  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 distribution
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 adaptively estimate this
parameter using a stochastic gradient descent method. In both these approaches,
we employ 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 study the performance of these
approaches and offer insights on the relevance of different decision statistics
in different SNR regimes. We also address the problem of designing sensing
matrices with small coherence by using designs from quantum information theory.
One such design, called SIC POVM, also has an additional structure which allows
exact computation of the post-change pdfs of some decision statistics even when
the support set of the sparse signal is unknown. We apply our detection
algorithms with SIC POVM based sequences to a massive random access problem and
show their superior performance over conventional Gold codes.
