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

arXiv: https://arxiv.org/abs/2603.20070 · HTML: https://mlsanity.com/paper/2603.20070/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Mar 2026.

## Built on by (most cited first, 1 of 1)

- [High-Dimensional Statistics: Reflections on Progress and Open Problems](https://mlsanity.com/paper/2605.05076/) (2026)

## Abstract (arXiv)

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 citefranz1995recipes,franz1997phase, while the mathematically rigorous literature characterizes hardness via the limitations of restricted algorithmic classes, most notably low-degree polynomial estimators citehopkins2017efficient. 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.