{"ID":2850368,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.22401","arxiv_id":"2510.22401","title":"Johnson-Lindenstrauss Lemma Beyond Euclidean Geometry","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.","short_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 pre...","url_abs":"https://arxiv.org/abs/2510.22401","url_pdf":"https://arxiv.org/pdf/2510.22401v1","authors":"[\"Chengyuan Deng\",\"Jie Gao\",\"Kevin Lu\",\"Feng Luo\",\"Cheng Xin\"]","published":"2025-10-25T18:53:55Z","proceeding":"cs.DS","tasks":"[\"cs.DS\"]","methods":"[]","has_code":false}