A concentration theorem for projections

2012/06/27 by Sanjoy Dasgupta, Dasgupta, Sanjoy, Daniel Hsu +3 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.1206.6813

openalex publication_date 2012/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

X in R^D has mean zero and finite second moments. We show that there is a precise sense in which almost all linear projections of X into R^d (for d < D) look like a scale-mixture of spherical Gaussians -- specifically, a mixture of distributions N(0, sigma² I_d) where the weight of the particular sigma component is P (| X |² = sigma² D). The extent of this effect depends upon the ratio of d to D, and upon a particular coefficient of eccentricity of X's distribution. We explore this result in a variety of experiments.

Citations

• A Two-round Variant of EM for Gaussian Mixtures

Cited by

• Balancing Utility and Privacy: Dynamically Private SGD with Random Projection
• Compressibility Barriers to Neighborhood-Preserving Data Visualizations

Related

• Concentration of Measure for Radial Distributions and Consequences for Statistical Modeling
• Concentration of measures via size biased couplings
• Second order concentration on the sphere
• Random projections beyond zero overlap
• Concentration of Measure Techniques and Applications