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Exponentially Consistent Low-Complexity Outlier Hypothesis Testing for Continuous Sequences

Lina Zhu, Lin Zhou

Abstract

In this work, we revisit outlier hypothesis testing and propose exponentially consistent, low-complexity fixed-length tests that achieve a better tradeoff between detection performance and computational complexity than existing exhaustive-search methods. In this setting, the goal is to identify outlying sequences from a set of observed sequences, where most sequences are i.i.d. from a nominal distribution and outliers are i.i.d. from a different anomalous distribution. While prior work has primarily focused on discrete-valued sequences, we extend the results of Bu et al. (TSP 2019) to continuous-valued sequences and develop a distribution-free test based on the MMD metric. Our framework handles both known and unknown numbers of outliers. In the unknown-count case, we bound the detection performance and characterize the tradeoff among the exponential decay rates of three types of error probabilities. Finally, we quantify the performance penalty incurred when the number of outliers is unknown.

Exponentially Consistent Low-Complexity Outlier Hypothesis Testing for Continuous Sequences

Abstract

In this work, we revisit outlier hypothesis testing and propose exponentially consistent, low-complexity fixed-length tests that achieve a better tradeoff between detection performance and computational complexity than existing exhaustive-search methods. In this setting, the goal is to identify outlying sequences from a set of observed sequences, where most sequences are i.i.d. from a nominal distribution and outliers are i.i.d. from a different anomalous distribution. While prior work has primarily focused on discrete-valued sequences, we extend the results of Bu et al. (TSP 2019) to continuous-valued sequences and develop a distribution-free test based on the MMD metric. Our framework handles both known and unknown numbers of outliers. In the unknown-count case, we bound the detection performance and characterize the tradeoff among the exponential decay rates of three types of error probabilities. Finally, we quantify the performance penalty incurred when the number of outliers is unknown.
Paper Structure (17 sections, 3 theorems, 43 equations, 2 figures, 2 algorithms)

This paper contains 17 sections, 3 theorems, 43 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Basics
  3. Problem Formulation
  4. Case of Known Number of Outliers
  5. Case of Unknown Number of Outliers
  6. MMD Metric
  7. Main Results
  8. Main Results for the Known Number of Outliers
  9. Test Design and Asymptotic Intuition
  10. Theoretical Results and Discussions
  11. Main Results for the Unknown Number of Outliers
  12. Test Design and Asymptotic Intuition
  13. Theoretical Results and Discussions
  14. Numerical Results
  15. Conclusion
  16. ...and 2 more sections

Key Result

Theorem 1

For any choice of distributions $(f_\mathrm{N},f_\mathrm{A})\in\mathcal{P}(\mathbb{R})^2$ and for every candidate outliers set $\mathcal{B}\in\mathcal{S}_s$, misclassification error exponent of the fixed-length testing procedure described in Algorithm LB_algorithm satisfies When $s<\frac{M}{3}$, it is satisfied that for $l\to\infty$.

Figures (2)

  • Figure 1: Plot of the sum of simulated misclassification and the false reject probabilities as a function of sequence length for the test in Algorithm \ref{['LB_algorithm']} and \ref{['unknown_algorithm']}, the joint MMD tests proposed in We and the tests in MMD when $M = 10$, $s=2$, the threshold $\lambda$ in the unknown case is defined by $\lambda=0.3\mathrm{MMD}^2(f_\mathrm{N},f_\mathrm{A})$.
  • Figure 2: Plot of the error probability under each non-null hypothesis and null hypothesis as a function of $\lambda$ for the test in Algorithm \ref{['unknown_algorithm']} when $M = 10$, $n=60$,$s=2$. The threshold $\lambda$ in the unknown case is defined by $\lambda=\alpha\mathrm{MMD}^2(f_\mathrm{N},f_\mathrm{A})$, where $\alpha$ is selected from $0.1$ to $0.9$.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Lemma 1