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

What the math gives to ML

The paper identifies a concrete failure mode of Johnson–Lindenstrauss guarantees: preserving large pairwise distances does not imply preserving the small centered fluctuations that determine rankings, nearest neighbors, or covariance shape. Its Gaussian calculations provide quantitative signal-retention laws: a rank-m sketch retains at most an m/d fraction of the variance of any feature of one squared distance, while ranking correlation scales only as sqrt(m/d). These formulas can become practical dimension-budget rules and training diagnostics for projection bottlenecks, compressed embeddings, and low-dimensional retrieval representations. The most useful transfer is a centered-distance-aware projection objective and a dimension selector based on ranking quality rather than the JL event alone.