155 citations · 987 across the 104 of their papers we have counts for
9 papers · 1 filter
A polynomial dimension-dependence analysis of Bramble--Pasciak--Xu preconditioners
Boou Jiang, Jongho Park, Jinchao Xu
We investigate the dimension dependence of Bramble--Pasciak--Xu (BPX) preconditioners for high-dimensional partial differential equations and establish that the condition numbers o…
Condition Numbers and Eigenvalue Spectra of Shallow Networks on Spheres
Xinliang Liu, Tong Mao, Jinchao Xu
We present an estimation of the condition numbers of the \emph{mass} and \emph{stiffness} matrices arising from shallow ReLU neural networks defined on the unit sphere~$\mathbb…
On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions
Yulong Lu, Tong Mao, Jinchao Xu +1
Deep neural networks have been widely used as universal approximators for functions with inherent physical structures, including permutation symmetry. In this paper, we construct s…
Configuration-Dependent Lower Bounds for Approximation by Shallow ReLU Networks on the Sphere
Tong Mao, Jinchao Xu
We establish two related but logically distinct results for shallow ReLU neural networks on the unit sphere $\SS^d$. First, for an arbitrary set of inner neural-network paramet…
Auxiliary space theory for the analysis of iterative methods for semidefinite linear systems
Jongho Park, Jinchao Xu
We present an auxiliary space theory that provides a unified framework for analyzing various iterative methods for solving linear systems that may be semidefinite. By interpreting…
Self-composing neural operators for high-frequency and multiscale PDE surrogates
Juncai He, Xinliang Liu, Jinchao Xu
Addressing the computational challenges of high-frequency and multiscale partial differential equations (PDEs), this work introduces a self-composing neural operator (SC-NO) framew…
Randomized subspace correction methods for convex optimization
Boou Jiang, Jongho Park, Jinchao Xu
This paper introduces an abstract framework for randomized subspace correction methods for convex optimization, which unifies and generalizes a broad class of existing algorithms,…
Integral Representations of Sobolev Spaces via ReLU Activation Function and Optimal Error Estimates for Linearized Networks
Xinliang Liu, Tong Mao, Jinchao Xu
This paper presents two main theoretical results concerning shallow neural networks with ReLU activation functions. We establish a novel integral representation for Sobolev spa…
Subspace correction as a general framework for convex optimization algorithms
Boou Jiang, Jongho Park, Jinchao Xu
This paper shows that subspace correction methods provide a common algorithmic and theoretical foundation for several classes of convex optimization algorithms, including operator…