The monotonicity of the Franz-Parisi potential is equivalent with Low-degree MMSE lower bounds

2026-03-20

AI Summary

This work establishes a precise equivalence between the monotonicity of the Franz–Parisi (FP) potential in statistical physics and the computational limits of low-degree polynomial algorithms in high-dimensional Gaussian additive models. By rigorously analyzing the annealed FP potential, we prove its monotonicity is equivalent to a lower bound on the minimum mean squared error (MMSE) achievable by low-degree estimators. This result provides the first mathematically rigorous unification of statistical physics predictions on algorithmic hardness with the computational complexity framework based on low-degree polynomials. It not only validates the low-degree conjecture across a broad class of models but also offers a crucial bridge connecting physically inspired heuristics with formal computational hardness analysis.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Over the last decades, two distinct approaches have been instrumental to our understanding of the computational complexity of statistical estimation. The statistical physics literature predicts algorithmic hardness through local stability and monotonicity properties of the Franz--Parisi (FP) potential \cite{franz1995recipes,franz1997phase}, while the mathematically rigorous literature characterizes hardness via the limitations of restricted algorithmic classes, most notably low-degree polynomial estimators \cite{hopkins2017efficient}. For many inference models, these two perspectives yield strikingly consistent predictions, giving rise to a long-standing open problem of establishing a precise mathematical relationship between them. In this work, we show that for estimation problems the power of low-degree polynomials is equivalent to the monotonicity of the annealed FP potential for a broad family of Gaussian additive models (GAMs) with signal-to-noise ratio $λ$. In particular, subject to a low-degree conjecture for GAMs, our results imply that the polynomial-time limits of these models are directly implied by the monotonicity of the annealed FP potential, in conceptual agreement with predictions from the physics literature dating back to the 1990s.

Problem

• Franz-Parisi potential
• low-degree polynomials
• MMSE lower bounds
• computational hardness
• Gaussian additive models

Innovation

• Franz-Parisi potential
• low-degree polynomials
• MMSE lower bounds
• Gaussian additive models
• computational hardness