arXiv:2504. 11302v5 Announce Type: replace-cross Abstract: The Hausdorff dimension of a set can be detected using the Riesz energy.
arXiv:2607. 10751v1 Announce Type: new Abstract: Following Alon, Hanneke, Holzman, and Moran (FOCS 2021), we define a partial concept class (PCC) as a family of partial functions \(f: V\to\{0,1,\ast\}\); equivalently, its concepts partition the ground set into black ($f^{-1}(1)$), grey ($f^{-1}(\ast)$), and white parts ($f^{-1}(0)$).
arXiv:2310. 09149v3 Announce Type: replace-cross Abstract: We study the approximation of probability measures in the Wasserstein-$p$ distance by structured classes of approximators, motivated by applications in imaging, machine learning, and physical measurement under sensor constraints.
arXiv:2607. 21761v1 Announce Type: cross Abstract: We prove function-theoretic analogues of a quantitative result of Hodges on extracting the order property from a sufficiently large 2-tree coded in a binary relation.
arXiv:2606. 06148v1 Announce Type: new Abstract: In recent years, list replicability has emerged as a framework for formalizing reproducibility in learning theory.
arXiv:2609.23163v1 Announce Type: cross Abstract: Comparing probability measures in machine learning trades transport geometry against computational cost: Wasserstein distances encode the geometry of...
arXiv:2609. 06050v1 Announce Type: cross Abstract: For a compact set $K\subset\mathbb{C}$, let $\vartheta(K)$ be the infimum of the planar areas of the unit lemniscates of all monic polynomials with zeros in $K$, allowing arbitrary degree and repeated zeros.
arXiv:2607. 03112v1 Announce Type: cross Abstract: We revisit random projections for reducing the dimension of high-dimensional polygonal curves.
arXiv:2608. 03368v1 Announce Type: new Abstract: For $n$ unit vectors $x_1,\ldots,x_n \in \mathbb{R}^d$, we study the continuous ReLU derivative Gram matrix $H$, whose entries are obtained by averaging pairwise gated inner products over a standard Gaussian direction.
The paper develops a theory for relocating a finite number of compact sets in ℝ^n to arbitrary target domains using diffeomorphisms of ℝ^n. It proves that any such collection can be embedded differentiably into ℝ^{n+1} so that the images become linearly separable. The authors apply this result to show that compact datasets in ℝ^n can be made linearly separable by width‑n deep neural networks with Leaky‑ReLU, ELU, or SELU activations, and that mutually disjoint compact datasets can be separated in ℝ^{n+1} by a width‑(n+1) DNN.
arXiv:2608.30254v1 Announce Type: new Abstract: We resolve the threshold part of Question 4 of the COLT 2025 open problem "Data Selection for Regression Tasks" of Hanneke, Moran, Shlimovich and Yehud...
arXiv:2605. 09916v2 Announce Type: replace-cross Abstract: We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, designed to bypass the computational intractability of exact optimal transport in large-scale, non-Euclidean datasets.
arXiv:2608. 02533v1 Announce Type: cross Abstract: We construct unambiguous DNFs having width $O(n)$ but $0$-certificate complexity $\Omega(n^2)$.