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The purpose of this short article is to prove some potential estimates that naturally arise in the study of subelliptic Sobolev inequalites for functions. This will allow us to prove a local subelliptic Sobolev inequality with the optimal…

Although statistical learning theory provides a robust framework to understand supervised learning, many theoretical aspects of deep learning remain unclear, in particular how different architectures may lead to inductive bias when trained…

Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on…

Semisupervised methods inevitably invoke some assumption that links the marginal distribution of the features to the regression function of the label. Most commonly, the cluster or manifold assumptions are used which imply that the…

Practitioners often face the challenge of deploying prediction models in new environments with shifted distributions of covariates and responses. With observational data, such shifts are often driven by unobserved confounding, and can in…

Variable projection solves structured optimization problems by completely minimizing over a subset of the variables while iterating over the remaining variables. Over the last 30 years, the technique has been widely used, with empirical and…

We study the non-parametric estimation of an unknown density f with support on R+ based on an i.i.d. sample with multiplicative measurement errors. The proposed fully-data driven procedure consists of the estimation of the Mellin transform…

We present the first optimal rates for infinite-dimensional vector-valued ridge regression on a continuous scale of norms that interpolate between $L_2$ and the hypothesis space, which we consider as a vector-valued reproducing kernel…

We propose regularizing the empirical loss for semi-supervised learning by acting on both the input (data) space, and the weight (parameter) space. We show that the two are not equivalent, and in fact are complementary, one affecting the…

When optimizing over-parameterized models, such as deep neural networks, a large set of parameters can achieve zero training error. In such cases, the choice of the optimization algorithm and its respective hyper-parameters introduces…

Accurate estimation of spatial derivatives from discrete and noisy data is central to scientific machine learning and numerical solutions of PDEs. We extend kinetic-based regularization (KBR), a localized multidimensional kernel regression…

One central theme in machine learning is function estimation from sparse and noisy data. An example is supervised learning where the elements of the training set are couples, each containing an input location and an output response. In the…

The reconstruction of an unknown quantity from noisy measurements is a mathematical problem relevant in most applied sciences, for example, in medical imaging, radar inverse scattering, or astronomy. This underlying mathematical problem is…

Inverse scattering problems, such as those in electromagnetic imaging using phaseless data (PD-ISPs), involve imaging objects using phaseless measurements of wave scattering. Such inverse problems can be highly non-linear and ill-posed…

In this paper, we study the problem of early stopping for iterative learning algorithms in a reproducing kernel Hilbert space (RKHS) in the nonparametric regression framework. In particular, we work with the gradient descent and (iterative)…

In this study, we present and analyze a novel variant of the stochastic gradient descent method, referred as Stochastic data-driven Bouligand Landweber iteration tailored for addressing the system of non-smooth ill-posed inverse problems.…

We construct least squares formulations of PDEs with inhomogeneous essential boundary conditions, where boundary residuals are not measured in unpractical fractional Sobolev norms, but which formulations nevertheless are shown to yield a…

Inverse problems are key issues in several scientific areas, including signal processing and medical imaging. Data-driven approaches for inverse problems aim for learning model and regularization parameters from observed data samples, and…

So-called unsupervised anomaly detection is better described as semi-supervised, as it assumes all training data are nominal. This assumption simplifies training but requires manual data curation, introducing bias and limiting adaptability.…

We consider the nonparametric regression estimation problem of recovering an unknown response function f on the basis of spatially inhomogeneous data when the design points follow a known compactly supported density g with a finite number…