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Comparing distributions: $\ell_1$ geometry improves kernel twosample testing
Meyer Scetbon · Gael Varoquaux
Are two sets of observations drawn from the same distribution? This
problem is a twosample test.
Kernel methods lead to many appealing properties. Indeed stateoftheart
approaches use the $L^2$ distance between kernelbased
distribution representatives to derive their test statistics. Here, we show that
$L^p$ distances (with $p\geq 1$) between these
distribution representatives give metrics on the space of distributions that are
wellbehaved to detect differences between distributions as they
metrize the weak convergence. Moreover, for analytic kernels,
we show that the $L^1$ geometry gives improved testing power for
scalable computational procedures. Specifically, we derive a finite
dimensional approximation of the metric given as the $\ell_1$ norm of a vector which captures differences of expectations of analytic functions evaluated at spatial locations or frequencies (i.e, features). The features can be chosen to
maximize the differences of the distributions and give interpretable
indications of how they differs. Using an $\ell_1$ norm gives better detection
because differences between representatives are dense
as we use analytic kernels (nonzero almost everywhere). The tests are consistent, while
much faster than stateoftheart quadratictime kernelbased tests. Experiments
on artificial
and realworld problems demonstrate
improved power/time tradeoff than the state of the art, based on
$\ell_2$ norms, and in some cases, better outright power than even the most
expensive quadratictime tests. This performance gain is retained even in high dimensions.
Author Information
Meyer Scetbon (CRESTENSAE)
Gael Varoquaux (Parietal Team, INRIA)
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2019 Poster: Comparing distributions: $\ell_1$ geometry improves kernel twosample testing »
Fri. Dec 13th 01:00  03:00 AM Room East Exhibition Hall B + C #6
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