Related papers: Quantitative Universal Approximation Bounds for De…

This paper extends the proof of density of neural networks in the space of continuous (or even measurable) functions on Euclidean spaces to functions on compact sets of probability measures. By doing so the work parallels a more then a…

This work establishes computable bounds between f-divergences for probability measures within a generalized quasi-$\varepsilon_{(M,m)}$-neighborhood framework. We make the following key contributions. (1) a unified characterization of local…

Uncertainty quantification (UQ) in scientific machine learning is increasingly critical as neural networks are widely adopted to tackle complex problems across diverse scientific disciplines. For physics-informed neural networks (PINNs), a…

This paper concerns the universal approximation property with neural networks in variable Lebesgue spaces. We show that, whenever the exponent function of the space is bounded, every function can be approximated with shallow neural networks…

Recent performance breakthroughs in Artificial intelligence (AI) and Machine learning (ML), especially advances in Deep learning (DL), the availability of powerful, easy-to-use ML libraries (e.g., scikit-learn, TensorFlow, PyTorch.), and…

While Bayesian neural networks (BNNs) hold the promise of being flexible, well-calibrated statistical models, inference often requires approximations whose consequences are poorly understood. We study the quality of common variational…

Quantum photonic neural networks are variational photonic circuits that can be trained to implement high-fidelity quantum operations. However, work-to-date has assumed idealized components, including a perfect $\pi$ Kerr nonlinearity. Here,…

The classical Universal Approximation Theorem holds for neural networks of arbitrary width and bounded depth. Here we consider the natural `dual' scenario for networks of bounded width and arbitrary depth. Precisely, let $n$ be the number…

We provide finite-sample distribution approximations, that are uniform in the parameter, for inference in linear mixed models. Focus is on variances and covariances of random effects in cases where existing theory fails because their…

Aimed at explaining the surprisingly good generalization behavior of overparameterized deep networks, recent works have developed a variety of generalization bounds for deep learning, all based on the fundamental learning-theoretic…

As a fundamental piece of multi-object Bayesian inference, multi-object density has the ability to describe the uncertainty of the number and values of objects, as well as the statistical correlation between objects, thus perfectly matches…

Albeit worryingly underrated in the recent literature on machine learning in general (and, on deep learning in particular), multivariate density estimation is a fundamental task in many applications, at least implicitly, and still an open…

Neural networks activated by the rectified linear unit (ReLU) play a central role in the recent development of deep learning. The topic of approximating functions from H\"older spaces by these networks is crucial for understanding the…

Bayesian Neural Networks (BNNs) provide principled uncertainty quantification but suffer from substantial computational and memory overhead compared to deterministic networks. While quantization techniques have successfully reduced resource…

This paper deals with a method for the approximation of a spectral density function among the solutions of a generalized moment problem a` la Byrnes/Georgiou/Lindquist. The approximation is pursued with respect to the Kullback-Leibler…

The problem of establishing out-of-sample bounds for the values of an unkonwn ground-truth function is considered. Kernels and their associated Hilbert spaces are the main formalism employed herein along with an observational model where…

We prove Carl's type inequalities for the error of approximation of compact sets K by deep and shallow neural networks. This in turn gives lower bounds on how well we can approximate the functions in K when requiring the approximants to…

We study norm-based uniform convergence bounds for neural networks, aiming at a tight understanding of how these are affected by the architecture and type of norm constraint, for the simple class of scalar-valued one-hidden-layer networks,…

We study empirical Bayes (EB) predictive density estimation in linear mixed models (LMMs) with large number of units, which induce a high dimensional random effects space. Focusing on Kullback Leibler (KL) risk minimization, we develop a…

We can compare the expressiveness of neural networks that use rectified linear units (ReLUs) by the number of linear regions, which reflect the number of pieces of the piecewise linear functions modeled by such networks. However,…