Johnson-Lindenstrauss Lemma Beyond Euclidean Geometry

TL;DR

This paper extends the Johnson-Lindenstrauss lemma to non-Euclidean data by providing theoretical guarantees for pseudo-Euclidean spaces and dissimilarity matrices, enabling effective dimensionality reduction beyond Euclidean geometry.

Contribution

It introduces two approaches: applying JL transform to pseudo-Euclidean spaces and representing dissimilarity matrices as generalized power distances, broadening JL's applicability.

Findings

The JL transform can be applied to pseudo-Euclidean vectors with guarantees based on their norms.

Symmetric hollow dissimilarity matrices can be approximated using generalized power distances.

Experimental validation shows effectiveness on synthetic and real-world datasets.

Abstract

The Johnson-Lindenstrauss (JL) lemma is a cornerstone of dimensionality reduction in Euclidean space, but its applicability to non-Euclidean data has remained limited. This paper extends the JL lemma beyond Euclidean geometry to handle general dissimilarity matrices that are prevalent in real-world applications. We present two complementary approaches: First, we show the JL transform can be applied to vectors in pseudo-Euclidean space with signature $(p,q)$, providing theoretical guarantees that depend on the ratio of the $(p, q)$ norm and Euclidean norm of two vectors, measuring the deviation from Euclidean geometry. Second, we prove that any symmetric hollow dissimilarity matrix can be represented as a matrix of generalized power distances, with an additional parameter representing the uncertainty level within the data. In this representation, applying the JL transform yields multiplicative approximation with a controlled additive error term proportional to the deviation from Euclidean geometry. Our theoretical results provide fine-grained performance analysis based on the degree to which the input data deviates from Euclidean geometry, making practical and meaningful reduction in dimensionality accessible to a wider class of data. We validate our approaches on both synthetic and real-world datasets, demonstrating the effectiveness of extending the JL lemma to non-Euclidean settings.