Infinity Search：在 q-Metric 空间上用投影实现近似向量搜索

View PDF HTML (experimental)

Abstract:An ultrametric space or infinity-metric space is defined by a dissimilarity function that satisfies a strong triangle inequality in which every side of a triangle is not larger than the larger of the other two. We show that search in ultrametric spaces with a vantage point tree has worst-case complexity equal to the depth of the tree. Since datasets of interest are not ultrametric in general, we employ a projection operator that transforms an arbitrary dissimilarity function into an ultrametric space while preserving nearest neighbors. We further learn an approximation of this projection operator to efficiently compute ultrametric distances between query points and points in the dataset. We proceed to solve a more general problem in which we consider projections in $q$-metric spaces -- in which triangle sides raised to the power of $q$ are smaller than the sum of the $q$-powers of the other two. Notice that the use of learned approximations of projected $q$-metric distances renders the search pipeline approximate. We show in experiments that increasing values of $q$ result in faster search but lower recall. Overall, search in q-metric and infinity metric spaces is competitive with existing search methods.

Subjects:	Information Retrieval (cs.IR); Machine Learning (cs.LG); Signal Processing (eess.SP); Metric Geometry (math.MG)
Cite as:	arXiv:2506.06557 [cs.IR]
	(or arXiv:2506.06557v3 [cs.IR] for this version)
	https://doi.org/10.48550/arXiv.2506.06557 arXiv-issued DOI via DataCite

Submission history

From: Antonio Pariente [view email]
[v1] Fri, 6 Jun 2025 22:09:44 UTC (7,035 KB)
[v2] Sat, 7 Feb 2026 04:21:36 UTC (37,868 KB)
[v3] Thu, 1 Oct 2026 21:04:57 UTC (59,537 KB)