The paper introduces a method for approximate vector search in ultrametric (infinity-metric) spaces by using a projection operator that transforms arbitrary dissimilarity functions into ultrametric ones while preserving nearest neighbors. It extends this approach to q-metric spaces, where triangle inequalities involve q-th powers, and learns an efficient approximation of the projected distances. Experiments show that higher q values speed up search at the cost of recall, yet the proposed methods remain competitive with existing search techniques.
arXiv:2509. 20848v2 Announce Type: replace-cross Abstract: In the classic point location problem, one is given an arbitrary dataset $X \subset \mathbb{R}^d$ of $n$ points with query access to an unknown halfspace $f : \mathbb{R}^d \to \{0,1\}$, and the goal is to learn the label of every point in $X$.
arXiv:2607. 09909v1 Announce Type: cross Abstract: We study nearest neighbor search from the perspective of data-driven algorithm design: given a dataset $P \subset \mathbb{R}^d$ of size $n$ and sample access to a query distribution over $\mathbb{R}^d$, the goal is to learn a data structure optimized for queries drawn from that specific distribution.
arXiv:2609. 02143v1 Announce Type: cross Abstract: Most vector databases rely on graph-based indexes, notably HNSW and Vamana, for approximate nearest neighbor search.
arXiv:2606. 11780v1 Announce Type: cross Abstract: We establish conditions for embedding a corpus of $N$ documents as $d$-dimensional vectors such that every $k$-subset $S \subseteq [N]$ is realizable as a result of top-$k$ retrieval by some query vector.
arXiv:2605. 25303v3 Announce Type: replace-cross Abstract: The $2 \rightarrow q$ norm of a matrix $X \in \mathbb{R}^{n \times d}$ is defined as $\lVert X \rVert_{2 \rightarrow q} = \sup_{\lVert v \rVert_2 = 1} \lVert Xv \rVert_q$.
CertEx introduces a GPU‑based exact k‑nearest‑neighbor search that relies on certified exclusion, a technique that conservatively eliminates vectors using low‑precision bounds before computing exact distances. By storing a half‑precision lower‑bounding projection and embedding a rounding certificate into the tensor‑core dot product, CertEx only evaluates distances for a small subset of candidates, guaranteeing exact results. Experiments on sixteen datasets with up to 100 million vectors show that CertEx achieves 7.1× to 18.0× the throughput of the fastest cuVS search on an H100 and outperforms quantized and graph indexes when exactness is required.
arXiv:2605. 14981v2 Announce Type: replace Abstract: Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system.
arXiv:2603. 06660v2 Announce Type: replace-cross Abstract: Approximate Nearest Neighbor Search (ANNS) is fundamental to modern AI applications.
arXiv:2606. 07782v1 Announce Type: cross Abstract: We propose a new framework for optimisation over non-Archimedean spaces inspired by Berkovich geometry.
arXiv:2606. 18520v1 Announce Type: cross Abstract: Computing geometric representations of data is a cornerstone of modern machine learning, typically achieved by training dual encoders which map queries and documents into a shared embedding space.
The paper introduces View distance, a novel metric that projects high‑dimensional data onto all pairwise two‑dimensional planes and sums the Euclidean distances across these projections. It satisfies metric axioms, couples features, suppresses redundancy, and captures anisotropic geometry. To make it scalable, the authors propose a plane‑selection strategy using iterative Maximum Weight Matching, reducing complexity from ω(n²) to ω(k) and demonstrating competitive performance on twelve datasets.
arXiv:2603. 07221v2 Announce Type: replace Abstract: Margin-based learning, exemplified by linear and kernel methods, is one of the few classical settings where generalization guarantees are independent of the number of parameters.