Nonparametric Least Squares Regression for Image Reconstruction on the Sphere

This paper addresses the problem of interpolating signals defined on a 2-d sphere from non-uniform samples. We present an interpolation method based on locally weighted linear and nonlinear regression, which takes into account the differences in importance of neighboring samples for signal reconstruction. We show that for optimal kernel function variance, the proposed method performs interpolation more accurately than the nearest neighbor method, especially in noisy conditions. Moreover, this method does not have memory limitations which set the upper bound on the possible interpolation points number

Manifold structured prediction

Structured prediction provides a general framework to deal with supervised problems where the outputs have semantically rich structure. While classical approaches consider finite, albeit potentially huge, output spaces, in this paper we discuss how structured prediction can be extended to a continuous scenario. Specifically, we study a structured prediction approach to manifold-valued regression. We characterize a class of problems for which the considered approach is statistically consistent and study how geometric optimization can be used to compute the corresponding estimator. Promising experimental results on both simulated and real data complete our study

Optimal rates of convergence for convex set estimation from support functions

We present a minimax optimal solution to the problem of estimating a compact, convex set from finitely many noisy measurements of its support function. The solution is based on appropriate regularizations of the least squares estimator. Both fixed and random designs are considered.Comment: Published in at http://dx.doi.org/10.1214/11-AOS959 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org