Exponentially Consistent Low Complexity Tests for Outlier Hypothesis Testing

TL;DR

This paper introduces new low complexity, exponentially consistent tests for outlier hypothesis testing that outperform existing methods in detection accuracy and computational efficiency, applicable in fixed-length and sequential scenarios.

Contribution

The paper proposes novel fixed-length and sequential outlier tests that improve detection performance and computational complexity, extending previous work with new scoring functions and performance bounds.

Findings

New tests achieve better tradeoff between detection accuracy and complexity.

Sequential tests demonstrate the benefits of sequentiality in outlier detection.

Performance bounds show the penalty of unknown number of outliers.

Abstract

We revisit outlier hypothesis testing, propose exponentially consistent low complexity fixed-length and sequential tests and show that our tests achieve better tradeoff between detection performance and computational complexity than existing tests that use exhaustive search. Specifically, in outlier hypothesis testing, one is given a list of observed sequences, most of which are generated i.i.d. from a nominal distribution while the rest sequences named outliers are generated i.i.d. from another anomalous distribution. The task is to identify all outliers when both the nominal and anomalous distributions are unknown. There are two basic settings: fixed-length and sequential. In the fixed-length setting, the sample size of each observed sequence is fixed a priori while in the sequential setting, the sample size is a random number that can be determined by the test designer to ensure reliable decisions. For the fixed-length setting, we strengthen the results of Bu \emph{et. al} (TSP 2019) by i) allowing for scoring functions beyond KL divergence and further simplifying the test design when the number of outliers is known and ii) proposing a new test, explicitly bounding the detection performance of the test and characterizing the tradeoff among exponential decay rates of three error probabilities when the number of outliers is unknown. For the sequential setting, our tests for both cases are novel and enable us to reveal the benefit of sequentiality. Finally, for both fixed-length and sequential settings, we demonstrate the penalty of not knowing the number of outliers on the detection performance.