Spectral method guarantees approximate label recovery on noisy signed graphs
Provable Guarantees for Spectral Structured Prediction
Machine Learning
Summary
Structured prediction is about guessing many related labels at once, like figuring out multiple parts of a picture or sentence. This paper looks at a problem where labels are nodes in a network with positive and negative links that can be noisy. The authors show that looking at the main eigenvector (a mathematical summary) of this noisy network can still approximately recover the true labels, with guarantees on how close this recovery is. Their work provides new math tools that explain how well this works depending on network size, noise, and structure.
What this means in practice
- •For natural language processing teams: Use spectral methods with theoretical confidence to infer structured labels from noisy graph-based representations of language data.
- •For computer vision developers: Apply provable spectral algorithms to recover multiple interacting labels in visual data modeled as noisy signed graphs.
Tested on simulated data.
Authors
Violet Zheng, Jean Honorio
Abstract
Structured prediction is the simultaneous prediction of multiple labels, and is widely used in various fields, such as natural language processing and computer vision. In this paper, we study binary node label recovery on signed graphs with edge-flip noise, a model introduced by (Globerson et al., 2015), via a simple spectral method that decodes node labels from the signs of the principal eigenvector of the noisy signed adjacency matrix. We develop graph structure-agnostic theoretical guarantees for approximate inference of node labels as well as guarantees for maximum angle deviation with respect to the ground truth node labels. By leveraging tools from matrix concentration theory and eigenvector perturbation analysis, we derive new concentration inequalities that explicitly quantify the effect of the spectral gap of the adjacency matrix, number of nodes, degree distribution, and noise level. As a corollary, we relate our general results to the Cheeger constant and provide results for different classes of graphs. We perform several synthetic experiments to validate our theory. To the best of our knowledge, we are the first to provide theoretical guarantees for the spectral-based approach. As a byproduct of our analysis, we derive technical results that might be of independent interest and useful for other machine learning problems.