The focus of this paper is on detection theory for union of subspaces (UoS). To this end, generalized likelihood ratio tests (GLRTs) are presented for detection of signals conforming to the UoS model and detection of the corresponding “active” subspace. One of the main contributions of this paper is bounds on the performances of these GLRTs in terms of geometry of subspaces under various assumptions on the observation noise.

Recovery of certain piecewise continuous signals from noisy observations has been a major challenge in sciences and engineering. In this paper, in a tight-dimensional representation space, we exploit sparsity hidden in a class of possibly discontinuous signals named finite-dimensional piecewise continuous (FPC) signals. More precisely, we propose a tight-dimensional linear transformation which reveals a certain sparsity in discrete samples of the FPC signals. This transformation is designed by exploiting the fact that most of the consecutive samples are contained in special subspaces.