Infinity Search: Approximate Vector Search with Projections on q-Metric Spaces

Despite the ubiquity of vector search applications, prevailing search algorithms overlook the metric structure of vector embeddings, treating it as a constraint rather than exploiting its underlying properties. In this paper, we demonstrate that in $q$-metric spaces, metric trees can leverage a stronger version of the triangle inequality to reduce comparisons for exact search. Notably, as $q$ approaches infinity, the search complexity becomes logarithmic. Therefore, we propose a novel projection method that embeds vector datasets with arbitrary dissimilarity measures into $q$-metric spaces while preserving the nearest neighbor. We propose to learn an approximation of this projection to efficiently transform query points to a space where euclidean distances satisfy the desired properties. Our experimental results with text and image vector embeddings show that learning $q$-metric approximations enables classic metric tree algorithms -- which typically underperform with high-dimensional data -- to achieve competitive performance against state-of-the-art search methods.