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

Konstantinos Tsirkas, Leda Wang, Ilias Zadik

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.

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

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 . 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.
Paper Structure (87 sections, 79 theorems, 675 equations, 1 figure, 1 algorithm)

This paper contains 87 sections, 79 theorems, 675 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. MMSE lower bounds against low-degree polynomials
  3. Monotonicity of the Franz-Parisi potential
  4. A mathematical relation? Obstacles and prior results
  5. An equivalence for detection settings by another FP criterion
  6. Contributions
  7. Main Result: Equivalence between FP monotonicity and low-degree MMSE
  8. Low Degree correlation upper bound from decreasing FP
  9. Low Degree correlation lower bound from increasing FP
  10. The equivalence for discrete priors
  11. Low Degree correlation upper bound from decreasing FP (discrete case)
  12. Low Degree correlation lower bound from increasing FP (discrete case)
  13. Proof ideas
  14. Applications
  15. Tensor PCA models
  16. ...and 72 more sections

Key Result

Theorem 1.1

For any low-order cumulant-nonnegative GAM with prior $P_0$, for any degree $0 < D \le \mathrm{poly}(\log n)$ and any SNR $\lambda>0$, the optimal degree-$D$ correlation satisfies where the approximation hides only polylogarithmic factors in $n$, and $q(D)$ denotes the $e^{-D}$-quantile of the overlap $|\langle X,X' \rangle|$ (see Definition def:quantile_ov).

Figures (1)

  • Figure 1: Pictorial representations of our equivalence.

Theorems & Definitions (150)

  • Theorem 1.1: Informal, see Theorems \ref{['thm:mainthm']}, \ref{['thm:corr-lower-bound']}.
  • Corollary 1.2: Informal
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1: Quantile function
  • Definition 2.2: Sub-Weibull$(\phi)$
  • Remark 2.3: Examples of sub-Weibull random variables
  • Remark 2.5: On Assumption (1)
  • Remark 2.6: On Assumption (4)
  • Theorem 2.7: Increasing FP potential implies Low-Degree hard
  • ...and 140 more