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Chernoff Bounds for Tensor Expanders on Riemannian Manifolds Using Graph Laplacian Approximation

Shih-Yu Chang

TL;DR

This work addresses concentration bounds for high-dimensional, tensor-valued random objects on curved spaces by linking manifold geometry to graph spectral theory. It introduces a Burago‑style weighted graph approximation $\mathscr{G}_{\mathscr{M}}(\epsilon,\mu,\kappa)$ that faithfully matches the Laplace–Beltrami spectrum up to controllable error, enabling the transfer of tensor expander Chernoff bounds from graphs to manifolds. The main contributions include a spectral closeness bound between graph Laplacians and manifold Laplace–Beltrami operators, a tensor expander Chernoff bound for random walks on manifolds expressed via a minimization over $t>0$ and spectral terms, and corollaries for nonnegative coefficient polynomials with an illustrative $S^n(R)$ example; the framework extends to $T$-product tensors. This enables robust concentration analysis for high-dimensional data on non-Euclidean domains, with implications for manifold learning, diffusion processes, and geometric data analysis.

Abstract

This paper addresses the advancement of probability tail bound analysis, a crucial statistical tool for assessing the probability of large deviations of random variables from their expected values. Traditional tail bounds, such as Markov's, Chebyshev's, and Chernoff bounds, have proven valuable across numerous scientific and engineering fields. However, as data complexity grows, there is a pressing need to extend tail bound estimation from scalar variables to high-dimensional random objects. Existing studies often rely on the assumption of independence among high-dimensional random objects, an assumption that may not always be valid. Building on the work of researchers like Garg et al. and Chang, who employed random walks to model high-dimensional ensembles, this study introduces a more generalized approach by exploring random walks over manifolds. To address the challenges of constructing an appropriate underlying graph for a manifold, we propose a novel method that enhances random walks on graphs approximating the manifold. This approach ensures spectral similarity between the original manifold and the approximated graph, including matching eigenvalues, eigenvectors, and eigenfunctions. Leveraging graph approximation technique proposed by Burago et al. for manifolds, we derive the tensor Chernoff bound and establish its range for random walks on a Riemannian manifold according to the underlying manifold's spectral characteristics.

Chernoff Bounds for Tensor Expanders on Riemannian Manifolds Using Graph Laplacian Approximation

TL;DR

This work addresses concentration bounds for high-dimensional, tensor-valued random objects on curved spaces by linking manifold geometry to graph spectral theory. It introduces a Burago‑style weighted graph approximation that faithfully matches the Laplace–Beltrami spectrum up to controllable error, enabling the transfer of tensor expander Chernoff bounds from graphs to manifolds. The main contributions include a spectral closeness bound between graph Laplacians and manifold Laplace–Beltrami operators, a tensor expander Chernoff bound for random walks on manifolds expressed via a minimization over and spectral terms, and corollaries for nonnegative coefficient polynomials with an illustrative example; the framework extends to -product tensors. This enables robust concentration analysis for high-dimensional data on non-Euclidean domains, with implications for manifold learning, diffusion processes, and geometric data analysis.

Abstract

This paper addresses the advancement of probability tail bound analysis, a crucial statistical tool for assessing the probability of large deviations of random variables from their expected values. Traditional tail bounds, such as Markov's, Chebyshev's, and Chernoff bounds, have proven valuable across numerous scientific and engineering fields. However, as data complexity grows, there is a pressing need to extend tail bound estimation from scalar variables to high-dimensional random objects. Existing studies often rely on the assumption of independence among high-dimensional random objects, an assumption that may not always be valid. Building on the work of researchers like Garg et al. and Chang, who employed random walks to model high-dimensional ensembles, this study introduces a more generalized approach by exploring random walks over manifolds. To address the challenges of constructing an appropriate underlying graph for a manifold, we propose a novel method that enhances random walks on graphs approximating the manifold. This approach ensures spectral similarity between the original manifold and the approximated graph, including matching eigenvalues, eigenvectors, and eigenfunctions. Leveraging graph approximation technique proposed by Burago et al. for manifolds, we derive the tensor Chernoff bound and establish its range for random walks on a Riemannian manifold according to the underlying manifold's spectral characteristics.
Paper Structure (3 sections, 2 theorems, 18 equations)

This paper contains 3 sections, 2 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Graph Approximation of a Riemannian Manifold
  3. Tensor Expander Chernoff Bounds over a Riemannian Manifold

Key Result

Theorem 1

Let $\mathscr{G}_{\mathscr{M}}(\epsilon,\mu,\kappa)=(\mathscr{V}, \mathscr{E}, \mathscr{W})$ be the approximation graph for $\mathscr{M}$ whose transition matrix has second largest eigenvalue $\lambda_{\bm{P}_{\mathscr{G}_{\mathscr{M}}},\tilde{i}}$ for some $\tilde{i} \in \{1,2,\ldots,N\}$, and let Then, we have where $\overline{\lambda_{\bm{P}_{\mathscr{G}_{\mathscr{M}}},\tilde{i}}} = 1 - \lamb

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Example 1
  • Remark 1