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

TL;DR

This paper establishes a precise mathematical link between the monotonicity of the Franz-Parisi potential and the low-degree polynomial lower bounds in Gaussian additive models, connecting physics-based predictions with rigorous computational complexity results.

Contribution

It proves that for Gaussian additive models, the low-degree polynomial lower bounds are equivalent to the monotonicity of the annealed Franz-Parisi potential, bridging physics and computational complexity.

Findings

Low-degree polynomial bounds match FP potential monotonicity predictions

Results apply to a broad family of Gaussian additive models

Supports physics-based conjectures on computational hardness

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 $\lambda$. 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.