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\citet{farrell2021deep} establish non-asymptotic high-probability bounds for general deep feedforward neural network (with rectified linear unit activation function) estimators, with \citet[Theorem 1]{farrell2021deep} achieving a suboptimal…

Econometrics · Economics 2025-12-11 Zhaoji Tang

We analyze rates of approximation by quantized, tensor-structured representations of functions with isolated point singularities in ${\mathbb R}^3$. We consider functions in countably normed Sobolev spaces with radial weights and analytic-…

Numerical Analysis · Mathematics 2019-12-18 Carlo Marcati , Maxim Rakhuba , Christoph Schwab

We present a theory for simultaneous approximation of the score function and its derivatives, enabling the handling of data distributions with low-dimensional structure and unbounded support. Our approximation error bounds match those in…

Numerical Analysis · Mathematics 2025-12-30 Konstantin Yakovlev , Nikita Puchkin

In this paper, we develop a framework for showing that neural networks can overcome the curse of dimensionality in different high-dimensional approximation problems. Our approach is based on the notion of a catalog network, which is a…

Numerical Analysis · Mathematics 2021-10-13 Patrick Cheridito , Arnulf Jentzen , Florian Rossmannek

Deep learning techniques are increasingly popular in the textual entailment task, overcoming the fragility of traditional discrete models with hard alignments and logics. In particular, the recently proposed attention models (Rockt\"aschel…

Computation and Language · Computer Science 2017-09-05 Kai Zhao , Liang Huang , Mingbo Ma

Many computational problems can be formulated in terms of high-dimensional functions. Simple representations of such functions and resulting computations with them typically suffer from the "curse of dimensionality", an exponential cost…

Numerical Analysis · Mathematics 2022-09-16 Ruojing Peng , Johnnie Gray , Garnet Kin-Lic Chan

We introduce a neural network that represents sentences by composing their words according to induced binary parse trees. We use Tree-LSTM as our composition function, applied along a tree structure found by a fully differentiable natural…

Computation and Language · Computer Science 2020-01-16 Jean Maillard , Stephen Clark , Dani Yogatama

This paper develops a randomized approach for incrementally building deep neural networks, where a supervisory mechanism is proposed to constrain the random assignment of the weights and biases, and all the hidden layers have direct links…

Machine Learning · Computer Science 2018-03-19 Dianhui Wang , Ming Li

We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule…

Methodology · Statistics 2009-12-19 Herve Cardot , Jan Johannes

We introduce a novel deep learning algorithm for computing convex conjugates of differentiable convex functions, a fundamental operation in convex analysis with various applications in different fields such as optimization, control theory,…

Machine Learning · Computer Science 2026-01-21 Aleksey Minabutdinov , Patrick Cheridito

We enhance the approximation capabilities of algebraic polynomials by composing them with homeomorphisms. This composition yields families of functions that remain dense in the space of continuous functions, while enabling more accurate…

Numerical Analysis · Mathematics 2025-12-17 Álvaro Fernández Corral , Yahya Saleh

By selecting different filter functions, spectral algorithms can generate various regularization methods to solve statistical inverse problems within the learning-from-samples framework. This paper combines distributed spectral algorithms…

Machine Learning · Statistics 2025-02-18 Jiading Liu , Lei Shi

Constructing neural networks for function approximation is a classical and longstanding topic in approximation theory. In this paper, we aim at constructing deep neural networks (deep nets for short) with three hidden layers to approximate…

Information Theory · Computer Science 2020-01-14 Xia Liu

We study the expressive power of deep ReLU neural networks for approximating functions in dilated shift-invariant spaces, which are widely used in signal processing, image processing, communications and so on. Approximation error bounds are…

Machine Learning · Computer Science 2023-12-05 Yunfei Yang , Zhen Li , Yang Wang

Areas of computational mechanics such as uncertainty quantification and optimization usually involve repeated evaluation of numerical models that represent the behavior of engineering systems. In the case of complex nonlinear systems…

Machine Learning · Computer Science 2024-10-03 A. O. M. Kilicsoy , J. Liedmann , M. A. Valdebenito , F. -J. Barthold , M. G. R. Faes

Recently, deep learning has been widely applied in functional data analysis (FDA) with notable empirical success. However, the infinite dimensionality of functional data necessitates an effective dimension reduction approach for functional…

Machine Learning · Statistics 2025-05-13 Zhongjie Shi , Jun Fan , Linhao Song , Ding-Xuan Zhou , Johan A. K. Suykens

We examine the closedness of sets of realized neural networks of a fixed architecture in Sobolev spaces. For an exactly $m$-times differentiable activation function $\rho$, we construct a sequence of neural networks $(\Phi_n)_{n \in…

Machine Learning · Statistics 2021-01-29 Scott Mahan , Emily King , Alex Cloninger

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of…

Probability · Mathematics 2021-09-21 Côme Huré , Huyên Pham , Achref Bachouch , Nicolas Langrené

We study the approximation of functions which are invariant with respect to certain permutations of the input indices using flow maps of dynamical systems. Such invariant functions includes the much studied translation-invariant ones…

Machine Learning · Computer Science 2022-08-19 Qianxiao Li , Ting Lin , Zuowei Shen

We investigate a Tikhonov regularization scheme specifically tailored for shallow neural networks within the context of solving a classic inverse problem: approximating an unknown function and its derivatives within a unit cubic domain…

Numerical Analysis · Mathematics 2024-12-17 Yuanyuan Li , Shuai Lu
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