Approximating the covariance ellipsoid

Abstract

We explore ways in which the covariance ellipsoid [Formula: see text] of a centered random vector [Formula: see text] in [Formula: see text] can be approximated by a simple set. The data one is given for constructing the approximating set is [Formula: see text] that are independent and distributed as [Formula: see text]. We present a general method that can be used to construct such approximations and implement it for two types of approximating sets. We first construct a set [Formula: see text] defined by a union of intersections of slabs [Formula: see text] (and therefore [Formula: see text] is actually the output of a simple neural network). We show that under minimal assumptions on [Formula: see text] (e.g. [Formula: see text] can be heavy-tailed) it suffices that [Formula: see text] to ensure that [Formula: see text]. In some cases (e.g. if [Formula: see text] is rotation invariant and has marginals that are well behaved in some weak sense), a smaller sample size suffices: [Formula: see text]. We then show that if the slabs are replaced by well-chosen ellipsoids, the same degree of approximation is true when [Formula: see text]. The construction is based on the small-ball method.