  We consider statistical models of estimation of a rank-one matrix (the spike)
corrupted by an additive gaussian noise matrix in the sparse limit. In this
limit the underlying hidden vector (that constructs the rank-one matrix) has a
number of non-zero components that scales sub-linearly with the total dimension
of the vector, and the signal strength tends to infinity at an appropriate
speed. We prove explicit low-dimensional variational formulas for the
asymptotic mutual information between the spike and the observed noisy matrix
in suitable sparse limits. For Bernoulli and Bernoulli-Rademacher distributed
vectors, and when the sparsity and signal strength satisfy an appropriate
scaling relation, these formulas imply sharp 0-1 phase transitions for the
asymptotic minimum mean-square-error. A similar phase transition was analyzed
recently in the context of sparse high-dimensional linear regression
(compressive sensing).
