We provide explicit bounds on the number of sample points required to estimate tangent spaces and intrinsic dimensions of (smooth, compact) Euclidean submanifolds via local principal component analysis. Our approach directly estimates covariance matrices locally, which simultaneously allows estimating both the tangent spaces and the intrinsic dimension of a manifold. The key arguments involve a matrix concentration inequality, a Wasserstein bound for flattening a manifold, and a Lipschitz relation for the covariance matrix with respect to the Wasserstein distance.

翻译：我们通过当地主要组成部分分析,对估计(软、紧、紧、紧)Euclidean子元件的正切空间和内在维度所需的抽样点数目规定了明确的界限。我们的方法是直接估计本地的共变矩阵,这既可以估计正切空间,也可以估计多个元件的内在维度。关键的论点包括矩阵浓度不平等、一个瓦塞斯坦用于平整一个元件的瓦塞斯特因距离的瓦塞斯特因关系矩阵和利普施奇茨关系。