Geometría en Ciencias de Datos

How well can we estimate the latent orthogonal factors of a probability distribution from random samples? This question sits at the intersection of optimal transport theory and factor analysis. In this talk, we show how to recover the orthogonal frame that best approximates the distribution in Wasserstein distance, via empirical risk minimization and an online stochastic subgradient method on the Stiefel manifold. A central difficulty is that the empirical objective is non-smooth, so standard M-estimation arguments do not apply. We circumvent this with a Clarke subdifferential analysis combined with a geometric covering argument on the sphere. The result is near-parametric convergence rates with polynomial, rather than exponential, dependence on the dimension, escaping the curse of dimensionality inherent in Wasserstein approximation. We close with numerical examples and real-world data applications.