Related papers: Minimax Boundary Estimation and Estimation with Bo…
We focus on the problem of manifold estimation: given a set of observations sampled close to some unknown submanifold $M$, one wants to recover information about the geometry of $M$. Minimax estimators which have been proposed so far all…
We find the minimax rate of convergence in Hausdorff distance for estimating a manifold M of dimension d embedded in R^D given a noisy sample from the manifold. We assume that the manifold satisfies a smoothness condition and that the noise…
Many algorithms in machine learning and computational geometry require, as input, the intrinsic dimension of the manifold that supports the probability distribution of the data. This parameter is rarely known and therefore has to be…
Many economic parameters are identified by ``thin sets'' (submanifolds with Lebesgue measure zero) and hence difficult to recover from data in an ambient space. This paper provides a unified theory for estimation and inference of such…
The creation and justification of the methods for minimax estimation of parameters of the external boundary value problems for the Helmholtz equation in unbounded domains are considered. When observations are distributed in subdomains, the…
Given an $n$-sample drawn on a submanifold $M \subset \mathbb{R}^D$, we derive optimal rates for the estimation of tangent spaces $T\_X M$, the second fundamental form $II\_X^M$, and the submanifold $M$.After motivating their study, we…
Given a random sample of points from some unknown density, we propose a data-driven method for estimating density level sets under the r-convexity assumption. This shape condition generalizes the convexity property. However, the main…
Distribution regression seeks to estimate the conditional distribution of a multivariate response given a continuous covariate. This approach offers a more complete characterization of dependence than traditional regression methods.…
Often, high dimensional data lie close to a low-dimensional submanifold and it is of interest to understand the geometry of these submanifolds. The homology groups of a manifold are important topological invariants that provide an algebraic…
We consider the problem of optimality in manifold reconstruction. A random sample $\mathbb{X}_n = \left\{X_1,\ldots,X_n\right\}\subset \mathbb{R}^D$ composed of points close to a $d$-dimensional submanifold $M$, with or without outliers…
Statistical inference from high-dimensional data with low-dimensional structures has recently attracted lots of attention. In machine learning, deep generative modeling approaches implicitly estimate distributions of complex objects by…
We investigate density estimation from a $n$-sample in the Euclidean space $\mathbb R^D$, when the data is supported by an unknown submanifold $M$ of possibly unknown dimension $d < D$ under a reach condition. We study nonparametric kernel…
This paper characterizes the minimax linear estimator of the value of an unknown function at a boundary point of its domain in a Gaussian white noise model under the restriction that the first-order derivative of the unknown function is…
In this paper, we derive minimax rates for estimating both parametric and nonparametric components in partially linear additive models with high dimensional sparse vectors and smooth functional components. The minimax lower bound for…
The paper deals with the problem of nonparametric estimating the $L_p$--norm, $p\in (1,\infty)$, of a probability density on $R^d$, $d\geq 1$ from independent observations. The unknown density %to be estimated is assumed to belong to a ball…
We develop an approach for estimating models described via conditional moment restrictions, with a prototypical application being non-parametric instrumental variable regression. We introduce a min-max criterion function, under which the…
This paper presents minimax rates for density estimation when the data dimension $d$ is allowed to grow with the number of observations $n$ rather than remaining fixed as in previous analyses. We prove a non-asymptotic lower bound which…
Nonlinear dimensionality reduction or, equivalently, the approximation of high-dimensional data using a low-dimensional nonlinear manifold is an active area of research. In this paper, we will present a thematically different approach to…
Consider the problem of estimating the $\gamma$-level set $G^*_{\gamma}=\{x:f(x)\geq\gamma\}$ of an unknown $d$-dimensional density function $f$ based on $n$ independent observations $X_1,...,X_n$ from the density. This problem has been…
The paper presents analytic expressions of minimax (worst-case) estimates for solutions of linear abstract Neumann problems in Hilbert space with uncertain (not necessarily bounded!) inputs and boundary conditions given incomplete…