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Stochastic Localization with Non-Gaussian Tilts and Applications to Tensor Ising Models

Dan Mikulincer, Arianna Piana

Abstract

We present generalizations and modifications of Eldan's Stochastic Localization process, extending it to incorporate non-Gaussian tilts, making it useful for a broader class of measures. As an application, we introduce new processes that enable the decomposition and analysis of non-quadratic potentials on the Boolean hypercube, with a specific focus on quartic polynomials. Using this framework, we derive new spectral gap estimates for tensor Ising models under Glauber dynamics, resulting in rapid mixing.

Stochastic Localization with Non-Gaussian Tilts and Applications to Tensor Ising Models

Abstract

We present generalizations and modifications of Eldan's Stochastic Localization process, extending it to incorporate non-Gaussian tilts, making it useful for a broader class of measures. As an application, we introduce new processes that enable the decomposition and analysis of non-quadratic potentials on the Boolean hypercube, with a specific focus on quartic polynomials. Using this framework, we derive new spectral gap estimates for tensor Ising models under Glauber dynamics, resulting in rapid mixing.

Paper Structure

This paper contains 33 sections, 26 theorems, 179 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Sampling via Glauber Dynamics.
  3. Two questions related to SL.
  4. Tensor Ising models and a naïve approach for fourth-order tensors.
  5. Contributions.
  6. Further related works.
  7. Overview.
  8. Conventions.
  9. Acknowledgments.
  10. Fourth-order tensors
  11. Fourth-order tensors as multi-linear forms
  12. Fourth-order tensors as matrix operators
  13. Concentration of the injective norm
  14. Tensorized Stochastic Localization
  15. Decomposing lg-tensors into tensor products of matrices
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $\varphi: \mathcal{C}_n \rightarrow {\mathbb{R}}$ be a test function and let $\mu$ be a measure on $\mathcal{C}_n$ given by where $T$ is a positive definite symmetric fourth-order tensor with zero diagonal entries. Then, for every $\delta>0$, there exists a decomposition of $\mu$ as where with $u, v, w,\ell \in {\mathbb{R}}^n$ and $\psi:\mathcal{C}_n\to {\mathbb{R}}$, and $\mu_{u, v, w, \ell

Figures (3)

  • Figure 1: Decomposition of a fourth-order tensor.
  • Figure 2: Decomposition of a order $8$ tensor $T_8$.
  • Figure 3: Decomposition of a tensor of degree 16, $T_{16}$. First, it is decomposed into $T_8^{(1)} \otimes T_8^{(2)} + T_8^{(3)}$. Each $T_8^{(i)}$ is then further decomposed as described in the previous section. The final decomposition consists of rank-1 tensors $T_1^{(i,j,k)}$ with $i, j, k = 1, 2, 3$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 37 more