Estimation of multiple mean vectors in high dimension
Gilles Blanchard Jean-Baptiste Fermanian Hannah Marienwald
We endeavour to estimate numerous multi-dimensional means of various
probability distributions on a common space based on independent samples. Our
approach involves forming estimators through convex combinations of empirical
means derived from these samples. We introduce two strategies to find
appropriate data-dependent convex combination weights: a first one employing a
testing procedure to identify neighbouring means with low variance, which
results in a closed-form plug-in formula for the weights, and a second one
determining weights via minimization of an upper confidence bound on the
quadratic risk.Through theoretical analysis, we evaluate the improvement in
quadratic risk offered by our methods compared to the empirical means. Our
analysis focuses on a dimensional asymptotics perspective, showing that our
methods asymptotically approach an oracle (minimax) improvement as the
effective dimension of the data increases.We demonstrate the efficacy of our
methods in estimating multiple kernel mean embeddings through experiments on
both simulated and real-world datasets.