Constrained Epsilon-Minimax Test for Simultaneous Detection and Classification

A constrained epsilon-minimax test is proposed to detect and classify nonorthogonal vectors in Gaussian noise, with a general covariance matrix, and in presence of linear interferences. This test is epsilon-minimax in the sense that it has a small loss of optimality with respect to the purely theoretical and incalculable constrained minimax test which minimizes the maximum classification error probability subject to a constraint on the false alarm probability. This loss is even more negligible as the signal-to-noise ratio is large. Furthermore, it is also an epsilon-equalizer test since its classification error probabilities are equalized up to a negligible difference. When the signal-to-noise ratio is sufficiently large, an asymptotically equivalent test with a very simple form is proposed. This equivalent test coincides with the generalized likelihood ratio test when the vectors to classify are strongly separated in term of Euclidean distance. Numerical experiments on active user identification in a multiuser system confirm the theoretical findings.