# Local convexity of the TAP free energy and AMP convergence for Z2-synchronization

@article{Celentano2021LocalCO, title={Local convexity of the TAP free energy and AMP convergence for Z2-synchronization}, author={Michael Celentano and Zhou Fan and Song Mei}, journal={ArXiv}, year={2021}, volume={abs/2106.11428} }

We study mean-field variational Bayesian inference using the TAP approach, for Z2-synchronization as a prototypical example of a high-dimensional Bayesian model. We show that for any signal strength λ > 1 (the weak-recovery threshold), there exists a unique local minimizer of the TAP free energy functional near the mean of the Bayes posterior law. Furthermore, the TAP free energy in a local neighborhood of this minimizer is strongly convex. Consequently, a natural-gradient/mirror-descent…

## 5 Citations

### Sudakov-Fernique post-AMP, and a new proof of the local convexity of the TAP free energy

- Mathematics, Computer ScienceArXiv
- 2022

An asymptotic comparison inequality is derived, which is called the Sudakov-Fernique post-AMP inequality, which, in a certain class of problems involving a GOE matrix, is able to probe properties of an optimization landscape locally around the iterates of an approximate message passing (AMP) algorithm.

### Approximate Message Passing for orthogonally invariant ensembles: Multivariate non-linearities and spectral initialization

- Mathematics, Computer Science
- 2021

A BayesOAMP algorithm that uses as its non-linearity the posterior mean conditional on all preceding AMP iterates, which derives the forms of the Onsager debiasing coefficients and corresponding AMP state evolution, which depend on the free cumulants of the noise spectral distribution.

### The TAP free energy for high-dimensional linear regression

- Computer Science, Mathematics
- 2022

This work rigorously establishes the Thouless-Anderson-Palmer (TAP) approximation arising from spin glass theory, and proves a conjecture of [23] in the special case of the spherical prior (at sufficiently high temperature).

### Minimum 𝓁1-norm interpolators: Precise asymptotics and multiple descent

- Computer ScienceArXiv
- 2021

This paper considers the noisy sparse regression model under Gaussian design, focusing on linear sparsity and high-dimensional asymptotics (so that both the number of features and the sparsity level scale proportionally with the sample size), and provides rigorous theoretical justification for a curious multi-descent phenomenon.

### Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization

- Computer Science, MathematicsArXiv
- 2022

This work proves that, for any inverse temperature β < 1/2, there exists an algorithm with complexity O(n) that samples from a distribution μ which is close in normalized Wasserstein distance to μ, and introduces a suitable “stability” property for sampling algorithms, which is verified by many standard techniques.

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