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On the existence of Parseval frames for vector bundles

Samuel A. Ballas, Tom Needham, Clayton Shonkwiler

TL;DR

This work extends Parseval frame theory to vector bundles by recasting the problem as sections of the $n$-Parseval bundle $\mathcal{P}^n(E)$ and employing obstruction theory and classifying-space techniques. It proves constructive existence results: for a rank-$k$ orientable real bundle over a $d$-manifold with a Riemannian structure, Parseval frames of size $n=d+k$ exist; for complex bundles with Hermitian structure, frames exist of size $n=\lfloor d/2\rfloor+k$, with frame-existence equivalent to Parseval-existence via a deformation retraction of frame spaces. The paper also analyzes tangent bundles, showing that stably parallelizable manifolds yield size $d+1$ Parseval frames, and provides both positive and negative results depending on topological data like Euler class and $H^1(M,\mathbb{Z}/2)$. A classifying-space perspective clarifies when Parseval bundles are trivial, and the work includes numerical evidence suggesting Parseval frames improve reconstruction robustness under noise. Collectively, these findings lay foundations for signal representations on parameterized vector spaces and highlight rich interactions between topology and frame theory with potential practical impact in geometry-aware signal processing.

Abstract

Frames in finite-dimensional vector spaces are spanning sets of vectors which provide redundant representations of signals. The Parseval frames are particularly useful and important, since they provide a simple reconstruction scheme and are maximally robust against certain types of noise. In this paper we describe a theory of frames on arbitrary vector bundles -- this is the natural setting for signals which are realized as parameterized families of vectors rather than as single vectors -- and discuss the existence of Parseval frames in this setting. Our approach is phrased in the language of $G$-bundles, which allows us to use many tools from classical algebraic topology. In particular, we show that orientable vector bundles always admit Parseval frames of sufficiently large size and provide an upper bound on the necessary size. We also give sufficient conditions for the existence of Parseval frames of smaller size for tangent bundles of several families of manifolds, and provide some numerical evidence that Parseval frames on vector bundles share the desirable reconstruction properties of classical Parseval frames.

On the existence of Parseval frames for vector bundles

TL;DR

This work extends Parseval frame theory to vector bundles by recasting the problem as sections of the -Parseval bundle and employing obstruction theory and classifying-space techniques. It proves constructive existence results: for a rank- orientable real bundle over a -manifold with a Riemannian structure, Parseval frames of size exist; for complex bundles with Hermitian structure, frames exist of size , with frame-existence equivalent to Parseval-existence via a deformation retraction of frame spaces. The paper also analyzes tangent bundles, showing that stably parallelizable manifolds yield size Parseval frames, and provides both positive and negative results depending on topological data like Euler class and . A classifying-space perspective clarifies when Parseval bundles are trivial, and the work includes numerical evidence suggesting Parseval frames improve reconstruction robustness under noise. Collectively, these findings lay foundations for signal representations on parameterized vector spaces and highlight rich interactions between topology and frame theory with potential practical impact in geometry-aware signal processing.

Abstract

Frames in finite-dimensional vector spaces are spanning sets of vectors which provide redundant representations of signals. The Parseval frames are particularly useful and important, since they provide a simple reconstruction scheme and are maximally robust against certain types of noise. In this paper we describe a theory of frames on arbitrary vector bundles -- this is the natural setting for signals which are realized as parameterized families of vectors rather than as single vectors -- and discuss the existence of Parseval frames in this setting. Our approach is phrased in the language of -bundles, which allows us to use many tools from classical algebraic topology. In particular, we show that orientable vector bundles always admit Parseval frames of sufficiently large size and provide an upper bound on the necessary size. We also give sufficient conditions for the existence of Parseval frames of smaller size for tangent bundles of several families of manifolds, and provide some numerical evidence that Parseval frames on vector bundles share the desirable reconstruction properties of classical Parseval frames.
Paper Structure (21 sections, 28 theorems, 5 equations, 1 figure)

This paper contains 21 sections, 28 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Previous work
  3. Main results
  4. Outline of the paper
  5. Frames in vector spaces
  6. The space of frames
  7. The space of Parseval frames.
  8. Geometric structure of Parseval space
  9. Complex frame spaces.
  10. Frames on vector bundles
  11. The frame and Parseval bundles.
  12. Topology and geometry of Parseval space
  13. Complex vector bundles.
  14. Evidence for the optimality of Parseval frames
  15. Existence and Obstructions for Parseval frames
  16. ...and 6 more sections

Key Result

Theorem 1

Let $E\to M$ be a rank-$k$ orientable real vector bundle over a $d$-dimensional smooth manifold. Suppose further that $E\to M$ is equipped with a Riemannian structure. Then there exists a Parseval frame of size $d+k$ for $E\to M$.

Figures (1)

  • Figure 1: Left: plots of the empirical CDFs of the MSE distributions for the Parseval frame (dark blue) and the random frames (light orange). Right: histograms of the MSE distributions for the Parseval frame (dark blue) and for the best of the random frames (lighter orange).

Theorems & Definitions (60)

  • Theorem : Theorem \ref{['thm:parseval_sections_exist']}
  • Theorem : Theorem \ref{['thm:parseval_sections_exist_complex']}
  • Theorem : Theorem \ref{['thm:parseval_deformation_retract']} and Corollary \ref{['cor:parseval_if_and_only_if_frame']}
  • Theorem : Theorem \ref{['thm:stably_parallel']}
  • Theorem : Theorem \ref{['thm:obstruction_to_generic_sections']}
  • Definition 2.1: Frame
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 50 more