  This paper investigates the fundamental limits for detecting a
high-dimensional sparse matrix contaminated by white Gaussian noise from both
the statistical and computational perspectives. We consider $p\times p$
matrices whose rows and columns are individually $k$-sparse. We provide a tight
characterization of the statistical and computational limits for sparse matrix
detection, which precisely describe when achieving optimal detection is easy,
hard, or impossible, respectively. Although the sparse matrices considered in
this paper have no apparent submatrix structure and the corresponding
estimation problem has no computational issue at all, the detection problem has
a surprising computational barrier when the sparsity level $k$ exceeds the
cubic root of the matrix size $p$: attaining the optimal detection boundary is
computationally at least as hard as solving the planted clique problem.
  The same statistical and computational limits also hold in the sparse
covariance matrix model, where each variable is correlated with at most $k$
others. A key step in the construction of the statistically optimal test is a
structural property for sparse matrices, which can be of independent interest.
