[Paper Review] Supervised Linear Regression for Graph Learning from Graph Signals

Arun Venkitaraman, Hermina Petric Maretić+2 more|arXiv (Cornell University)|Nov 5, 2018

Advanced Graph Neural Networks15 references4 citations

TL;DR

This paper proposes a supervised linear regression framework for learning graph structures from graph signals, where edge weights are predicted as a function of node signal values using convex optimization with graph-Laplacian-based regularization. The method outperforms unsupervised graph learning under noisy or outlier-contaminated data by adaptively weighting signals via learned regression coefficients.

ABSTRACT

We propose a supervised learning approach for predicting an underlying graph from a set of graph signals. Our approach is based on linear regression. In the linear regression model, we predict edge-weights of a graph as the output, given a set of signal values on nodes of the graph as the input. We solve for the optimal regression coefficients using a relevant optimization problem that is convex and uses a graph-Laplacian based regularization. The regularization helps to promote a specific graph spectral profile of the graph signals. Simulation experiments demonstrate that our approach predicts well even in presence of outliers in input data.

Motivation & Objective

• To address the lack of supervised learning approaches for graph structure inference from graph signals.
• To develop a method that leverages labeled training data (signals and corresponding graphs) to predict underlying graph structures.
• To improve robustness to noisy or non-smooth signals in graph learning by allowing differential weighting of input signals through regression.
• To formulate a convex optimization problem that promotes desired spectral properties in the learned graph signals.
• To demonstrate superior performance over unsupervised graph learning methods under challenging conditions like limited data and high outlier ratios.

Proposed method

• Model edge weights of the adjacency matrix as a linear function of input graph signal values using a regression framework.
• Formulate a convex optimization problem to learn regression coefficients that minimize prediction error on training data.
• Incorporate a graph-Laplacian-based regularization term expressed as a second-order polynomial of the Laplacian matrix to control the spectral profile of the predicted signals.
• Use the regularization to promote smoothness in the graph signals and to implicitly encode prior knowledge about the graph structure.
• Apply thresholding to the estimated adjacency matrix to enforce sparsity and connectivity in the final graph.
• Optimize hyperparameters (α and β) via cross-validation to balance regularization and data fidelity.

Experimental results

Research questions

• RQ1 Can a supervised learning approach improve graph structure inference when training data includes noisy or non-smooth signals?
• RQ2 How does the proposed regression model handle outliers in the input graph signals compared to traditional unsupervised graph learning?
• RQ3 To what extent do learned regression coefficients reflect the spectral characteristics of the input signals?
• RQ4 Does the method maintain consistent performance across varying signal-to-noise ratios and limited training data sizes?
• RQ5 How does the regularization based on the graph-Laplacian influence the quality and spectral properties of the predicted graphs?

Key findings

• The proposed method achieves significantly lower normalized mean square error (NMSE) than unsupervised graph learning (Kalofolias et al.) when 25% of input signals are outliers, especially for small training set sizes.
• With 10% outliers, the method outperforms the baseline for small M (number of signals), and performance converges to the baseline as M increases.
• The regression coefficients clearly differentiate between smooth signals and high-frequency outliers, indicating effective signal-specific weighting.
• F-scores for the predicted graphs increase with M and remain comparable between the proposed method and the baseline, indicating consistent structural quality.
• The method maintains stable performance across different graph types (e.g., Erdős–Rényi), suggesting robustness to underlying network topology.
• The use of polynomial regularization on the graph-Laplacian enables effective control of the spectral profile of the learned graph signals.