  We introduce the notion of multiscale covariance tensor fields (CTF)
associated with Euclidean random variables as a gateway to the shape of their
distributions. Multiscale CTFs quantify variation of the data about every point
in the data landscape at all spatial scales, unlike the usual covariance tensor
that only quantifies global variation about the mean. Empirical forms of
localized covariance previously have been used in data analysis and
visualization, but we develop a framework for the systematic treatment of
theoretical questions and computational models based on localized covariance.
We prove strong stability theorems with respect to the Wasserstein distance
between probability measures, obtain consistency results, as well as estimates
for the rate of convergence of empirical CTFs. These results ensure that CTFs
are robust to sampling, noise and outliers. We provide numerous illustrations
of how CTFs let us extract shape from data and also apply CTFs to manifold
clustering, the problem of categorizing data points according to their noisy
membership in a collection of possibly intersecting, smooth submanifolds of
Euclidean space. We prove that the proposed manifold clustering method is
stable and carry out several experiments to validate the method.
