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Reconstructing Graph Signals from Noisy Dynamical Samples

Akram Aldroubi, Victor Bailey, Ilya Krishtal, Brendan Miller, Armenak Petrosyan

TL;DR

This work derives necessary and sufficient conditions for space-time sampling that enable the reconstruction of an initial band-limited signal on a graph and develops and test numerical algorithms for approximating the optimal placement of sensors on the graph to minimize the mean squared error when recovering signals from time-space measurements corrupted by i.i.d.~additive noise.

Abstract

We investigate the dynamical sampling space-time trade-off problem within a graph setting. Specifically, we derive necessary and sufficient conditions for space-time sampling that enable the reconstruction of an initial band-limited signal on a graph. Additionally, we develop and test numerical algorithms for approximating the optimal placement of sensors on the graph to minimize the mean squared error when recovering signals from time-space measurements corrupted by i.i.d.~additive noise. Our numerical experiments demonstrate that our approach outperforms previously proposed algorithms for related problems.

Reconstructing Graph Signals from Noisy Dynamical Samples

TL;DR

This work derives necessary and sufficient conditions for space-time sampling that enable the reconstruction of an initial band-limited signal on a graph and develops and test numerical algorithms for approximating the optimal placement of sensors on the graph to minimize the mean squared error when recovering signals from time-space measurements corrupted by i.i.d.~additive noise.

Abstract

We investigate the dynamical sampling space-time trade-off problem within a graph setting. Specifically, we derive necessary and sufficient conditions for space-time sampling that enable the reconstruction of an initial band-limited signal on a graph. Additionally, we develop and test numerical algorithms for approximating the optimal placement of sensors on the graph to minimize the mean squared error when recovering signals from time-space measurements corrupted by i.i.d.~additive noise. Our numerical experiments demonstrate that our approach outperforms previously proposed algorithms for related problems.

Paper Structure

This paper contains 14 sections, 4 theorems, 31 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Basic definitions and notation.
  4. Reconstruction in Paley-Wiener Spaces
  5. Optimal node sampling for space-time problem on graphs
  6. Problem setup
  7. Numerical Algorithms
  8. Greedy Algorithm
  9. Norm Penalty
  10. Exponential Penalty
  11. Numerical experiments
  12. Consistency across various graph types and eigenvalue distributions
  13. Comparison to the brute force algorithm for small graphs
  14. Relative accuracy comparison between algorithms

Key Result

Theorem 1

Let $\Omega \subset \{1, \dots, d\}$ and $\{b_i: i \in \Omega \}$ vectors in $\mathbb{C}^d$. Let $D$ be a diagonal matrix and $r_i$ the degree of the $D$-annihilator of $b_i$. Set $l_i=r_i-1$. Then $\{D^{j}b_i: \; i\in \Omega, \, j=0, \dots, l_i\}$ is a frame of $\mathbb{C}^d$ if and only if $\{P_j

Figures (6)

  • Figure 1: Greedy algorithm selecting 16 sampling locations (in red) to minimize the MSE of signal reconstruction over $PW_{\lambda_{10}}$ on the Minnesota roadmap graph (left) and a random connected graph (right), where $\lambda_{10}$ is the 10-th smallest eigenvalue of the graph operator.
  • Figure 2: Clustered Graph
  • Figure 3: Four figures showing the output of each algorithm varying the type of graph and distribution of eigenvalues of graph operator.
  • Figure 4: Median Trace Inverse and CPU time of each algorithm (left) and CPU time for brute force algorithm (right).
  • Figure 5: Average trace inverse of the output of each algorithm (left) and the comparative time taken (right) run on random graphs.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Theorem 1: ACMT17
  • Definition 3
  • Theorem 2
  • proof
  • Remark 1
  • Proposition 1
  • Proposition 2
  • proof
  • ...and 2 more