Transformer-based large models have demonstrated remarkable generalization abilities across different tasks by leveraging a context-aware attention module for in-context learning. With richer context, transformers adapt more effectively to the current use case without any parameter updates. However, the quadratic computational and memory complexity with respect to context length significantly slows data processing in softmax transformers. Linear transformers were proposed to address this issue by reducing the complexity to linear dependence on context length, but the design and understanding of the feature mapping in linear attention, from a theoretical viewpoint, remain unclear. In this paper, we investigate the approximation and generalization abilities of linear transformers under a two-staged sampling process from domain generalization. We show that linear transformers perform in-context learning as learning a mapping from context distributions to response functions. A dimension-independent convergence rate is obtained for our generalization analysis, which also exhibits the tradeoff between the regularities of data distributions and latent features. Guided by our theoretical framework, we propose a new perspective on activation and loss design for linearizing pretrained softmax large language models.

Autoencoders (AEs) have emerged as powerful tools for non-linear dimensionality reduction, often surpassing traditional linear methods such as Proper Orthogonal Decomposition (POD) in scenarios characterized by slowly decaying Kolmogorov $n$-widths. In the realm of Reduced-Order Modelling (ROM), these models are increasingly utilized to learn low-dimensional representations of solution manifolds associated with parametric Partial Differential Equations (PDEs). However, the high expressivity of AEs presents a challenge: although trained networks typically minimize reconstruction error, they often struggle to capture the essential properties necessary for building accurate and robust ROMs. Recent works by arXiv:2307.15288v2 and arXiv:2506.11641v1 have tackled this challenge in fully connected AEs by proposing representation-consistent architectures, which preserve some of the properties belonging to POD. This study builds upon that concept by extending representation consistency for convolutional layers. We introduce a novel class of symmetric Convolutional AutoEncoders (CAEs) designed to embody the primary properties of manifold parametrization mappings. When integrated into a ROM framework, this architecture demonstrates significantly improved predictive capabilities. Specifically, we compared the performance of the ROMs based on classical and symmetric CAEs on three one dimensional academic test cases, namely the Linear Advection, the Viscous Burger and the Kuramoto Sivashinsky equation. Numerical results demonstrate that our proposed symmetric approach consistently yields more accurate latent trajectories, lower reconstruction errors, and enhanced model robustness.

Despite the remarkable successes of general-purpose neural networks, such as MLPs and Transformers, we find that they exhibit notable shortcomings in modeling and reasoning about periodic phenomena, achieving only marginal performance within the training domain and failing to generalize effectively to out-of-domain (OOD) scenarios. Periodicity is ubiquitous throughout nature and science. Therefore, neural networks should be equipped with the essential ability to model and handle periodicity. In this work, we propose FAN, a novel neural network that effectively addresses periodicity modeling challenges while offering broad applicability similar to MLP with fewer parameters and FLOPs. Periodicity is naturally integrated into FAN's structure and computational processes by introducing the Fourier Principle. Unlike existing Fourier-based networks, which possess particular periodicity modeling abilities but face challenges in scaling to deeper networks and are typically designed for specific tasks, our approach overcomes this challenge to enable scaling to large-scale models and maintains the capability to be applied to more types of tasks. Through extensive experiments, we demonstrate the superiority of FAN in periodicity modeling tasks and the effectiveness and generalizability of FAN across a range of real-world tasks. Moreover, we reveal that compared to existing Fourier-based networks, FAN accommodates both periodicity modeling and general-purpose modeling well.

In classification tasks, softmax functions are ubiquitously used as output activations to produce predictive probabilities. Such outputs only capture aleatoric uncertainty. To capture epistemic uncertainty, approximate Gaussian inference methods have been proposed. We develop a common formalism to describe such methods, which we view as outputting Gaussian distributions over the logit space. Predictives are then obtained as the expectations of the Gaussian distributions pushed forward through the softmax.

We study the task of agnostically learning general (as opposed to homogeneous) ReLUs under the Gaussian distribution with respect to the squared loss. In the passive learning setting, recent work gave a computationally efficient algorithm that uses poly(d,1/ϵ)labeled examples and outputs a hypothesis with error O(opt)+ϵ, where optis the squared loss of the best fit ReLU. Here we focus on the interactive setting, where the learner has some form of query access to the labels of unlabeled examples. Our main result is the first computationally efficient learner that uses dpolylog(1/ϵ)+ O(min{1/p,1/ϵ})black-box label queries, where pis the bias of the target function, and achieves error O(opt)+ϵ. We complement our algorithmic result by showing that its query complexity bound is qualitatively near-optimal, even ignoring computational constraints. Finally, we establish that query access is essentially necessary to improve on the label complexity of passive learning. Specifically, for pool-based active learning, any active learner requires Ω(d/ϵ) labels, unless it draws a super-polynomial number of unlabeled examples.

