  This paper develops several average-case reduction techniques to show new
hardness results for three central high-dimensional statistics problems,
implying a statistical-computational gap induced by robustness, a
detection-recovery gap and a universality principle for these gaps. A main
feature of our approach is to map to these problems via a common intermediate
problem that we introduce, which we call Imbalanced Sparse Gaussian Mixtures.
We assume the planted clique conjecture for a version of the planted clique
problem where the position of the planted clique is mildly constrained, and
from this obtain the following computational lower bounds: (1) a $k$-to-$k^2$
statistical-computational gap for robust sparse mean estimation, providing the
first average-case evidence for a conjecture of Li (2017) and Balakrishnan et
al. (2017); (2) a tight lower bound for semirandom planted dense subgraph,
which shows that a semirandom adversary shifts the detection threshold in
planted dense subgraph to the conjectured recovery threshold; and (3) a
universality principle for $k$-to-$k^2$ gaps in a broad class of sparse mixture
problems that includes many natural formulations such as the spiked covariance
model.
  Our main approach is to introduce several average-case techniques to produce
structured and Gaussianized versions of an input graph problem, and then to
rotate these high-dimensional Gaussians by matrices carefully constructed from
hyperplanes in $\mathbb{F}_r^t$. For our universality result, we introduce a
new method to perform an algorithmic change of measure tailored to sparse
mixtures. We also provide evidence that the mild promise in our variant of
planted clique does not change the complexity of the problem.
