Learning Laplacian Matrix in Smooth Graph Signal Representations

Abstract

The construction of a meaningful graph plays a crucial role in the success of many graph-based representations and algorithms for handling structured data, especially in the emerging field of graph signal processing. However, a meaningful graph is not always readily available from the data, nor easy to define depending on the application domain. In particular, it is often desirable in graph signal processing applications that a graph is chosen such that the data admit certain regularity or smoothness on the graph. In this paper, we address the problem of learning graph Laplacians, which is equivalent to learning graph topologies, such that the input data form graph signals with smooth variations on the resulting topology. To this end, we adopt a factor analysis model for the graph signals and impose a Gaussian probabilistic prior on the latent variables that control these signals. We show that the Gaussian prior leads to an efficient representation that favors the smoothness property of the graph signals. We then propose an algorithm for learning graphs that enforces such property and is based on minimizing the variations of the signals on the learned graph. Experiments on both synthetic and real world data demonstrate that the proposed graph learning framework can efficiently infer meaningful graph topologies from signal observations under the smoothness prior.

References in corpus (2)

• Discrete Signal Processing on Graphs
• Learning parametric dictionaries for graph signals

Cited by in corpus (13)

• Connecting the Dots: Identifying Network Structure via Graph Signal Processing
• Autoregressive Moving Average Graph Filtering
• Kernel-based Reconstruction of Graph Signals
• Signal Processing on Graphs: Causal Modeling of Unstructured Data
• Local-set-based Graph Signal Reconstruction
• Semi-Blind Inference of Topologies and Dynamical Processes over Graphs
• Large Scale Graph Learning from Smooth Signals
• How to learn a graph from smooth signals
• Bayesian Estimation of Graph Signals
• Scalable Probabilistic Matrix Factorization with Graph-Based Priors
• Bias-Variance Tradeoff of Graph Laplacian Regularizer
• Local Measurement and Reconstruction for Noisy Graph Signals
• Simultaneous Low-rank Component and Graph Estimation for High-dimensional Graph Signals: Application to Brain Imaging