# Invariance principle of random projection for the norm

arXiv: https://arxiv.org/abs/2112.00300 · HTML: https://mlsanity.com/paper/2112.00300/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 22 references that are in the map.
- First posted on arXiv 1 Dec 2021.

## Builds on (most cited first, 3 of 3)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Sketching as a Tool for Numerical Linear Algebra](https://mlsanity.com/paper/1411.4357/) (2014)
- [Random-projection ensemble classification](https://mlsanity.com/paper/1504.04595/) (2015)

## Abstract (arXiv)

Johnson-Lindenstrauss guarantees certain topological structure is preserved under random projections when project high dimensional deterministic vectors to low dimensional vectors. In this work, we try to understand how random matrix affect norms of random vectors. In particular we prove the distribution of the norm of random vector X ∈ ℝn, whose entries are i.i.d. random variables, is preserved by random projection S:ℝn → ℝm. More precisely, [ fracXTSTSX - mn√σ2 m2n+2mn2 xrightarrow[quad m/n→ 0 quad ] m,n→ ∞  N(0,1) ] We also prove a concentration of the random norm transformed by either random projection or random embedding. Overall, our results showed random matrix has low distortion for the norm of random vectors with i.i.d. entries.