Asymptotics for Outlier Hypothesis Testing

TL;DR

This paper derives fundamental limits and proposes an optimal threshold-based test for outlier hypothesis testing with unknown distributions, multiple outliers, and tradeoffs among error probabilities, demonstrating asymptotic optimality.

Contribution

It introduces a new threshold-based test for complex outlier detection scenarios with unknown distributions and multiple outliers, establishing bounds and asymptotic optimality.

Findings

Exponential decay of misclassification and false alarm probabilities.

Bounds on false reject probability as a function of threshold.

Asymptotic optimality under the generalized Neyman-Pearson criterion.

Abstract

We revisit the outlier hypothesis testing framework of Li \emph{et al.} (TIT 2014) and derive fundamental limits for the optimal test. In outlier hypothesis testing, one is given multiple observed sequences, where most sequences are generated i.i.d. from a nominal distribution. The task is to discern the set of outlying sequences that are generated according to anomalous distributions. The nominal and anomalous distributions are \emph{unknown}. We consider the case of multiple outliers where the number of outliers is unknown and each outlier can follow a different anomalous distribution. Under this setting, we study the tradeoff among the probabilities of misclassification error, false alarm and false reject. Specifically, we propose a threshold-based test that ensures exponential decay of misclassification error and false alarm probabilities. We study two constraints on the false reject probability, with one constraint being that it is a non-vanishing constant and the other being that it has an exponential decay rate. For both cases, we characterize bounds on the false reject probability, as a function of the threshold, for each tuple of nominal and anomalous distributions. Finally, we demonstrate the asymptotic optimality of our test under the generalized Neyman-Pearson criterion.