Generalized Multivariate Signs for Nonparametric Hypothesis Testing in High Dimensions

TL;DR

This paper introduces generalized multivariate sign transformations for high-dimensional data, enabling robust, powerful nonparametric mean tests that adapt to data geometry and are computationally efficient.

Contribution

It develops a new class of generalized sign vectors for high-dimensional hypothesis testing, improving power and flexibility over existing methods.

Findings

Tests using generalized signs have higher power than existing methods.

The proposed tests maintain nominal type-I error rates.

Applications demonstrate effectiveness on image and genomic data.

Abstract

High-dimensional data, where the dimension of the feature space is much larger than sample size, arise in a number of statistical applications. In this context, we construct the generalized multivariate sign transformation, defined as a vector divided by its norm. For different choices of the norm function, the resulting transformed vector adapts to certain geometrical features of the data distribution. Building up on this idea, we obtain one-sample and two-sample testing procedures for mean vectors of high-dimensional data using these generalized sign vectors. These tests are based on U-statistics using kernel inner products, do not require prohibitive assumptions, and are amenable to a fast randomization-based implementation. Through experiments in a number of data settings, we show that tests using generalized signs display higher power than existing tests, while maintaining nominal type-I error rates. Finally, we provide example applications on the MNIST and Minnesota Twin Studies genomic data.