Graph Laplacian Learning with Exponential Family Noise

TL;DR

This work tackles learning a graph Laplacian when the underlying graph is unknown by extending graph signal processing to exponential-family noise. It proposes GLEN, an alternating framework that jointly estimates the Laplacian and the latent smooth signals, with extensions to variational inference (GLEN-VI) and to time-vertex data (GLEN-TV). Concrete instantiations for Poisson and Bernoulli noise demonstrate the method's flexibility beyond Gaussian noise, while experiments on synthetic and real data show improved structure and weight reconstruction under noise-model mismatch. The approach enables rigorous Laplacian learning for diverse data types, including discrete counts and binary signals, with practical impact in domains ranging from neuroscience to crime analytics.

Abstract

Graph signal processing (GSP) is a prominent framework for analyzing signals on non-Euclidean domains. The graph Fourier transform (GFT) uses the combinatorial graph Laplacian matrix to reveal the spectral decomposition of signals in the graph frequency domain. However, a common challenge in applying GSP methods is that in many scenarios the underlying graph of a system is unknown. A solution in such cases is to construct the unobserved graph from available data, which is commonly referred to as graph or network inference. Although different graph inference methods exist, these are restricted to learning from either smooth graph signals or simple additive Gaussian noise. Other types of noisy data, such as discrete counts or binary digits, are rather common in real-world applications, yet are underexplored in graph inference. In this paper, we propose a versatile graph inference framework for learning from graph signals corrupted by exponential family noise. Our framework generalizes previous methods from continuous smooth graph signals to various data types. We propose an alternating algorithm that jointly estimates the graph Laplacian and the unobserved smooth representation from the noisy signals. We also extend our approach to a variational form to account for the inherent stochasticity of the latent smooth representation. Finally, since real-world graph signals are frequently non-independent and temporally correlated, we further adapt our original setting to a time-vertex formulation. We demonstrate on synthetic and real-world data that our new algorithms outperform competing Laplacian estimation methods that suffer from noise model mismatch.

Figures (5)

• Figure 1: Graph Laplacian learning with Exponential Family Noise (GLEN). A. Illustration of GLEN. When only noisy signals $\mathbf{X}$ are available, GLEN (i) learns a graph from the estimation of unobserved smooth signals $\mathbf{Y}$ and (ii) denoises $\mathbf{X}$ using the newly obtained graph estimation to improve the quality of $\mathbf{Y}$ estimation and alternates these two steps. B. Example of a Poisson graph signal $\mathbf{x}$. Red star counts the events. C. Example of a smooth graph signal $\mathbf{y}$ underlying the noisy Poisson observations. D. Example of temporal correlated smooth graph signals.
• Figure 2: Graph Laplacians estimated by different methods and the ground truth Erdős-Rényi graph Laplacian (p=0.3).
• Figure 5: Graphs (normalized) inferred from the Chicago crime dataset using original CGL and GLEN. Nodes correspond to crime types. Width of edges correspond to the edge weights.
• Figure 6: Graphs (normalized) inferred from animal dataset using original CGL and GLEN. The nodes correspond to the species. The widths of the edges correspond to the weights of the edges.
• Figure 7: Average of raw neural spiking data compared against average denoised signals by GLEN-TV, for each target direction shown on the left. Brighter indicates higher firing rate.