  We determine statistical and computational limits for estimation of a
rank-one matrix (the spike) corrupted by an additive gaussian noise matrix, in
a sparse limit, where 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-to-noise ratio 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 and analyze the approximate message passing algorithm in
the sparse regime. For Bernoulli and Bernoulli-Rademacher distributed vectors,
and when the sparsity and signal strength satisfy an appropriate scaling
relation, we find all-or-nothing phase transitions for the asymptotic minimum
and algorithmic mean-square errors. These jump from their maximum possible
value to zero, at well defined signal-to-noise thresholds whose asymptotic
values we determine exactly. In the asymptotic regime the
statistical-to-algorithmic gap diverges indicating that sparse recovery is hard
for approximate message passing.
