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.

中文翻译：

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逼近协方差椭球

我们探索[公式：见文本]中的居中随机向量[公式：见文本]的协方差椭圆体[公式：见文本]可以通过简单集合近似的方法。为构造近似集给出的数据是[公式：参见文本]，它们是独立的并且分布为[公式：参见文本]。我们提出了一种通用方法，可用于构造此类近似值并针对两种类型的近似集实现它。我们首先构造一个集合 [Formula: see text]，由 slabs [Formula: see text] 的交集定义（因此 [Formula: see text] 实际上是一个简单神经网络的输出）。我们表明，在对[公式：见文本]的最小假设下（例如，[公式：见文本]可能是重尾的），[公式：见文本]足以确保[公式：见文本]。在某些情况下（例如 如果 [公式：参见文本] 是旋转不变的，并且具有在某种弱意义上表现良好的边缘），则较小的样本量就足够了：[公式：参见文本]。然后我们证明，如果用精心挑选的椭圆体代替平板，则在 [公式：参见文本] 时，相同程度的近似是正确的。该结构基于小球方法。