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Frame expansions with erasures: an approach through the non-commutative operator theory

Roman Vershynin

TL;DR

The paper addresses the erasure-resilience of frame expansions by proving that for any $n$-dimensional uniform tight frame, any source vector $x$ can be linearly reconstructed from $k$ randomly picked frame coefficients with small error, uniformly over $x$, provided $k \ge C\,(n/\varepsilon^2)\log(n/\varepsilon^2)$. The main approach connects frame theory to Rudelson's selection theorem via the noncommutative Khinchine inequality, recasting frames as decompositions of the identity and exploiting isotropic-position techniques. The key contributions are nonasymptotic bounds showing that frame expansions withstand random coefficient losses better than trivial basis expansions, with tail guarantees of the form $1 - Ce^{-t^2}$ and a proof strategy built from moments, symmetrization, and Bernstein concentration. The results have implications for real-time communication and MDC-like settings, demonstrating universal, practically relevant bounds on the number of received coefficients needed for accurate reconstruction.

Abstract

In modern communication systems such as the Internet, random losses of information can be mitigated by oversampling the source. This is equivalent to expanding the source using overcomplete systems of vectors (frames), as opposed to the traditional basis expansions. Dependencies among the coefficients in frame expansions often allow for better performance comparing to bases under random losses of coefficients. We show that for any n-dimensional frame, any source can be linearly reconstructed from only (n log n) randomly chosen frame coefficients, with a small error and with high probability. Thus every frame expansion withstands random losses better (for worst case sources) than the orthogonal basis expansion, for which the (n log n) bound is attained. The proof reduces to M.Rudelson's selection theorem on random vectors in the isotropic position, which is based on the non-commutative Khinchine's inequality.

Frame expansions with erasures: an approach through the non-commutative operator theory

TL;DR

The paper addresses the erasure-resilience of frame expansions by proving that for any -dimensional uniform tight frame, any source vector can be linearly reconstructed from randomly picked frame coefficients with small error, uniformly over , provided . The main approach connects frame theory to Rudelson's selection theorem via the noncommutative Khinchine inequality, recasting frames as decompositions of the identity and exploiting isotropic-position techniques. The key contributions are nonasymptotic bounds showing that frame expansions withstand random coefficient losses better than trivial basis expansions, with tail guarantees of the form and a proof strategy built from moments, symmetrization, and Bernstein concentration. The results have implications for real-time communication and MDC-like settings, demonstrating universal, practically relevant bounds on the number of received coefficients needed for accurate reconstruction.

Abstract

In modern communication systems such as the Internet, random losses of information can be mitigated by oversampling the source. This is equivalent to expanding the source using overcomplete systems of vectors (frames), as opposed to the traditional basis expansions. Dependencies among the coefficients in frame expansions often allow for better performance comparing to bases under random losses of coefficients. We show that for any n-dimensional frame, any source can be linearly reconstructed from only (n log n) randomly chosen frame coefficients, with a small error and with high probability. Thus every frame expansion withstands random losses better (for worst case sources) than the orthogonal basis expansion, for which the (n log n) bound is attained. The proof reduces to M.Rudelson's selection theorem on random vectors in the isotropic position, which is based on the non-commutative Khinchine's inequality.

Paper Structure

This paper contains 8 sections, 7 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Frames as decompositions of identity and their random parts
  3. Noncommutative Khinchine's inequality and Rudelson's theorem
  4. Proof of Theorem \ref{['main full']}
  5. Moments and tails
  6. Symmetrization
  7. Bounding the moments
  8. Replacing the average size of the random set by its actual size

Key Result

Theorem 1.1

For any uniform tight frame in $\mathbb{R}^n$ and any source vector $x$, the linear reconstruction from $k$ random coefficients of $x$ yields an approximation error at most $\varepsilon$ with probability $1-\varepsilon$, provided $k \ge C(\varepsilon) n \log n$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 2.1
  • Corollary 2.2
  • Lemma 3.1: M.Rudelson
  • Theorem 3.2: Non-commutative Khinchine's inequality LP, P, see R2
  • Lemma 4.1
  • Corollary 4.2