arXiv Machine Learning August 10, 2026 By Xinliang Liu, Tong Mao, Jinchao Xu

Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samples

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arXiv:2608. 06687v1 Announce Type: cross Abstract: We develop a rigorous theory of discrete residual least-squares approximation for elliptic spectral equations $\mathfrak L_\beta u=f$ using linearized ReLU$^k$ neural networks on the sphere, where $\mathfrak L_\beta$ is a positive elliptic spectral multiplier of order $\beta$.

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arXiv Machine Learning

5d ago

Active-Trace Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling

arXiv:2608. 13467v1 Announce Type: new Abstract: We study the Moreau--Yosida unadjusted Langevin algorithm (MYULA) for the nonsmooth composite target \[ \pi(dx)\propto \exp\{-f(x)-g(x)\}\,dx, \qquad x\in\mathbb R^d, \] where $f$ is $m$-strongly convex with $L_f$-Lipschitz gradient and $g$ is convex and $G$-Lipschitz.

By Yuchen Xin, Zhihua Zhang

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arXiv Machine Learning

Aug 4

Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation

arXiv:2605. 08170v2 Announce Type: replace Abstract: Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces.

By Nicole Hao

arXiv Machine Learning

Jun 16

Convergence Rate Analysis of the AdamW-style Shampoo: Unifying One-Sided and Two-Sided Preconditioning

arXiv:2601. 07326v4 Announce Type: replace-cross Abstract: This paper studies AdamW-style Shampoo, an effective variant of the classical Shampoo that won the external tuning track of the AlgoPerf neural network training competition.

By Huan Li, Yiming Dong, Zhouchen Lin

arXiv Machine Learning

Aug 12

Dual Space Preconditioning for Gradient Descent in the Overparameterized Regime

arXiv:2603. 10485v3 Announce Type: replace-cross Abstract: In this work, we study the convergence properties of the Dual Space Preconditioned Gradient Descent, encompassing optimizers such as Normalized Gradient Descent and Gradient Clipping.

By Reza Ghane, Danil Akhtiamov, Babak Hassibi

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arXiv Machine Learning

Jul 27

Approximation and learning of anisotropic and mixed smooth functions by deep ReLU neural networks

arXiv:2605. 31152v2 Announce Type: replace-cross Abstract: This paper studies how efficiently deep ReLU neural networks can approximate and learn smooth functions.

By Yunfei Yang, Jun Fan

rag

arXiv Machine Learning

Aug 4

Improved convergence rate of kNN graph Laplacians: differentiable self-tuned affinity

arXiv:2410. 23212v3 Announce Type: replace-cross Abstract: In graph-based data analysis, $k$-nearest neighbor ($k$NN) graphs are widely used due to their adaptivity to local data densities.

By Xiuyuan Cheng, Yixuan Tan, Nan Wu

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Related stories

Active-Trace Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling

Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation

Convergence Rate Analysis of the AdamW-style Shampoo: Unifying One-Sided and Two-Sided Preconditioning

Dual Space Preconditioning for Gradient Descent in the Overparameterized Regime

Approximation and learning of anisotropic and mixed smooth functions by deep ReLU neural networks

Improved convergence rate of kNN graph Laplacians: differentiable self-tuned affinity