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Offline detection of change-points in the mean for stationary graph signals

Alejandro de la Concha, Nicolas Vayatis, Argyris Kalogeratos

TL;DR

The paper addresses offline detection of mean-change points in streams of graph signals defined on a known graph. It exploits stationarity in the Graph Fourier domain to obtain a sparse spectral representation and formulates a penalized cost that combines a weighted least-squares term with $\ell_1$ sparsity penalties, enabling dynamic-programming-based change-point identification. Two model-selection strategies, LGS and VSGS, are proposed; the latter uses slope heuristics to adapt penalties and sparsity, and both come with non-asymptotic oracle-type guarantees. Empirical results on synthetic graphs demonstrate accurate change-point recovery and robustness to PSD estimation, highlighting the approach's practical utility and potential generalization to other sparse bases such as graph wavelets.

Abstract

This paper addresses the problem of segmenting a stream of graph signals: we aim to detect changes in the mean of a multivariate signal defined over the nodes of a known graph. We propose an offline method that relies on the concept of graph signal stationarity and allows the convenient translation of the problem from the original vertex domain to the spectral domain (Graph Fourier Transform), where it is much easier to solve. Although the obtained spectral representation is sparse in real applications, to the best of our knowledge this property has not been sufficiently exploited in the existing related literature. Our change-point detection method adopts a model selection approach that takes into account the sparsity of the spectral representation and determines automatically the number of change-points. Our detector comes with a proof of a non-asymptotic oracle inequality. Numerical experiments demonstrate the performance of the proposed method.

Offline detection of change-points in the mean for stationary graph signals

TL;DR

The paper addresses offline detection of mean-change points in streams of graph signals defined on a known graph. It exploits stationarity in the Graph Fourier domain to obtain a sparse spectral representation and formulates a penalized cost that combines a weighted least-squares term with sparsity penalties, enabling dynamic-programming-based change-point identification. Two model-selection strategies, LGS and VSGS, are proposed; the latter uses slope heuristics to adapt penalties and sparsity, and both come with non-asymptotic oracle-type guarantees. Empirical results on synthetic graphs demonstrate accurate change-point recovery and robustness to PSD estimation, highlighting the approach's practical utility and potential generalization to other sparse bases such as graph wavelets.

Abstract

This paper addresses the problem of segmenting a stream of graph signals: we aim to detect changes in the mean of a multivariate signal defined over the nodes of a known graph. We propose an offline method that relies on the concept of graph signal stationarity and allows the convenient translation of the problem from the original vertex domain to the spectral domain (Graph Fourier Transform), where it is much easier to solve. Although the obtained spectral representation is sparse in real applications, to the best of our knowledge this property has not been sufficiently exploited in the existing related literature. Our change-point detection method adopts a model selection approach that takes into account the sparsity of the spectral representation and determines automatically the number of change-points. Our detector comes with a proof of a non-asymptotic oracle inequality. Numerical experiments demonstrate the performance of the proposed method.

Paper Structure

This paper contains 9 sections, 6 theorems, 48 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Basic concepts and notations
  3. Change-point detection for a stream of graph signals
  4. Problem formulation
  5. Penalized cost function for an SGS with sparse GFT representation.
  6. Model selection approach
  7. Numerical experiments
  8. Conclusion
  9. Proof of Theorems 1 and 2

Key Result

Theorem 1

Assume that: where $c_1 \geq 6 \sqrt{2} \epsilon^2$, $c_2 \geq 3 \sqrt{2}\epsilon^2$, and $L$ is such that $L > \log 2$. Then, there exists an absolute constant $C>0$ satisfying: where $\left\lVert\tilde{\mu}\right\rVert_{[\tau]}:=\frac{1}{T}\sum_{l=1}^{d} I_{\tau_l} \left\lVert\tilde{\mu}_{\tau_l}\right\rVert_1$, $\mathcal{T}$ is the set of all possible segmentations of the SGS, $\gamma= \frac{

Figures (2)

  • Figure 1: Example stream of graph signals (SGS) with five change-points in the mean (according our problem formulation, the end of the sequence is always a change-point). Successive segments are shown with different colors. The color of the graph nodes represents the mean of the signal during the first observed segment. The signal observed at each node evolves through time, as shown in the line plots next to them. At some timestamps, the mean of the graph signals exhibits a change in a subset of nodes, which also signifies changes in the spectral representation of the signals.
  • Figure 2: Instance of Scenario III (after the first change-point: $10$ regions; after the second change-point: $20$ random nodes). The node colors indicate their expected values. First two rows: In the first row, the figures show the true mean in each of the 3 segments induced by the $2$ change-points. The red contour around a node indicates that its mean has changed compared to the previous segment. In the second row, the two plots show the true difference between the GFT of the mean of two consecutive segments (1st and 2nd, 2nd and 3rd). Last two rows: The estimated mean is shown. The red contour around a node indicates that it was detected by our algorithm as having a change in its mean. In the last row, we show the difference between the GFT of the estimated mean of two consecutive segments.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Definition 5
  • Definition 6
  • Theorem 3
  • ...and 2 more