Minimax Robust Detection: Classic Results and Recent Advances

Abstract

This paper provides an overview of results and concepts in minimax robust hypothesis testing for two and multiple hypotheses. It starts with an introduction to the subject, highlighting its connection to other areas of robust statistics and giving a brief recount of the most prominent developments. Subsequently, the minimax principle is introduced and its strengths and limitations are discussed. The first part of the paper focuses on the two-hypothesis case. After briefly reviewing the basics of statistical hypothesis testing, uncertainty sets are introduced as a generic way of modeling distributional uncertainty. The design of minimax detectors is then shown to reduce to the problem of determining a pair of least favorable distributions, and different criteria for their characterization are discussed. Explicit expressions are given for least favorable distributions under three types of uncertainty: $\epsilon$-contamination, probability density bands, and $f$-divergence balls. Using examples, it is shown how the properties of these least favorable distributions translate to properties of the corresponding minimax detectors. The second part of the paper deals with the problem of robustly testing multiple hypotheses, starting with a discussion of why this is fundamentally different from the binary problem. Sequential detection is then introduced as a technique that enables the design of strictly minimax optimal tests in the multi-hypothesis case. Finally, the usefulness of robust detectors in practice is showcased using the example of ground penetrating radar. The paper concludes with an outlook on robust detection beyond the minimax principle and a brief summary of the presented material.

arXiv comment35 pages, 19 figures, 1 table. Published as an Overview Paper in the IEEE Transactions on Signal Processing, vol. 69, pp. 2252-2283, 2021

References in corpus (20)

• f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization
• On the (Statistical) Detection of Adversarial Examples
• Training generative neural networks via Maximum Mean Discrepancy optimization
• $QD$-Learning: A Collaborative Distributed Strategy for Multi-Agent Reinforcement Learning Through Consensus + Innovations
• Minimax Robust Quickest Change Detection
• On surrogate loss functions and $f$-divergences
• Robust Hypothesis Testing with a Relative Entropy Tolerance
• Universal and Composite Hypothesis Testing via Mismatched Divergence
• On $f$-Divergences: Integral Representations, Local Behavior, and Inequalities
• Old Bands, New Tracks---Revisiting the Band Model for Robust Hypothesis Testing
• Robust Hypothesis Testing with $\alpha$-Divergence
• Robust Hypothesis Testing Using Wasserstein Uncertainty Sets
• Wasserstein Distributionally Robust Kalman Filtering
• Robust Sequential Detection in Distributed Sensor Networks
• Optimal sequential multiple hypothesis tests
• Minimax Robust Landmine Detection Using Forward-Looking Ground-Penetrating Radar
• Model Selection and Multiple Testing - A Bayesian and Empirical Bayes Overview and some New Results
• Minimax Optimal Sequential Hypothesis Tests for Markov Processes
• On the Minimization of Convex Functionals of Probability Distributions Under Band Constraints
• Fading Boundaries: On a Nonparametric Variant of the Kiefer--Weiss Problem

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