Graph Laplacian Learning with Exponential Family Noise.

Area of Science:

• Graph Signal Processing (GSP)
• Network Science
• Machine Learning

Background:

• Graph Signal Processing (GSP) analyzes data on non-Euclidean domains using the graph Fourier transform (GFT).
• A key challenge is inferring the underlying graph structure when it's unknown.
• Existing graph inference methods are limited to smooth signals or Gaussian noise, neglecting common discrete data types.

Purpose of the Study:

• To develop a versatile graph inference framework capable of handling graph signals corrupted by exponential family noise.
• To generalize existing graph inference techniques to various data types beyond smooth signals.
• To adapt the framework for non-independent and temporally correlated graph signals.

Main Methods:

• Proposed a novel graph inference framework utilizing an alternating algorithm.
• The algorithm jointly estimates the graph Laplacian and the unobserved smooth signal representation.
• Extended the framework to incorporate an offset variable for node-specific variations and a time-vertex formulation for temporal data.

Main Results:

• The proposed framework successfully generalizes graph inference to diverse data types, including discrete counts and binary digits.
• The joint estimation algorithm effectively recovers the graph Laplacian and underlying smooth signals.
• The time-vertex formulation addresses temporal correlations in real-world graph signals.

Conclusions:

• The developed graph inference framework offers a versatile solution for learning network structures from various noisy data types.
• Outperforms existing methods, particularly when dealing with noise models that do not match the data distribution.
• The approach is robust and applicable to both synthetic and real-world datasets with complex signal characteristics.