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Maximum Mean Discrepancy Meets Neural Networks: The Radon-Kolmogorov-Smirnov Test
Maximum mean discrepancy (MMD) refers to a general class of nonparametric two-sample tests that are based on maximizing the mean difference over samples from one distribution $P$ versus another $Q$, over all choices of data transformations $f$ living in some function space $\mathcal{F}$.
The Voronoigram: Minimax Estimation of Bounded Variation Functions From Scattered Data
We study an estimator that forms the Voronoi diagram of the design points, and then solves an optimization problem that regularizes according to a certain discrete notion of total variation (TV): the sum of weighted absolute differences of parameters $\theta_i,\theta_j$ (which estimate the function values $f_0(x_i), f_0(x_j)$) at all neighboring cells $i, j$ in the Voronoi diagram.
Minimax Optimal Regression over Sobolev Spaces via Laplacian Eigenmaps on Neighborhood Graphs
We also show that PCR-LE is \emph{manifold adaptive}: that is, we consider the situation where the design is supported on a manifold of small intrinsic dimension $m$, and give upper bounds establishing that PCR-LE achieves the faster minimax estimation ($n^{-2s/(2s + m)}$) and testing ($n^{-4s/(4s + m)}$) rates of convergence.
Minimax Optimal Regression over Sobolev Spaces via Laplacian Regularization on Neighborhood Graphs
In this paper we study the statistical properties of Laplacian smoothing, a graph-based approach to nonparametric regression.
