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Estimating rank-one matrices with mismatched prior and noise: universality and large deviations

Alice Guionnet, Justin Ko, Florent Krzakala, Lenka Zdeborová

TL;DR

This work addresses rank-one matrix estimation under mismatched priors and channels, establishing a universality principle that reduces general inference to a Gaussian SK-type model with parameters \bar{\beta} determined by generalized Fisher information. It proves an almost-sure large deviation principle for overlaps (the Franz–Parisi potential) and shows that these deviations—and hence the free-energy limit—are universal across noise models, including non-Gaussian channels. The core methodology combines Taylor expansion universality, Guerra interpolation, the cavity method, and the Ghirlanda–Guerra identities, culminating in a Parisi-type variational formula for the limiting free energy. The results unify a broad class of problems (e.g., spiked Wigner, submatrix localization, BBP) under a common variational framework and illuminate information–algorithmic gaps via the Franz–Parisi potential, with implications for understanding phase diagrams in mismatched inference. Overall, the paper provides a rigorous bridge from general mismatched inference to Gaussian-based analysis, delivering precise asymptotics for both the free energy and the overlaps.

Abstract

We prove a universality result that reduces the free energy of rank-one matrix estimation problems in the setting of mismatched prior and noise to the computation of the free energy for a modified Sherrington-Kirkpatrick spin glass. Our main result is an almost sure large deviation principle for the overlaps between the truth signal and the estimator for both the Bayes-optimal and mismatched settings. Through the large deviations principle, we recover the limit of the free energy in mismatched inference problems and the universality of the overlaps.

Estimating rank-one matrices with mismatched prior and noise: universality and large deviations

TL;DR

This work addresses rank-one matrix estimation under mismatched priors and channels, establishing a universality principle that reduces general inference to a Gaussian SK-type model with parameters \bar{\beta} determined by generalized Fisher information. It proves an almost-sure large deviation principle for overlaps (the Franz–Parisi potential) and shows that these deviations—and hence the free-energy limit—are universal across noise models, including non-Gaussian channels. The core methodology combines Taylor expansion universality, Guerra interpolation, the cavity method, and the Ghirlanda–Guerra identities, culminating in a Parisi-type variational formula for the limiting free energy. The results unify a broad class of problems (e.g., spiked Wigner, submatrix localization, BBP) under a common variational framework and illuminate information–algorithmic gaps via the Franz–Parisi potential, with implications for understanding phase diagrams in mismatched inference. Overall, the paper provides a rigorous bridge from general mismatched inference to Gaussian-based analysis, delivering precise asymptotics for both the free energy and the overlaps.

Abstract

We prove a universality result that reduces the free energy of rank-one matrix estimation problems in the setting of mismatched prior and noise to the computation of the free energy for a modified Sherrington-Kirkpatrick spin glass. Our main result is an almost sure large deviation principle for the overlaps between the truth signal and the estimator for both the Bayes-optimal and mismatched settings. Through the large deviations principle, we recover the limit of the free energy in mismatched inference problems and the universality of the overlaps.
Paper Structure (25 sections, 31 theorems, 349 equations)

This paper contains 25 sections, 31 theorems, 349 equations.

Table of Contents

  1. Introduction
  2. Setting and main theorems
  3. Main results
  4. Examples
  5. Sherrington-Kirkpatrick model with soft spins PVS
  6. Sherrington-Kirkpatrick model with ferromagnetic interaction chenferro
  7. Sherrington-Kirkpatrick model with Mattis interaction Camillimismatch
  8. Symmetric Rank 1 Matrix Estimation lelarge2017fundamental
  9. Maximum-average Submatrix Problem erbasubmatrix
  10. The BBP Transition BBP
  11. Outline of the Paper
  12. Universality
  13. Weak Large deviation Upper Bound
  14. Large deviation lower bound
  15. Large Deviations under the reference measure
  16. ...and 10 more sections

Key Result

Proposition 2.4

If Hypothesis hypcompact, hypg, and hypderiv hold, then the free energy of the vector spin models satisfy for $N$ large enough where $\bar{\beta}=({\beta},{\beta_{SNR}},{\beta_{S}})$ is given by More generally, for any sequence of measurable sets $A = A_N({\bm{x}}^0) \subset {\mathbb R}^N$ such that $\liminf_{N \to \infty} F_{N}(0:A)>-\infty$,

Theorems & Definitions (60)

  • Proposition 2.4: Universality
  • Remark 2.5
  • Theorem 2.6: Limit of the Free Energy
  • Theorem 2.7
  • Remark 2.8
  • Corollary 2.9
  • Corollary 2.10
  • Theorem 2.11
  • Remark 2.12
  • Remark 2.13
  • ...and 50 more