Borsuk-Ulam and Replicable Learning of Large-Margin Halfspaces

Recent advances in learning theory have established that, for total concepts, list replicability, global stability, differentially private (DP) learnability, and shared-randomness replicability coincide precisely with the finiteness of the Littlestone dimension. Does the same hold for partial concept classes? We answer this question by studying the large-margin half-spaces class, which has bounded Littlestone dimension and is purely DP-learnable and shared-randomness replicable even in high dimensions. We prove that the list replicability number of $\gamma$-margin half-spaces satisfies \[ \frac{d}{2} + 1 \le \mathrm{LR}(H_{\gamma}^d) \le d, \] which increases with the dimension $d$. This reveals a surprising separation for partial concepts: list replicability and global stability do not follow from bounded Littlestone dimension, DP-learnability, or shared-randomness replicability. By applying our main theorem, we also answer the following open problems. - We prove that any disambiguation of an infinite-dimensional large-margin half-space to a total concept class has unbounded Littlestone dimension, answering an open question of Alon et al. (FOCS '21). - We prove that the maximum list-replicability number of any *finite* set of points and homogeneous half-spaces in $d$-dimensional Euclidean space is $d$, resolving a problem of Chase et al. (FOCS '23). - We prove that any disambiguation of the Gap Hamming Distance problem in the large gap regime has unbounded public-coin randomized communication complexity. This answers an open problem of Fang et al. (STOC '25). We prove the lower bound via a topological argument involving the local Borsuk-Ulam theorem of Chase et al. (STOC '24). For the upper bound, we design a learning rule that relies on certain triangulations of the cross-polytope and recent results on the generalization properties of SVM.

A Bottom-Up Approach to Class-Agnostic Image Segmentation

Class-agnostic image segmentation is a crucial component in automating image editing workflows, especially in contexts where object selection traditionally involves interactive tools. Existing methods in the literature often adhere to top-down formulations, following the paradigm of class-based approaches, where object detection precedes per-object segmentation. In this work, we present a novel bottom-up formulation for addressing the class-agnostic segmentation problem. We supervise our network directly on the projective sphere of its feature space, employing losses inspired by metric learning literature as well as losses defined in a novel segmentation-space representation. The segmentation results are obtained through a straightforward mean-shift clustering of the estimated features. Our bottom-up formulation exhibits exceptional generalization capability, even when trained on datasets designed for class-based segmentation. We further showcase the effectiveness of our generic approach by addressing the challenging task of cell and nucleus segmentation. We believe that our bottom-up formulation will offer valuable insights into diverse segmentation challenges in the literature.