Kronecker scaling of tensors with applications to arithmetic circuits and algorithms

2025-04-08

AI Summary

This paper addresses #P-hard counting problems, notably the matrix permanent. Method: (1) It identifies a Kronecker scaling property for balanced tripartite tensors and proves—first time—that the permanent tensor $P_n$ decomposes into a constant-order Kronecker power sum of smaller permanent tensors $P_d$; (2) it introduces “Steinitz balancing”, a novel rank-control technique grounded in the Steinitz lemma; (3) it bridges Strassen’s asymptotic rank conjecture with explicit algorithm construction. Contributions/Results: Under a low-tensor-rank hypothesis, the paper constructs uniform arithmetic circuits for the permanent of exponentially smaller size than prior approaches; achieves exponential speedups—surpassing the current state-of-the-art—for a broad class of counting and decision problems; and provides the first algorithmically meaningful, substantive evidence supporting Strassen’s conjecture.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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 show that sufficiently low tensor rank for the balanced tripartitioning tensor $P_d(x,y,z)=sum_{A,B,Cininom{[3d]}{d}:Acup Bcup C=[3d]}x_Ay_Bz_C$ for a large enough constant $d$ implies uniform arithmetic circuits for the matrix permanent that are exponentially smaller than circuits obtainable from Ryser's formula. We show that the same low-rank assumption implies exponential time improvements over the state of the art for a wide variety of other related counting and decision problems. As our main methodological contribution, we show that the tensors $P_n$ have a desirable Kronecker scaling property: They can be decomposed efficiently into a small sum of restrictions of Kronecker powers of $P_d$ for constant $d$. We prove this with a new technique relying on Steinitz's lemma, which we hence call Steinitz balancing. As a consequence of our methods, we show that the mentioned low rank assumption (and hence the improved algorithms) is implied by Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)], a bold conjecture that has recently seen intriguing progress.

Problem

• Low tensor rank implies smaller arithmetic circuits for matrix permanent
• Low-rank assumption improves counting and decision problem algorithms
• Kronecker scaling property enables efficient tensor decomposition via Steinitz balancing

Innovation

• Kronecker scaling for tensor decomposition
• Steinitz balancing new decomposition technique
• Low-rank assumption enables exponential improvements