Related papers: Sample complexity and effective dimension for regr…

Existing theories on deep nonparametric regression have shown that when the input data lie on a low-dimensional manifold, deep neural networks can adapt to the intrinsic data structures. In real world applications, such an assumption of…

In this paper, we establish minimax optimal rates of convergence for prediction in a semi-functional linear model that consists of a functional component and a less smooth nonparametric component. Our results reveal that the smoother…

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…

Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to…

A neural network with the widely-used ReLU activation has been shown to partition the sample space into many convex polytopes for prediction. However, the parameterized way a neural network and other machine learning models use to partition…

We consider the theory of bond discounts, defined as the difference between the terminal payoff of the contract and its current price. Working in the setting of finite-dimensional realizations in the HJM framework, under suitable notions of…

In statistical learning, identifying underlying structures of true target functions based on observed data plays a crucial role to facilitate subsequent modeling and analysis. Unlike most of those existing methods that focus on some…

Kernel ridge regression is an important nonparametric method for estimating smooth functions. We introduce a new set of conditions, under which the actual rates of convergence of the kernel ridge regression estimator under both the L_2 norm…

The universality properties of kernels characterize the class of functions that can be approximated in the associated reproducing kernel Hilbert space and are of fundamental importance in the theoretical underpinning of kernel methods in…

We perform a study on kernel regression for large-dimensional data (where the sample size $n$ is polynomially depending on the dimension $d$ of the samples, i.e., $n\asymp d^{\gamma}$ for some $\gamma >0$ ). We first build a general tool to…

In this paper, the flexibility, versatility and predictive power of kernel regression are combined with now lavishly available network data to create regression models with even greater predictive performances. Building from previous work…

In this paper, we propose a novel lower dimensional representation of a shape sequence. The proposed dimension reduction is invertible and computationally more efficient in comparison to other related works. Theoretically, the differential…

Nonlinear kernels can be approximated using finite-dimensional feature maps for efficient risk minimization. Due to the inherent trade-off between the dimension of the (mapped) feature space and the approximation accuracy, the key problem…

We propose a robust and scalable procedure for general optimization and inference problems on manifolds leveraging the classical idea of `median-of-means' estimation. This is motivated by ubiquitous examples and applications in modern data…

We study non-linear data-dimension reduction. We are motivated by the classical linear framework of Principal Component Analysis. In nonlinear case, we introduce instead a new kernel-Principal Component Analysis, manifold and feature space…

We develop a new theoretical framework to analyze the generalization error of deep learning, and derive a new fast learning rate for two representative algorithms: empirical risk minimization and Bayesian deep learning. The series of…

We propose a metric learning paradigm, Regression-based Elastic Metric Learning (REML), which optimizes the elastic metric for geodesic regression on the manifold of discrete curves. Geodesic regression is most accurate when the chosen…

Two ubiquitous aspects of large-scale data analysis are that the data often have heavy-tailed properties and that diffusion-based or spectral-based methods are often used to identify and extract structure of interest. Perhaps surprisingly,…

Dimensionality reduction (DR) on the manifold includes effective methods which project the data from an implicit relational space onto a vectorial space. Regardless of the achievements in this area, these algorithms suffer from the lack of…

Exploiting the variational interpretation of kernel interpolation we exhibit a direct connection between interpolation and regression, where interpolation appears as a limiting case of regression. By applying this framework to point clouds…