MSGL+: Fast and Reliable Model Selection-Inspired Graph Metric Learning

The problem of learning graph-based data structures from data has attracted considerable attention in the past decade. Different types of data can be used to infer the graph structure, such as graphical Lasso, which is learned from multiple graph signals or graph metric learning based on node features. However, most existing methods that use node features to learn the graph face difficulties when the label signals of the data are incomplete. In particular, the pair-wise distance metric learning problem becomes intractable as the dimensionality of the node features increases. To address this challenge, we propose a novel method called MSGL+. MSGL+ is inspired from model selection, leverages recent advancements in graph spectral signal processing (GSP), and offers several key innovations: (1) Polynomial Interpretation: We use a polynomial function of a certain order on the graph Laplacian to represent the inverse covariance matrix of the graph nodes to rigorously formulate an optimization problem. (2) Convex Formulation: We formulate a convex optimization objective with a cone constraint that optimizes the coefficients of the polynomial, which makes our approach efficient. (3) Linear Constraints: We convert the cone constraint of the objective to a set of linear ones to further ensure the efficiency of our method. (4) Optimization Objective: We explore the properties of these linear constraints within the optimization objective, avoiding sub-optimal results by the removal of the box constraints on the optimization variables, and successfully further reduce the number of variables compared to our preliminary work, MSGL. (5) Efficient Solution: We solve the objective using the efficient linear-program-based Frank-Wolfe algorithm. Application examples, including binary classification, multi-class classification, binary image denoising, and time-series analysis, demonstrate that MSGL+ achieves competitive accuracy performance with a significant speed advantage compared to existing graphical Lasso and feature-based graph learning methods.