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

2026-02-24

AI Summary

This work addresses the absence of deterministic polynomial-time identity testing (PIT) and reconstruction algorithms for depth-4 arithmetic circuits with unbounded top fan-in and high-degree powering gates. Focusing on the generalized Waring decomposition model (Σ∧ΣΠ)—sums of high powers of low-degree sparse polynomials—the paper presents the first deterministic PIT and reconstruction algorithms for such depth-4 power circuits where both the top fan-in and bottom degree may grow polynomially, without relying on non-degeneracy or average-case assumptions. The approach leverages the function-field ABC theorem, Wronskian differential operators and their kernel structure, and a robust variant of the Klivans–Spielman hitting set. For $d > r^2$, an explicit hitting set of size $O(r^4 s^4 n^2 d \delta^3)$ is constructed; when $d = \Omega(r^4 \delta)$, reconstruction is achievable in $\mathrm{poly}(n, s, d)$ time over fields of characteristic zero or sufficiently large characteristic.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study deterministic polynomial identity testing (PIT) and reconstruction algorithms for depth-$4$ arithmetic circuits of the form \[ \Sigma^{[r]}\!\wedge^{[d]}\!\Sigma^{[s]}\!\Pi^{[\delta]}. \] 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 $\delta$. 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 $O(r^4 s^4 n^2 d \delta^3)$. The reconstruction algorithm runs in time $\textrm{poly}(n,s,d)$ under the condition $d=\Omega(r^4\delta)$, and in particular it tolerates polynomially large top fan-in $r$ and bottom degree $\delta$. 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.

Problem

• Polynomial Identity Testing
• Arithmetic Circuits
• Depth-4 Circuits
• Circuit Reconstruction
• Powering Circuits

Innovation

• Polynomial Identity Testing
• Depth-4 Arithmetic Circuits
• Deterministic Reconstruction
• Sparse Polynomials
• Wronskian Operators