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This talk is about multidimensional quantiles, which provide robust
notions of shape, depth and dispersion of a distribution or dataset. It
will focus on Tukey depth and depth curves, which are essentially the same as the convex floating body in convex geometry. These concepts can
be difficult to use because they are computationally intractable in
general. We develop a theory of algorithmic efficiency for these notions
for several broad and relevant families of distributions: symmetric
log-concave distributions and certain multivariate stable distributions
and power-law distributions. As an example of the power of these
results, we show how to solve the Independent Component Analysis problem
for power-law distributions, even when the first moment is infinite.