Consistent Semi-Supervised Graph Regularization for High Dimensional Data

Semi-supervised Laplacian regularization, a standard graph-based approach for learning from both labelled and unlabelled data, was recently demonstrated to have an insignificant high dimensional learning efficiency with respect to unlabelled data (Mai and Couillet 2018), causing it to be outperformed by its unsupervised counterpart, spectral clustering, given sufficient unlabelled data. Following a detailed discussion on the origin of this inconsistency problem, a novel regularization approach involving centering operation is proposed as solution, supported by both theoretical analysis and empirical results.

High Dimensional Classification via Empirical Risk Minimization: Improvements and Optimality

In this article, we investigate a family of classification algorithms defined by the principle of empirical risk minimization, in the high dimensional regime where the feature dimension $p$ and data number $n$ are both large and comparable. Based on recent advances in high dimensional statistics and random matrix theory, we provide under mixture data model a unified stochastic characterization of classifiers learned with different loss functions. Our results are instrumental to an in-depth understanding as well as practical improvements on this fundamental classification approach. As the main outcome, we demonstrate the existence of a universally optimal loss function which yields the best high dimensional performance at any given $n/p$ ratio.

A random matrix analysis and improvement of semi-supervised learning for large dimensional data

This article provides an original understanding of the behavior of a class of graph-oriented semi-supervised learning algorithms in the limit of large and numerous data. It is demonstrated that the intuition at the root of these methods collapses in this limit and that, as a result, most of them become inconsistent. Corrective measures and a new data-driven parametrization scheme are proposed along with a theoretical analysis of the asymptotic performances of the resulting approach. A surprisingly close behavior between theoretical performances on Gaussian mixture models and on real datasets is also illustrated throughout the article, thereby suggesting the importance of the proposed analysis for dealing with practical data. As a result, significant performance gains are observed on practical data classification using the proposed parametrization.