  In the general submatrix detection problem, the task is to detect the
presence of a small $k \times k$ submatrix with entries sampled from a
distribution $\mathcal{P}$ in an $n \times n$ matrix of samples from
$\mathcal{Q}$. This formulation includes a number of well-studied problems,
such as biclustering when $\mathcal{P}$ and $\mathcal{Q}$ are Gaussians and the
planted dense subgraph formulation of community detection when the submatrix is
a principal minor and $\mathcal{P}$ and $\mathcal{Q}$ are Bernoulli random
variables. These problems all seem to exhibit a universal phenomenon: there is
a statistical-computational gap depending on $\mathcal{P}$ and $\mathcal{Q}$
between the minimum $k$ at which this task can be solved and the minimum $k$ at
which it can be solved in polynomial time. Our main result is to tightly
characterize this computational barrier as a tradeoff between $k$ and the KL
divergences between $\mathcal{P}$ and $\mathcal{Q}$ through average-case
reductions from the planted clique conjecture. These computational lower bounds
hold given mild assumptions on $\mathcal{P}$ and $\mathcal{Q}$ arising
naturally from classical binary hypothesis testing. Our results recover and
generalize the planted clique lower bounds for Gaussian biclustering in Ma-Wu
(2015) and Brennan et al. (2018) and for the sparse and general regimes of
planted dense subgraph in Hajek et al. (2015) and Brennan et al. (2018). This
yields the first universality principle for computational lower bounds obtained
through average-case reductions.
