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

arXiv:2609.02155 · cs.LG, cs.IT, cs.NA, math.IT, math.NA, math.PR, math.ST, stat.TH · Submitted 2026-09-02 · Read on arXiv

cs.LG, cs.IT, cs.NA, math.IT, math.NA, math.PR, math.ST, stat.TH

Submitted: 2026-09-02

Updated: 2026-09-02

Code: https://github.com/piyush314/random-projection-geometry

Terminology

• Fisher Information — 11× in this paper · explained in 3 episodes
• Monte Carlo — 11× in this paper
• Anisotropy — 8× in this paper · explained in 3 episodes
• Benchmark — 6× in this paper · explained in 2 episodes
• Calibration — 6× in this paper · explained in 7 episodes
• Trace — 6× in this paper · explained in 2 episodes
• Ambient Dimension — 4× in this paper · explained in 1 episode
• Clustering — 4× in this paper · explained in 2 episodes
• Alignment — 3× in this paper · explained in 1 episode
• Exchangeability — 3× in this paper · explained in 2 episodes
• Quantization — 3× in this paper · explained in 12 episodes
• Verification — 3× in this paper · explained in 2 episodes
• Embeddings — 2× in this paper · explained in 2 episodes
• Foundations of Computer Science — 2× in this paper
• Hilbert Space — 2× in this paper · explained in 2 episodes
• Accuracy — 1× in this paper · explained in 1 episode
• Annual ACM Symposium on Theory of Computing — 1× in this paper
• Combinatorial Optimization — 1× in this paper · explained in 2 episodes
• Computational Biology — 1× in this paper · explained in 2 episodes
• Conference on Uncertainty in Artificial Intelligence — 1× in this paper
• Covariance Eigenspaces — 1× in this paper · explained in 2 episodes
• Curse of Dimensionality — 1× in this paper · explained in 6 episodes
• Distance Preservation — 1× in this paper · explained in 2 episodes
• Downstream Tasks — 1× in this paper · explained in 2 episodes
• Ensemble Approach — 1× in this paper · explained in 3 episodes
• International Conference on Machine Learning — 1× in this paper
• Mathematical Multifaceted Integrated Capability Center — 1× in this paper
• Office of Advanced Scientific Computing Research — 1× in this paper
• Point Clouds — 1× in this paper · explained in 2 episodes
• Representation Learning — 1× in this paper · explained in 2 episodes

Sources

• Optimality of the Johnson-Lindenstrauss Lemma
• On prescribing total preorders and linear orders to pairwise distances of points in Euclidean space
• Total orders realizable as the distances between two sets of points

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