arXiv:2609. 23529v1 Announce Type: new Abstract: Neural operators have emerged as powerful surrogates for solving partial differential equations (PDEs), yet their reliability under distribution shift remains a critical barrier to deployment.
arXiv:2602. 20971v3 Announce Type: replace-cross Abstract: Bubeck and Selke (2021) propose the connection between the Law of Robustness and robust generalization error as an open problem.
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:2109.02355v2 Announce Type: replace Abstract: The last decade of progress in machine learning (ML), especially the deep learning era, has raised a number of scientific questions that challenge...
The paper investigates how many linear samples are needed to learn Lipschitz operators under Gaussian measures. It establishes both lower and upper bounds on the Hermite polynomial approximation error and shows that the minimal worst‑case error cannot converge algebraically with the number of samples. However, if the covariance operator of the Gaussian measure decays rapidly, convergence rates arbitrarily close to any algebraic rate can be achieved.
The paper addresses the challenge of creating machine learning learners that can guarantee provably correct predictions in difficult test-time scenarios, such as adversarial attacks and natural distribution shifts. It introduces a reliable learner with optimal theoretical guarantees for these settings and discusses practical implementations. The authors demonstrate strong performance on examples like linear separators under log-concave distributions and smooth boundary classifiers under smooth probability distributions.
arXiv:2609.39512v1 Announce Type: new Abstract: The small-sample learning problem remains a fundamental challenge in machine learning because limited training data lead to unstable model estimation a...
arXiv:2607. 27995v1 Announce Type: cross Abstract: Adversarial training has emerged as a powerful approach for protecting models against adversarial attacks in a broad range of real-world applications.
arXiv:2504.18184v5 Announce Type: replace Abstract: We consider a class of statistical inverse problems involving the estimation of a regression operator from a Polish space to a separable Hilbert sp...
arXiv:2609.15355v2 Announce Type: replace-cross Abstract: We study the uniform approximation of smooth scalar-valued functionals on an infinite-dimensional separable Hilbert space by ReLU neural netw...
TestifAI is a deep learning testing framework that estimates model robustness against combinations of semantic input perturbations such as blur, brightness, and zoom. It allows users to define operational conditions as structured spaces with discrete severity levels and query robustness for any combination. By employing partial model tomography, TestifAI reconstructs higher‑order perturbation effects from low‑order tests, achieving less than 7% estimation error while reducing inference counts by 60‑80% across five image and language classification tasks.
The paper develops a statistical theory for minimum‑norm interpolation in high‑dimensional regression, showing how regularization geometry and signal sparsity affect generalization. It identifies regimes where sparsity‑promoting regularizers yield exact interpolation that is far more accurate than approximate fitting, and proves a zero–one generalization law for strongly overparameterized noiseless problems. The authors also characterize training and generalization errors along ρ‑regularization paths when feature dimension and sample size are proportional, demonstrating that generalization improves with more sparsity‑promoting norms and sparser targets, and that small changes in regularization strength can cause large shifts in generalization. whyItMatters":"The work provides a quantitative understanding of delayed generalization (grokking) and reveals a statistical instability in minimum‑norm interpolation, offering insights that could guide the design of regularizers for better generalization in overparameterized models."