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Generalizing Geometric Nonwindowed Scattering Transforms on Compact Riemannian Manifolds

Albert Chua, Yang Yang

Abstract

Let $\mathcal{M}$ be a compact, smooth, $n$-dimensional Riemannian manifold without boundary. In this paper, we generalize nonwindowed geometric scattering transforms, which we formulate as $\mathbf{L}^q(\mathcal{M})$ norms of a cascade of geometric wavelet transforms and modulus operators. We then provide weighted measures for these operators, prove that these operators are well-defined under specific conditions on the manifold, invariant to the action of isometries, and stable to diffeomorphisms for $λ$-bandlimited functions.

Generalizing Geometric Nonwindowed Scattering Transforms on Compact Riemannian Manifolds

Abstract

Let be a compact, smooth, -dimensional Riemannian manifold without boundary. In this paper, we generalize nonwindowed geometric scattering transforms, which we formulate as norms of a cascade of geometric wavelet transforms and modulus operators. We then provide weighted measures for these operators, prove that these operators are well-defined under specific conditions on the manifold, invariant to the action of isometries, and stable to diffeomorphisms for -bandlimited functions.
Paper Structure (13 sections, 18 theorems, 110 equations)

This paper contains 13 sections, 18 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Invariance and Stability
  3. A Review of the Geometric Scattering Transform on Manifolds
  4. Spectral Filters and the Geometric Wavelet Transform
  5. The Geometric Scattering Transform
  6. Generalization of the Littlewood Paley $g$-function
  7. Generalizing Nonwindowed Geometric Scattering
  8. The $2$-nonwindowed Geometric Scattering Norm
  9. The $q$-nonwindowed Geometric Scattering Norm
  10. Diffeomorphism Stability
  11. Stability Results for the $q$-nonwindowed Geometric Scattering Norm
  12. Conclusions and Future Work
  13. Acknowledgements

Key Result

Theorem 1

Let $G : [0, \infty) \to \mathbb{R}$ be nonnegative, decreasing, and continuous with $0 < G(0) = C$, $\lim_{x \to \infty}G(x) = 0$, and $\{\psi_j\}_{j \in \mathbb{Z}}$ is a set of wavelets generated by using the low pass filter $\hat{\phi}(k) = G(\lambda_k)$ in Equation eq: wavelet definition. Then

Theorems & Definitions (29)

  • Theorem 1
  • proof
  • Lemma 2: Geller_Pesenson_2014
  • Lemma 3: geller2009continuous
  • proof
  • Lemma 4
  • proof
  • Theorem 5: Coifman1971AnalyseHN
  • Lemma 6
  • proof
  • ...and 19 more