is a -th power of an -sparse polynomial of degree . This model also includes algebraic representations that arise in tensor decomposition and moment-based learning tasks such as mixture models and subspace learning. We give deterministic worst-case algorithms for PIT and reconstruction in this model. Our PIT construction applies when and yields explicit hitting sets of size . The reconstruction algorithm runs in time under the condition , and in particular it tolerates polynomially large top fan-in and bottom degree . Both results hold over fields of characteristic zero and over fields of sufficiently large characteristic. These algorithms provide the first polynomial-time deterministic solutions for depth- powering circuits with unbounded top fan-in. In particular, the reconstruction result improves upon previous work which required non-degeneracy or average-case assumptions. The PIT construction relies on the ABC theorem for function fields (Mason-Stothers theorem), which ensures linear independence of high-degree powers of sparse polynomials after a suitable projection. The reconstruction algorithm combines this with Wronskian-based differential operators, structural properties of their kernels, and a robust version of the Klivans-Spielman hitting set.
Paper Structure (24 sections, 24 theorems, 103 equations, 3 algorithms)

This paper contains 24 sections, 24 theorems, 103 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Prior Work and Related Models
  4. Polynomial Identity Testing for Small-Depth Circuits
  5. Reconstruction
  6. Proof Overview
  7. Reconstruction in the univariate case.
  8. Multivariate reconstruction.
  9. Organization
  10. Preliminaries
  11. The ABC theorem for function fields
  12. The Klivans–Spielman generator
  13. Wronskian and Differential Operators over Finite Fields
  14. Polynomial Factorization
  15. Hitting set for Sigma^[r] and^[d] Sigma Pi circuits with $r^2\le d$
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $n,d,r,s,\delta\in \mathbb{N}$ such that $d=\Omega(r^2)$. Let $\mathbb{F}$ be a field of characteristic $p=0$ or $p\ge rd\delta (s^2n+\delta)$. There is an explicit hitting set of size $\operatorname{poly}(s,n,d)$ for the class of $\Sigma^{[r]}\!\wedge^{[d]}\!\Sigma^{[s]}\!\Pi^{[\delta]}$ circ

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: vaserstein2003vanishing, Theorem 2.2(a)
  • Corollary 2.2
  • proof
  • Claim 2.3
  • proof
  • Theorem 2.4: vaserstein2003vanishing
  • Theorem 2.5: Corollary of Klivans–Spielman KS01
  • Corollary 2.6
  • ...and 55 more