  Say that we are given samples from a distribution $\psi$ over an
$n$-dimensional space. We expect or desire $\psi$ to behave like a product
distribution (or a $k$-wise independent distribution over its marginals for
small $k$). We propose the problem of enumerating/list-decoding all large
subcubes where the distribution $\psi$ deviates markedly from what we expect;
we refer to such subcubes as skewed subcubes. Skewed subcubes are certificates
of dependencies between small subsets of variables in $\psi$. We motivate this
problem by showing that it arises naturally in the context of algorithmic
fairness and anomaly detection.
  In this work we focus on the special but important case where the space is
the Boolean hypercube, and the expected marginals are uniform. We show that the
obvious definition of skewed subcubes can lead to intractable list sizes, and
propose a better definition of a minimal skewed subcube, which are subcubes
whose skew cannot be attributed to a larger subcube that contains it. Our main
technical contribution is a list-size bound for this definition and an
algorithm to efficiently find all such subcubes. Both the bound and the
algorithm rely on Fourier-analytic techniques, especially the powerful
hypercontractive inequality.
  On the lower bounds side, we show that finding skewed subcubes is as hard as
the sparse noisy parity problem, and hence our algorithms cannot be improved on
substantially without a breakthrough on this problem which is believed to be
intractable. Motivated by this, we study alternate models allowing query access
to $\psi$ where finding skewed subcubes might be easier.
