Exponentially Consistent Statistical Classification of Continuous Sequences with Distribution Uncertainty
This work addresses classification challenges in scenarios with distributional shifts, which is incremental as it extends existing discrete and perfect-match frameworks to continuous and uncertain settings.
The paper tackles the problem of classifying continuous sequences with distribution uncertainty, where testing and training distributions may deviate even under the true hypothesis, and proposes distribution-free tests that achieve exponentially decaying error probabilities for fixed-length, sequential, and two-phase designs.
In multiple classification, one aims to determine whether a testing sequence is generated from the same distribution as one of the M training sequences or not. Unlike most of existing studies that focus on discrete-valued sequences with perfect distribution match, we study multiple classification for continuous sequences with distribution uncertainty, where the generating distributions of the testing and training sequences deviate even under the true hypothesis. In particular, we propose distribution free tests and prove that the error probabilities of our tests decay exponentially fast for three different test designs: fixed-length, sequential, and two-phase tests. We first consider the simple case without the null hypothesis, where the testing sequence is known to be generated from a distribution close to the generating distribution of one of the training sequences. Subsequently, we generalize our results to a more general case with the null hypothesis by allowing the testing sequence to be generated from a distribution that is vastly different from the generating distributions of all training sequences.