We push each class to its supernode, then apply majority vote to determine the final class.

Machine learning models--including prominent zero-shot models--are often trained on datasets whose labels are only a small proportion of a larger label space. Such spaces are commonly equipped with a metric that relates the labels via distances between them.

Individual preference (IP) stability, introduced by Ahmadi et al. (ICML 2022), is a natural clustering objective inspired by stability and fairness constraints.

A fundamental theme throughout learning theory and statistics is that "smooth" functions ought to be

More generally, any finite subset of a hyperbolic space "looks like" a finite tree.

We now introduce the notation used in Theorems 3-4.

The of a metric space is a novelinvariant that provides a measure of the'effective size' of a space acrossmultiple scales, while also capturing numerous geometrical properties, such as curvature, density, or entropy.We develop a family of magnitude-based measures of the intrinsicdiversity of latent representations, formalising a novel notion ofdissimilarity between magnitude functions of finite metric spaces.Our measures are provably stable under perturbations of the data, can beefficiently calculated, and enable a rigorous multi-scale characterisation and comparison oflatent representations. We show their utility and superior performance across different domains and tasks, includingthe automated estimation of diversity,the detection of mode collapse, andthe evaluation of generative models for text, image, and graph data.

The metric $k$-center clustering problem with $z$ outliers, also known as $(k,z)$-center clustering, involves clustering a given point set $P$ in a metric space $(M,d)$ using at most $k$ balls, minimizing the maximum ball radius while excluding up to $z$ points from the clustering. This problem holds fundamental significance in various domains such as machine learning, data mining, and database systems.This paper addresses the fully dynamic version of the problem, where the point set undergoes continuous updates (insertions and deletions) over time. The objective is to maintain an approximate $(k,z)$-center clustering with efficient update times. We propose a novel fully dynamic algorithm that maintains a $(4+\epsilon)$-approximate solution to the $(k,z)$-center clustering problem that covers all but at most $(1+\epsilon)z$ points at any time in the sequence with probability $1-k/e^{\Omega(\log k)}$. The algorithm achieves an expected amortized update time of $\mathcal{O}(\epsilon^{-2} k^6\log(k) \log(\Delta))$, and is applicable to general metric spaces. Our dynamic algorithm presents a significant improvement over the recent dynamic $(14+\epsilon)$-approximation algorithm by Chan, Lattanzi, Sozio, and Wang for this problem.