The monotonicity of the Franz-Parisi potential is equivalent with Low-degree MMSE lower bounds
This work addresses a long-standing open problem in theoretical computer science and statistical physics by bridging two key frameworks for understanding computational complexity in inference models, potentially impacting researchers in these fields.
The paper tackles the problem of establishing a mathematical relationship between two distinct approaches for predicting computational hardness in statistical estimation: the Franz-Parisi potential from statistical physics and low-degree polynomial estimators from rigorous mathematics. It shows that for Gaussian additive models, the power of low-degree polynomials is equivalent to the monotonicity of the annealed Franz-Parisi potential, implying that polynomial-time limits are directly implied by this monotonicity, subject to a low-degree conjecture.
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.