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Spectral Algorithms on Manifolds through Diffusion

Weichun Xia, Lei Shi

TL;DR

This work develops a diffusion-space RKHS framework built from the heat kernel on a compact manifold to analyze spectral regression algorithms in high dimensions. By leveraging integral-operator techniques and the embedding properties of diffusion spaces, it achieves fast convergence rates that depend on the intrinsic dimension $m$ and demonstrate strong norm and derivative convergence, including minimax lower bounds that are tight in the $L^2$ setting. The exponential decay of heat-kernel eigenvalues provides improved rates over polynomial-decay kernels and enables $C^k$-convergence for derivatives via RKHS embeddings. Overall, the results provide a geometry-aware, theoretically grounded approach to high-dimensional regression on manifolds, with practical implications for diffusion-based spectral methods.

Abstract

The existing research on spectral algorithms, applied within a Reproducing Kernel Hilbert Space (RKHS), has primarily focused on general kernel functions, often neglecting the inherent structure of the input feature space. Our paper introduces a new perspective, asserting that input data are situated within a low-dimensional manifold embedded in a higher-dimensional Euclidean space. We study the convergence performance of spectral algorithms in the RKHSs, specifically those generated by the heat kernels, known as diffusion spaces. Incorporating the manifold structure of the input, we employ integral operator techniques to derive tight convergence upper bounds concerning generalized norms, which indicates that the estimators converge to the target function in strong sense, entailing the simultaneous convergence of the function itself and its derivatives. These bounds offer two significant advantages: firstly, they are exclusively contingent on the intrinsic dimension of the input manifolds, thereby providing a more focused analysis. Secondly, they enable the efficient derivation of convergence rates for derivatives of any k-th order, all of which can be accomplished within the ambit of the same spectral algorithms. Furthermore, we establish minimax lower bounds to demonstrate the asymptotic optimality of these conclusions in specific contexts. Our study confirms that the spectral algorithms are practically significant in the broader context of high-dimensional approximation.

Spectral Algorithms on Manifolds through Diffusion

TL;DR

This work develops a diffusion-space RKHS framework built from the heat kernel on a compact manifold to analyze spectral regression algorithms in high dimensions. By leveraging integral-operator techniques and the embedding properties of diffusion spaces, it achieves fast convergence rates that depend on the intrinsic dimension and demonstrate strong norm and derivative convergence, including minimax lower bounds that are tight in the setting. The exponential decay of heat-kernel eigenvalues provides improved rates over polynomial-decay kernels and enables -convergence for derivatives via RKHS embeddings. Overall, the results provide a geometry-aware, theoretically grounded approach to high-dimensional regression on manifolds, with practical implications for diffusion-based spectral methods.

Abstract

The existing research on spectral algorithms, applied within a Reproducing Kernel Hilbert Space (RKHS), has primarily focused on general kernel functions, often neglecting the inherent structure of the input feature space. Our paper introduces a new perspective, asserting that input data are situated within a low-dimensional manifold embedded in a higher-dimensional Euclidean space. We study the convergence performance of spectral algorithms in the RKHSs, specifically those generated by the heat kernels, known as diffusion spaces. Incorporating the manifold structure of the input, we employ integral operator techniques to derive tight convergence upper bounds concerning generalized norms, which indicates that the estimators converge to the target function in strong sense, entailing the simultaneous convergence of the function itself and its derivatives. These bounds offer two significant advantages: firstly, they are exclusively contingent on the intrinsic dimension of the input manifolds, thereby providing a more focused analysis. Secondly, they enable the efficient derivation of convergence rates for derivatives of any k-th order, all of which can be accomplished within the ambit of the same spectral algorithms. Furthermore, we establish minimax lower bounds to demonstrate the asymptotic optimality of these conclusions in specific contexts. Our study confirms that the spectral algorithms are practically significant in the broader context of high-dimensional approximation.
Paper Structure (8 sections, 17 theorems, 142 equations)

This paper contains 8 sections, 17 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Upper Bounds Analysis
  5. Bounding approximation error
  6. Bounding estimation error
  7. Proof of Theorem \ref{['Theorem: Upper Bound']}
  8. Lower Bound Analysis

Key Result

Theorem 1

Suppose that the regularization family $\{g_\lambda\}_{\lambda>0}$ has qualification $\xi\geq\frac{1}{2}$, the moment condition Moment Condition in Assumption Assumption: Moment Condition holds for some positive constants $\sigma, L$, and the source condition Source Condition in Assumption Assumptio Then, for all $t>0$, $\tau\geq1$, the spectral algorithm estimators $f_{D,\lambda_n}$ with respect

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 13 more