Algorithms for Change Detection With Sparse Signals

We consider the change detection problem where the pre-change observation vectors are purely noise and the post-change observation vectors are noise-corrupted compressive measurements of sparse signals with a common support, measured using a sensing matrix. In general, post-change probability density function (pdf) of the observations depends on parameters such as the support and variances of the sparse signal. When these parameters are unknown, we propose two approaches. In the first approach, we approximate the post-change pdf based on the known parameters such as mutual coherence of the sensing matrix and bounds on the signal variances. In the second approach, we parameterize the post-change pdf with an unknown parameter and try to estimate this parameter using two different methods, stochastic gradient descent and generalized likelihood ratio. In both these approaches, we employ the conventional cumulative sum (CUSUM) algorithm with various decision statistics such as the energy of the observations, correlation values with columns of the sensing matrix and the maximum value of such correlations. We analytically characterize the worst case detection delay and average run length to false alarm performance of pdf-approximation based approach. We also numerically study the performance and offer insights on the relevance of various decision statistics in different signal to noise ratio (SNR) regimes. We also address the problem of designing sensing matrices with small mutual coherence by using designs from quantum information theory. One such design using equi-angular lines has an additional structure which allows exact characterization of the post-change pdf of the correlation values, even when the support set of the sparse signal is unknown. We apply our detection algorithms with sensing matrix designed from equi-angular lines to a massive random access problem and show their superior performance over conventional Gold codes.