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We investigate the asymptotic risk of a general class of overparameterized likelihood models, including deep models. The recent empirical success of large-scale models has motivated several theoretical studies to investigate a scenario…
Deep neural networks have attained remarkable success across diverse classification tasks. Recent empirical studies have shown that deep networks learn features that are linearly separable across classes. However, these findings often lack…
The Lipschitz constant plays a crucial role in certifying the robustness of neural networks to input perturbations. Since calculating the exact Lipschitz constant is NP-hard, efforts have been made to obtain tight upper bounds on the…
We consider ill-posed inverse problems where the forward operator $T$ is unknown, and instead we have access to training data consisting of functions $f_i$ and their noisy images $Tf_i$. This is a practically relevant and challenging…
We consider deep neural networks with a Lipschitz continuous activation function and with weight matrices of variable widths. We establish a uniform convergence analysis framework in which sufficient conditions on weight matrices and bias…
Ensuring neural network robustness is essential for the safe and reliable operation of robotic learning systems, especially in perception and decision-making tasks within real-world environments. This paper investigates the robustness of…
Deep learning has been applied to various tasks in the field of machine learning and has shown superiority to other common procedures such as kernel methods. To provide a better theoretical understanding of the reasons for its success, we…
Datasets such as images, text, or movies are embedded in high-dimensional spaces. However, in important cases such as images of objects, the statistical structure in the data constrains samples to a manifold of dramatically lower…
Lipschitz constrained networks have gathered considerable attention in the deep learning community, with usages ranging from Wasserstein distance estimation to the training of certifiably robust classifiers. However they remain commonly…
Machine learning methods are commonly used to solve inverse problems, wherein an unknown signal must be estimated from few indirect measurements generated via a known acquisition procedure. In particular, neural networks perform well…
There are many surprising and perhaps counter-intuitive properties of optimization of deep neural networks. We propose and experimentally verify a unified phenomenological model of the loss landscape that incorporates many of them. High…
Latent space models have been widely adopted in modeling network data. Developing statistical inference for estimated model parameters enables quantifying associated uncertainty and is pivotal for downstream tasks. Despite recent progress…
We consider a class of statistical inverse problems involving the estimation of a regression operator from a Polish space to a separable Hilbert space, where the target lies in a vector-valued reproducing kernel Hilbert space induced by an…
The present work provides a systematic approach for the design of sampled-data observers to a wide class of 1-D, parabolic PDEs with non-local outputs. The studied class of parabolic PDEs allows the presence of globally Lipschitz nonlinear…
We study adaptive data-dependent dimensionality reduction in the context of supervised learning in general metric spaces. Our main statistical contribution is a generalization bound for Lipschitz functions in metric spaces that are…
We develop a theoretical analysis for special neural network architectures, termed operator recurrent neural networks, for approximating nonlinear functions whose inputs are linear operators. Such functions commonly arise in solution…
It has long been thought that high-dimensional data encountered in many practical machine learning tasks have low-dimensional structure, i.e., the manifold hypothesis holds. A natural question, thus, is to estimate the intrinsic dimension…
The classical development of neural networks has been primarily for mappings between a finite-dimensional Euclidean space and a set of classes, or between two finite-dimensional Euclidean spaces. The purpose of this work is to generalize…
Channel estimation is crucial in wireless communications. However, in many papers neural networks are frequently tested by training and testing on one example channel or similar channels. This is because data-driven methods often degrade on…
We introduce LiPopt, a polynomial optimization framework for computing increasingly tighter upper bounds on the Lipschitz constant of neural networks. The underlying optimization problems boil down to either linear (LP) or semidefinite…