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Exact Limits of Random Projections for Preserving Geometry: Distance Recovery, Nearest-Neighbor Rankings, and Covariance Shape in Gaussian Models

arXiv · AI, language, vision and robotics · article · Sep 2, 2026 · UTC

The Johnson-Lindenstrauss (JL) lemma guarantees that a random projection of $n$ points to $m=O(\varepsilon^{-2}\log n)$ dimensions preserves pairwise squared distances within relative error $\varepsilon$ with high probability, and this dimension order is asymptotically optimal. In high dimensions, however, distances concentrate around a baseline while key geometric information lies in much smaller fluctuations. We show that the JL bound can therefore be uninformative about retained geometry: an independent Gaussian replacement map can satisfy it even though the replacement cloud is independent

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