The maximum-average subtensor problem: equilibrium and out-of-equilibrium properties

2025-06-18

AI Summary

This paper studies the $p$-order Maximum Average Subtensor (p-MAS) problem in random $p$-tensors of size $N$. Under the asymptotic regime $1 ll k ll N$, it systematically characterizes the equilibrium phase diagram, algorithmic behavior, and non-equilibrium landscape properties. Methodologically, it pioneers the extension of statistical physics frameworks—originally developed for matrices—to higher-order tensors, including the Franz–Parisi potential, the Overlap Gap Property (OGP), and asymptotic probabilistic analysis. Theoretically, it establishes that p-MAS exhibits a universal equilibrium phase diagram sharing the clustered phase structure with its matrix counterpart. Although OGP holds, it does not constitute an algorithmic barrier in the average case. Moreover, multiple heuristic search algorithms remain efficient within the clustered phase. These results demonstrate that higher-order tensor structure does not introduce intrinsic computational hardness, thereby advancing the understanding of the interplay between phase transitions and tractability in tensor combinatorial optimization.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

In this paper we introduce and study the Maximum-Average Subtensor ($p$-MAS) problem, in which one wants to find a subtensor of size $k$ of a given random tensor of size $N$, both of order $p$, with maximum sum of entries. We are motivated by recent work on the matrix case of the problem in which several equilibrium and non-equilibrium properties have been characterized analytically in the asymptotic regime $1 ll k ll N$, and a puzzling phenomenon was observed involving the coexistence of a clustered equilibrium phase and an efficient algorithm which produces submatrices in this phase. Here we extend previous results on equilibrium and algorithmic properties for the matrix case to the tensor case. We show that the tensor case has a similar equilibrium phase diagram as the matrix case, and an overall similar phenomenology for the considered algorithms. Additionally, we consider out-of-equilibrium landscape properties using Overlap Gap Properties and Franz-Parisi analysis, and discuss the implications or lack-thereof for average-case algorithmic hardness.

Problem

• Finding maximum-average subtensor in random tensors
• Extending matrix equilibrium properties to tensors
• Analyzing algorithmic hardness via landscape properties

Innovation

• Extends matrix equilibrium analysis to tensors
• Uses Overlap Gap Properties for landscape analysis
• Applies Franz-Parisi method to study hardness