Related papers: Thin Sets Are Not Equally Thin: Minimax Learning o…
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 establish minimax convergence rates for classification of functional data and for nonparametric regression with functional design variables. The optimal rates are of logarithmic type under smoothness constraints on the functional density…
Nonparametric estimation of nonlocal interaction kernels is crucial in various applications involving interacting particle systems. The inference challenge, situated at the nexus of statistical learning and inverse problems, arises from the…
Given an observation $\mathbf Y \in \mathbb{R}^{d_1\times d_2}$ from the model $\mathbf Y = \mathbf X + \mathbf E$ where $\mathbf X$ is constant and $\mathbf E$ has i.i.d. $N(0,1)$ entries, we consider the problem of detecting a planted…
A common observation in data-driven applications is that high dimensional data has a low intrinsic dimension, at least locally. In this work, we consider the problem of estimating a $d$ dimensional sub-manifold of $\mathbb{R}^D$ from a…
This work studies the computational aspects of multivariate convex regression in dimensions $d \ge 5$. Our results include the \emph{first} estimators that are minimax optimal (up to logarithmic factors) with polynomial runtime in the…
We consider the regression problem of estimating functions on $\mathbb{R}^D$ but supported on a $d$-dimensional manifold $ \mathcal{M} \subset \mathbb{R}^D $ with $ d \ll D $. Drawing ideas from multi-resolution analysis and nonlinear…
We study the properties of nonparametric least squares regression using deep neural networks. We derive non-asymptotic upper bounds for the prediction error of the empirical risk minimizer of feedforward deep neural regression. Our error…
We study the problem of estimating an unknown function from noisy data using shallow ReLU neural networks. The estimators we study minimize the sum of squared data-fitting errors plus a regularization term proportional to the squared…
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…
Learning rates for least-squares regression are typically expressed in terms of $L_2$-norms. In this paper we extend these rates to norms stronger than the $L_2$-norm without requiring the regression function to be contained in the…
The central subspace of a pair of random variables $(y,x) \in \mathbb{R}^{p+1}$ is the minimal subspace $\mathcal{S}$ such that $y \perp \hspace{-2mm} \perp x\mid P_{\mathcal{S}}x$. In this paper, we consider the minimax rate of estimating…
Estimation of convex functions finds broad applications in engineering and science, while convex shape constraint gives rise to numerous challenges in asymptotic performance analysis. This paper is devoted to minimax optimal estimation of…
Previous analysis of regularized functional linear regression in a reproducing kernel Hilbert space (RKHS) typically requires the target function to be contained in this kernel space. This paper studies the convergence performance of…
A linear functional of an object from a convex symmetric set can be optimally estimated, in a worst-case sense, by a linear functional of observations made on the object. This well-known fact is extended here to a nonlinear setting: other…
We propose a general methodology for the construction and analysis of minimax estimators for a wide class of functionals of finite dimensional parameters, and elaborate on the case of discrete distributions, where the alphabet size $S$ is…
High-dimensional functional data have become increasingly prevalent in modern applications such as high-frequency financial data and neuroimaging data analysis. We investigate a class of high-dimensional linear regression models, where each…
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…
Coresets have emerged as a powerful tool to summarize data by selecting a small subset of the original observations while retaining most of its information. This approach has led to significant computational speedups but the performance of…
We study minimax estimation of two-dimensional totally positive distributions. Such distributions pertain to pairs of strongly positively dependent random variables and appear frequently in statistics and probability. In particular, for…