Robust Graph-Based Semi-Supervised Learning via $p$-Conductances

TL;DR

This paper introduces $p$-conductance learning, a robust semi-supervised learning method on graphs that generalizes existing techniques and performs well with scarce or corrupted labels.

Contribution

It proposes a new $p$-conductance framework, connects it to graph variational problems, and develops an efficient algorithm with strong empirical results.

Findings

Achieves state-of-the-art accuracy in low-label regimes

Demonstrates robustness to label corruption

Effective on computer vision and citation datasets

Abstract

We study the problem of semi-supervised learning on graphs in the regime where data labels are scarce or possibly corrupted. We propose an approach called $p$-conductance learning that generalizes the $p$-Laplace and Poisson learning methods by introducing an objective reminiscent of $p$-Laplacian regularization and an affine relaxation of the label constraints. This leads to a family of probability measure mincut programs that balance sparse edge removal with accurate distribution separation. Our theoretical analysis connects these programs to well-known variational and probabilistic problems on graphs (including randomized cuts, effective resistance, and Wasserstein distance) and provides motivation for robustness when labels are diffused via the heat kernel. Computationally, we develop a semismooth Newton-conjugate gradient algorithm and extend it to incorporate class-size estimates when converting the continuous solutions into label assignments. Empirical results on computer vision and citation datasets demonstrate that our approach achieves state-of-the-art accuracy in low label-rate, corrupted-label, and partial-label regimes.