Gokul Bhusal, Delanna Do, Kevin Miller, Ekaterina Merkurjev, Stephan Wojtowytsch
原文摘要
Active learning aims to reduce annotation cost by selecting the most informative examples for labeling. In the low-budget regime, where uncertainty-based methods often fail, coverage-based active learning methods such as ProbCover have been shown to be effective. However, these methods typically rely on Euclidean distance, which may not adequately capture the underlying geometry of data. In this article, we propose PC-Fermat, a coverage-based active learning method that replaces Euclidean coverage neighborhoods with neighborhoods defined by a Fermat distance, i.e., a data-adaptive distance that reflects the underlying distribution of the data points. To facilitate computation, we construct a weighted Euclidean neighborhood graph and use a truncated shortest-path algorithm to compute the Fermat coverage graph. We also establish sufficient conditions under which this construction exactly preserves the Fermat coverage computed on the complete graph. Finally, we evaluate PC-Fermat on eight classification datasets across varying labeling budgets, with particular emphasis on the very low-budget regime. Our experimental results demonstrate that PC-Fermat outperforms several existing active learning methods on the majority of the datasets.
Gokul Bhusal, Delanna Do, Ekaterina Merkurjev, Kevin Miller, Stephan Wojtowytsch
原文摘要
We compute the VC-dimension of metric ball classifiers in the model spaces of Riemannian geometry (Euclidean, spherical, hyperbolic) and present upper and lower bounds on the VC-dimension of metric balls in several other situations, including general normed vector spaces (potentially infinite VC-dimension), vector spaces with $\ell^p$-norms (VC-dimension grows at least linearly in the dimension, faster than Euclidean for $p=1$), flat tori, and general Riemannian manifolds (arbitrarily large VC-dimension after a conformal change). In several situations where the VC-dimension is large or even infinite, we prove generalization bounds under additional assumptions on the data distribution using covering arguments. Both in the VC-dimension and covering setting, we consider data on low-dimensional subsets of high-dimensional Euclidean spaces. As an application, we consider coverage-based algorithms in active learning.