  This paper deals with the problem of nonparametric independence testing, a
fundamental decision-theoretic problem that asks if two arbitrary (possibly
multivariate) random variables $X,Y$ are independent or not, a question that
comes up in many fields like causality and neuroscience. While quantities like
correlation of $X,Y$ only test for (univariate) linear independence, natural
alternatives like mutual information of $X,Y$ are hard to estimate due to a
serious curse of dimensionality. A recent approach, avoiding both issues,
estimates norms of an \textit{operator} in Reproducing Kernel Hilbert Spaces
(RKHSs). Our main contribution is strong empirical evidence that by employing
\textit{shrunk} operators when the sample size is small, one can attain an
improvement in power at low false positive rates. We analyze the effects of
Stein shrinkage on a popular test statistic called HSIC (Hilbert-Schmidt
Independence Criterion). Our observations provide insights into two recently
proposed shrinkage estimators, SCOSE and FCOSE - we prove that SCOSE is
(essentially) the optimal linear shrinkage method for \textit{estimating} the
true operator; however, the non-linearly shrunk FCOSE usually achieves greater
improvements in \textit{test power}. This work is important for more powerful
nonparametric detection of subtle nonlinear dependencies for small samples.
