Differentially private testing for relevant dependencies in high dimensions

TL;DR

This paper develops a differentially private testing method for detecting relevant dependencies among high-dimensional data components, focusing on composite hypotheses involving pairwise associations like Kendall's tau, with theoretical guarantees and medical data applications.

Contribution

It introduces a novel bootstrap-based testing approach for composite hypotheses under privacy constraints, advancing high-dimensional dependency detection methods.

Findings

Method performs well in sparse high-dimensional settings

Provides theoretical guarantees under mild assumptions

Demonstrates effectiveness on medical data applications

Abstract

We investigate the problem of detecting dependencies between the components of a high-dimensional vector. Our approach advances the existing literature in two important respects. First, we consider the problem under privacy constraints. Second, instead of testing whether the coordinates are pairwise independent, we are interested in determining whether certain pairwise associations between the components (such as all pairwise Kendall's $\tau$ coefficients) do not exceed a given threshold in absolute value. Considering hypotheses of this form is motivated by the observation that in the high-dimensional regime, it is rare and perhaps impossible to have a null hypothesis that can be modeled exactly by assuming that all pairwise associations are precisely equal to zero. The formulation of the null hypothesis as a composite hypothesis makes the problem of constructing tests already non-standard in the non-private setting. Additionally, under privacy constraints, state of the art procedures rely on permutation approaches that are rendered invalid under a composite null. We propose a novel bootstrap based methodology that is especially powerful in sparse settings, develop theoretical guarantees under mild assumptions and show that the proposed method enjoys good finite sample properties even in the high privacy regime. Additionally, we present applications in medical data that showcase the applicability of our methodology.