  Compressed sensing (CS) is a signal processing technique that enables the
efficient recovery of a sparse high-dimensional signal from low-dimensional
measurements. In the multiple measurement vector (MMV) framework, a set of
signals with the same support must be recovered from their corresponding
measurements. Here, we present the first exploration of the MMV problem where
signals are independently drawn from a sparse, multivariate Poisson
distribution. We are primarily motivated by a suite of biosensing applications
of microfluidics where analytes (such as whole cells or biomarkers) are
captured in small volume partitions according to a Poisson distribution. We
recover the sparse parameter vector of Poisson rates through maximum likelihood
estimation with our novel Sparse Poisson Recovery (SPoRe) algorithm. SPoRe uses
batch stochastic gradient ascent enabled by Monte Carlo approximations of
otherwise intractable gradients. By uniquely leveraging the Poisson structure,
SPoRe substantially outperforms a comprehensive set of existing and custom
baseline CS algorithms. Notably, SPoRe can exhibit high performance even with
one-dimensional measurements and high noise levels. This resource efficiency is
not only unprecedented in the field of CS but is also particularly potent for
applications in microfluidics in which the number of resolvable measurements
per partition is often severely limited. We prove the identifiability property
of the Poisson model under such lax conditions, analytically develop insights
into system performance, and confirm these insights in simulated experiments.
Our findings encourage a new approach to biosensing and are generalizable to
other applications featuring spatial and temporal Poisson signals.
