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The nonparametric estimation of the regression function of a real-valued random variable Y on a random object X valued in a closed Riemannian manifold M is considered. A regression estimator which generalizes kernel regression es-timators on Euclidean sample spaces is introduced. Under classical assumptions on the kernel and the bandwidth sequence , the… (More)
We consider the problem of estimating the gradient lines of a density, which can be used to cluster points sampled from that density, for example via the mean-shift algorithm of Fukunaga and Hostetler (1975). We prove general convergence bounds that we then specialize to kernel density estimation.
Remote sensing of ocean color from space, a problem that consists in retrieving spectral marine reflectance from spectral top-of-atmosphere reflectance, is considered as a collection of similar inverse problems continuously indexed by the angular variables influencing the observation process. A general solution is proposed in the form of a field of… (More)
A methodology is presented for retrieving phytoplankton chlorophyll-a concentration from space. The data to be inverted, namely, vectors of top-of-atmosphere reflectance in the solar spectrum, are treated as explanatory variables conditioned by angular geometry. This approach leads to a continuum of inverse problems, i.e., a collection of similar inverse… (More)
Despite a substantial literature on nonparametric two-sample goodness-of-fit testing in arbitrary dimensions, there is no mention there of any curse of dimensionality. In fact, in some publications, a parametric rate is derived. As we discuss below, this is because a directional alternative is considered. Indeed, even in dimension one, Ingster (1987) has… (More)
In the context of nonlinear regression, we consider the problem of explaining a variable y from a vector x of explanatory variables and from a vector t of conditionning variables, that influences the link function between y and x. A neural based solution is proposed in the form of a field of nonlinear regression models, by which it is meant that the… (More)
Maximum Variance Unfolding is one of the main methods for (nonlinear) dimensionality reduction. We study its large sample limit, providing specific rates of convergence under standard assumptions. We find that it is consistent when the underlying submanifold is isometric to a convex subset, and we provide some simple examples where it fails to be consistent.
We consider the linear inverse problem of reconstructing an unknown finite measure µ from a noisy observation of a generalized moment of µ defined as the integral of a continuous and bounded operator Φ with respect to µ. Motivated by various applications, we focus on the case where the operator Φ is unknown; instead, only an approximation Φ m to it is… (More)
We study the approximation of a continuous function field over a compact set T , by a continuous field of ridge ap-proximants over T , named ridge function fields. We first give general density results about function fields, and show how they apply to ridge function fields. We next discuss the parametrization of sets of ridge function fields, and give… (More)
Following Hartigan , a cluster is defined as a connected component of the t-level set of the underlying density, i.e., the set of points for which the density is greater than t. A clustering algorithm which combines a density estimate with spectral clustering techniques is proposed. Our algorithm is composed of two steps. First, a nonparametric… (More)