2012/09/05 by Guillermo D. Cañas, Lorenzo Rosasco, Canas, Guillermo D. +1 · 4 citations
Computer Science · Mathematics · #Machine Learning and Algorithms #Markov Chains and Monte Carlo Methods #Bayesian Modeling and Causal Inference
paper · pdf · doi:10.48550/arxiv.1209.1077
We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal quantization, and learning theory, we derive new probabilistic bounds for the performance of a classic algorithm in unsupervised learning (k-means), when used to produce a probability measure derived from the data. In the course of the analysis, we arrive at new lower bounds, as well as probabilistic upper bounds on the convergence rate of the empirical law of large numbers, which, unlike existing bounds, are applicable to a wide class of measures.