Related papers: Minimax adaptive estimation in manifold inference

We derive non-asymptotic minimax bounds for the Hausdorff estimation of $d$-dimensional submanifolds $M \subset \mathbb{R}^D$ with (possibly) non-empty boundary $\partial M$. The model reunites and extends the most prevalent…

Assume that we observe i.i.d.~points lying close to some unknown $d$-dimensional $\mathcal{C}^k$ submanifold $M$ in a possibly high-dimensional space. We study the problem of reconstructing the probability distribution generating the…

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 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…

We present the first minimax risk bounds for estimators of the spectral measure in multivariate linear factor models, where observations are linear combinations of regularly varying latent factors. Non-asymptotic convergence rates are…

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…

We consider a circular deconvolution problem, in which the density $f$ of a circular random variable $X$ must be estimated nonparametrically based on an i.i.d. sample from a noisy observation $Y$ of $X$. The additive measurement error is…

We start by considering the problem of estimating intrinsic distances on a smooth submanifold. We show that minimax optimality can be obtained via a reconstruction of the surface, and discuss the use of a particular mesh construction -- the…

We propose a generative model that achieves minimax-optimal convergence rates for estimating probability distributions supported on unknown low-dimensional manifolds. Building on Fefferman's solution to the geometric Whitney problem, our…

We consider a linear model where the coefficients - intercept and slopes - are random with a law in a nonparametric class and independent from the regressors. Identification often requires the regressors to have a support which is the whole…

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…

We find lower and upper bounds for the risk of estimating a manifold in Hausdorff distance under several models. We also show that there are close connections between manifold estimation and the problem of deconvolving a singular measure.

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…

We consider a problem of manifold estimation from noisy observations. Many manifold learning procedures locally approximate a manifold by a weighted average over a small neighborhood. However, in the presence of large noise, the assigned…

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.…

Quantile estimation in deconvolution problems is studied comprehensively. In particular, the more realistic setup of unknown error distributions is covered. Our plug-in method is based on a deconvolution density estimator and is minimax…

The task of estimating a matrix given a sample of observed entries is known as the \emph{matrix completion problem}. Most works on matrix completion have focused on recovering an unknown real-valued low-rank matrix from a random sample of…

For linear time-invariant systems with uncertain parameters belonging to a finite set, we present a purely deterministic approach to multiple-model estimation and propose an algorithm based on the minimax criterion using constrained…

While classical data analysis has addressed observations that are real numbers or elements of a real vector space, at present many statistical problems of high interest in the sciences address the analysis of data that consist of more…

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…