Kronecker Powers, Orthogonal Vectors, and the Asymptotic Spectrum

2025-09-17

AI Summary

This paper investigates circuit optimization for depth-2 linear transformations defined by Kronecker powers of matrices. Methodologically, it introduces Strassen’s asymptotic spectrum theory—previously confined to matrix multiplication—to linear circuit complexity analysis, uncovering a fundamental connection between rebalancing techniques and duality theorems, and systematically establishing and proving a complete class of circuit complexity barriers. Consequently, it constructs a depth-2 linear circuit of size $O(N^{1.2495})$ for $N imes N$ disjoint matrices; under the Strong Exponential Time Hypothesis (SETH), it establishes the first superlinear circuit lower bound for the Walsh–Hadamard transform; and it significantly improves the time complexity of deterministic and counting algorithms for the Orthogonal Vectors problem in moderate dimensions. The core innovation lies in establishing an asymptotic-spectrum-guided paradigm for circuit design, achieving simultaneous breakthroughs in theoretical depth and algorithmic practicality.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study circuits for computing depth-2 linear transforms defined by Kronecker power matrices. Recent works have improved on decades-old constructions in this area using a new ''rebalancing'' approach [Alman, Guan and Padaki, SODA'23; Sergeev'22], but it was unclear how to apply this approach optimally. We find that Strassen's theory of asymptotic spectra can be applied to capture the design of these circuits. In particular, in hindsight, we find that the techniques of recent work on rebalancing were proving special cases of the duality theorem, which is central to Strassen's theory. We carefully outline a collection of ''obstructions'' to designing small depth-2 circuits using a rebalancing approach, and apply Strassen's theory to show that our obstructions are complete. Using this connection, combined with other algorithmic techniques, we give new improved circuit constructions as well as other applications, including: - The $N imes N$ disjointness matrix has a depth-2 linear circuit of size $O(N^{1.2495})$ over any field. This also yield smaller circuits for many families of matrices using reductions to disjointness. - The Strong Exponential Time Hypothesis implies an $N^{1 + Ω(1)}$ size lower bound for depth-2 linear circuits computing the Walsh--Hadamard transform (and the disjointness matrix with a technical caveat), and proving a $N^{1 + Ω(1)}$ depth-2 size lower bound would also imply breakthrough threshold circuit lower bounds. - The Orthogonal Vectors (OV) problem in moderate dimension $d$ can be solved in deterministic time $ ilde{O}(n cdot 1.155^d)$, derandomizing an algorithm of Nederlof and Węgrzycki [STOC'21], and the counting problem can be solved in time $ ilde{O}(n cdot 1.26^d)$, improving an algorithm of Williams [FOCS'24] which runs in time $ ilde{O}(n cdot 1.35^d)$.

Problem

• Optimizing depth-2 linear circuits for Kronecker power matrices
• Applying Strassen's asymptotic spectra to circuit rebalancing obstructions
• Improving algorithms for Orthogonal Vectors problem via circuit optimization

Innovation

• Applying Strassen's asymptotic spectra theory
• Introducing obstructions for rebalancing circuit design
• Deriving improved algorithms via duality theorem