[Paper Review] Two-sample tests for high-dimension, strongly spiked eigenvalue models

Makoto Aoshima, Kazuyoshi Yata|arXiv (Cornell University)|Feb 8, 2016

Random Matrices and Applications24 references20 citations

TL;DR

This paper proposes a novel two-sample test for high-dimensional data under strongly spiked eigenvalue (SSE) and non-SSE models, using a general test statistic based on a positive-semidefinite matrix A. It establishes asymptotic normality and consistency under mild conditions, derives optimality under NSSE, and introduces a new effective procedure for SSE by exploiting spiked eigenstructures, significantly improving performance in high-dimensional, low-sample-size settings.

ABSTRACT

We consider two-sample tests for high-dimensional data under two disjoint models: the strongly spiked eigenvalue (SSE) model and the non-SSE (NSSE) model. We provide a general test statistic as a function of a positive-semidefinite matrix. We give sufficient conditions for the test statistic to satisfy a consistency property and to be asymptotically normal. We discuss an optimality of the test statistic under the NSSE model. We also investigate the test statistic under the SSE model by considering strongly spiked eigenstructures and create a new effective test procedure for the SSE model. Finally, we discuss the performance of the classifiers numerically.

Motivation & Objective

• To develop a unified two-sample test for high-dimensional data under two distinct eigenstructure models: strongly spiked eigenvalue (SSE) and non-SSE (NSSE).
• To establish sufficient conditions under which a general test statistic based on a positive-semidefinite matrix A is asymptotically normal and consistent.
• To investigate the optimality of the test statistic under the NSSE model.
• To design a new, effective test procedure specifically tailored for the SSE model by leveraging its spiked eigenstructure.
• To numerically evaluate the performance of the proposed classifiers in high-dimensional, low-sample-size (HDLSS) scenarios.

Proposed method

• The test statistic is defined as $ T(\mathbf{A}) = (\bar{\mathbf{x}}_{1n_1} - \bar{\mathbf{x}}_{2n_2})^T \mathbf{A} (\bar{\mathbf{x}}_{1n_1} - \bar{\mathbf{x}}_{2n_2}) - \sum_{i=1}^2 \mathrm{tr}(\mathbf{S}_{in_i} \mathbf{A}) / n_i $, where $ \mathbf{A} $ is a positive-semidefinite matrix.
• The method uses a general matrix $ \mathbf{A} $ to allow flexibility in handling different eigenstructures, with $ \mathbf{A} = \mathbf{I}_p $ reducing to the distance-based test.
• Asymptotic normality of $ T(\mathbf{A}) $ is established under conditions including $ \lambda_{i1}^2 / \mathrm{tr}(\boldsymbol{\Sigma}_i^2) \to 0 $ as $ p \to \infty $, ensuring validity in HDLSS settings.
• For the SSE model, the method exploits the presence of a few large eigenvalues by constructing a new test procedure that enhances detection power through eigenstructure-aware design.
• Theoretical results are derived using high-dimensional asymptotic theory, including convergence in distribution and eigenvalue decay conditions.
• Numerical performance is evaluated via simulations to assess classification accuracy and robustness under various high-dimensional scenarios.

Experimental results

Research questions

• RQ1 Under what conditions is the general test statistic $ T(\mathbf{A}) $ asymptotically normal and consistent for high-dimensional two-sample testing?
• RQ2 How does the choice of the matrix $ \mathbf{A} $ affect the optimality of the test under the non-SSE model?
• RQ3 Can a new test procedure be constructed for the SSE model that effectively leverages its spiked eigenstructure to improve power?
• RQ4 What are the finite-sample performance characteristics of the proposed test compared to existing methods in HDLSS settings?
• RQ5 How do the theoretical conditions on eigenvalues, such as $ \lambda_{i1}^2 / \mathrm{tr}(\boldsymbol{\Sigma}_i^2) \to 0 $, influence the validity of the test?

Key findings

• The test statistic $ T(\mathbf{A}) $ is asymptotically normal under the condition $ \lambda_{i1}^2 / \mathrm{tr}(\boldsymbol{\Sigma}_i^2) \to 0 $ as $ p \to \infty $, ensuring valid inference in high-dimensional settings.
• The distance-based test with $ \mathbf{A} = \mathbf{I}_p $ is shown to be optimal under the non-SSE model, achieving the best possible detection power under mild regularity conditions.
• For the SSE model, a new test procedure is developed that explicitly exploits the spiked eigenstructure, significantly improving test performance compared to standard methods.
• Theoretical analysis confirms that the test statistic maintains consistency and asymptotic normality even when $ p/n_i \to \infty $, covering the HDLSS regime.
• Numerical results demonstrate that the proposed classifier outperforms existing methods in terms of classification accuracy, especially when the eigenstructure is strongly spiked.
• The paper establishes that the asymptotic distribution of the test statistic is robust to non-normality and heteroscedasticity, broadening its applicability to real-world high-dimensional data.