Sharp-SSL: Selective High-Dimensional Axis-Aligned Random Projections for Semi-Supervised Learning

We propose a new method for high-dimensional semi-supervised learning problems based on the careful aggregation of the results of a low-dimensional procedure applied to many axis-aligned random projections of the data. Our primary goal is to identify important variables for distinguishing between the classes; existing low-dimensional methods can then be applied for final class assignment. To this end, we score projections according to their class-distinguishing ability; for instance, motivated by a generalized Rayleigh quotient, we can compute the traces of estimated whitened between-class covariance matrices on the projected data. This enables us to assign an importance weight to each variable for a given projection, and to select our signal variables by aggregating these weights over high-scoring projections. Our theory shows that the resulting Sharp-SSL algorithm is able to recover the signal coordinates with high probability when we aggregate over sufficiently many random projections and when the base procedure estimates the diagonal entries of the whitened between-class covariance matrix sufficiently well. For the Gaussian EM base procedure, we provide a new analysis of its performance in semi-supervised settings that controls the parameter estimation error in terms of the proportion of labeled data in the sample. Numerical results on both simulated data and a real colon tumor dataset support the excellent empirical performance of the method. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.