Graph Laplacian Learning with Exponential Family Noise

Changhao Shi, Gal Mishne

Abstract: Graph signal processing (GSP) has generalized classical Fourier analysis to signals lying on irregular structures such as networks. However, a common challenge in applying graph signal processing is that the underlying graph of a system is unknown. Well-established methods optimize a graph representation, usually the graph adjacency matrix or the graph Laplacian, so that the total variation of given signals will be minimal on the learned graph. These methods have been developed for continuous graph signals, however inferring the graph structure for other types of data, such as discrete counts or binary signal, remains underexplored. In this talk, I will address the problem of learning the graph Laplacian from noisy data and generalize a GSP framework for learning a graph from smooth graph signals to exponential family noise distributions, allowing for the modeling of various data types. We then propose Graph Learning with Exponential family Noise (GLEN), an alternating algorithm that estimates the underlying graph Laplacian as well as the unobserved smooth representation from the noisy signals. Furthermore, we extend GLEN to the time-vertex setting to handle data with temporal correlations, e.g., time-series on a network. We demonstrate in synthetic experiments with different graph models that GLEN outperforms competing Laplacian estimation methods under noise model mismatch. Furthermore, we apply GLEN to different types of real-world data (e.g., binary questionnaires, neural activity) to further demonstrate the efficacy of our methods.

Submission Type: Abstract