Recently, there has been a growing focus on determining the minimum width requirements for achieving the universal approximation property in deep, narrow Multi-Layer Perceptrons (MLPs). Among these challenges, one particularly challenging task is approximating a continuous function under the uniform norm, as indicated by the significant disparity between its lower and upper bounds. To address this problem, we propose a framework that simplifies finding the minimum width for deep, narrow MLPs into determining a purely geometrical function denoted as w(dx,dy). This function relies solely on the input and output dimensions, represented as dx and dy, respectively. To achieve this, we first demonstrate that deep, narrow MLPs, when provided with a small additional width, can approximate any C2-diffeomorphism. Subsequently, using this result, we prove that w(dx,dy) equates to the optimal minimum width required for deep, narrow MLPs to achieve universality. By employing the aforementioned framework and the Whitney embedding theorem, we provide an upper bound for the minimum width, given by max(2dx +1,dy)+α(σ), where 0 α(σ) 2represents a constant depending explicitly on the activation function. Furthermore, we provide novel optimal values for the minimum width in several settings, including w(2,2) = w(2,3) = 4.

Numerous studies have demonstrated that the Transformer architecture possesses the capability for in-context learning (ICL). In scenarios involving function approximation, context can serve as a control parameter for the model, endowing it with the universal approximation property (UAP). In practice, context is represented by tokens from a finite set, referred to as a vocabulary, which is the case considered in this paper, i.e., vocabulary in-context learning (VICL). We demonstrate that VICL in single-layer Transformers, without positional encoding, does not possess the UAP; however, it is possible to achieve the UAP when positional encoding is included. Several sufficient conditions for the positional encoding are provided. Our findings reveal the benefits of positional encoding from an approximation theory perspective in the context of ICL.

We study a shallow variant of XNet, a neural architecture whose activation functions are derived from the Cauchy integral formula. While prior work focused on deep variants, we show that even a single-layer XNet exhibits near-exponential approximation rates--exceeding the polynomial bounds of MLPs and spline-based networks such as Kolmogorov-Arnold Networks (KANs). Empirically, XNet reduces approximation error by over 600 on discontinuous functions, achieves up to 20,000 lower residuals in physics-informed PDEs, and improves policy accuracy and sample efficiency in PPO-based reinforcement learning--while maintaining comparable or better computational efficiency than KAN baselines. These results demonstrate that expressive approximation can stem from principled activation design rather than depth alone, offering a compact, theoretically grounded alternative for function approximation, scientific computing, and control.

Nonlinear activation functions are widely recognized for enhancing the expressivity of neural networks, which is the primary reason for their widespread implementation. In this work, we focus on ReLU activation and reveal a novel and intriguing property of nonlinear activations. By comparing enabling and disabling the nonlinear activations in the neural network, we demonstrate their specific effects on wide neural networks: (a) better feature separation, i.e., a larger angle separation for similar data in the feature space of model gradient, and (b) better NTK conditioning, i.e., a smaller condition number of neural tangent kernel (NTK). Furthermore, we show that the network depth (i.e., with more nonlinear activation operations) further amplifies these effects; in addition, in the infinite-width-then-depth limit, all data are equally separated with a fixed angle in the model gradient feature space, regardless of how similar they are originally in the input space. Note that, without the nonlinear activation, i.e., in a linear neural network, the data separation remains the same as for the original inputs and NTK condition number is equivalent to the Gram matrix, regardless of the network depth. Due to the close connection between NTK condition number and convergence theories, our results imply that nonlinear activation helps to improve the worst-case convergence rates of gradient based methods.

Equivariant neural networks outperform traditional deep neural networks on a number of tasks. The theoretical understanding of their generalization properties remains, however, limited. In this paper, we analyze the generalization capabilities of Group Convolutional Neural Networks (GCNNs) with ReLU activation function through the lens of Vapnik-Chervonenkis (VC) dimension theory. By deriving upper and lower bounds, we investigate how the network architecture affects the VC dimension.