arXiv:2606. 04390v1 Announce Type: new Abstract: We find the asymptotic ratio between the storage capacities when enforcing real pre-activations in a complex hypothesis class as opposed to complex ones in the same class.
arXiv:2609. 05263v1 Announce Type: cross Abstract: We investigate the best $L_2$ approximation of mixed Sobolev spaces by shallow neural networks with $n$ neurons and general activation functions.
arXiv:2608. 19584v1 Announce Type: new Abstract: We study landscapes for complex-parameterized networks.
arXiv:2610.18166v1 Announce Type: cross Abstract: Randomly initialised neural networks of width $m$ converge to Gaussian processes as $m \to \infty$, with two types of quantitative bounds characteris...
arXiv:2607. 10589v1 Announce Type: cross Abstract: In contrast to most studies on neural network approximation theory that characterize results through a single parameter, such as the total number of network parameters, \cite{shen2020deep} pioneered the characterization of approximation rates as a joint function of the width parameter $N$ and the depth parameter $L$, thereby granting greater architectural flexibility.
arXiv:2603. 00819v2 Announce Type: replace-cross Abstract: This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory.
arXiv:2106. 04770v2 Announce Type: replace Abstract: We study parameter nonuniqueness in continuous-width depth-two fully connected neural networks.
The paper investigates depth separation for neural networks in the context of $L^2$ approximation of radial functions under a standard Gaussian measure in $d$ dimensions. It demonstrates that a particular radial function can be approximated within a fixed small $L^2$ error by a depth‑3 ReLU network of polynomial size in $d$, yet any depth‑2 network approximating the same function with bounded weights must have size exponential in $d$. The authors prove a general result linking high‑frequency Hermite expansion mass to the limitations of depth‑2 networks with small width and bounded weights, and all results are formally verified in Lean 4.
arXiv:2607. 15702v2 Announce Type: replace-cross Abstract: We develop a non-asymptotic approximation, sampling, and finite-iteration optimization theory for variational physics-informed approximation of uniformly monotone nonlinear multiscale elliptic equations.
An input may activate few hidden units even when different inputs collectively use an entire network. We study the statistical complexity of this input-dependent sparsity in the one-hidden-layer ReLU model of Awasthi et al.
arXiv:2610.20229v1 Announce Type: cross Abstract: We investigate the minimal width of deep neural networks that can approximate all continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ in the t...
arXiv:2602. 19172v2 Announce Type: replace Abstract: Realizable online regression can behave very differently from online classification.
arXiv:2609. 09589v1 Announce Type: new Abstract: Deep neural networks exhibit regular macroscopic behavior despite highly nonlinear dynamics in vast parameter spaces.