Invariance principle of random projection for the norm
This work provides theoretical insights into the invariance of norms under random projections, which is incremental for applications in dimensionality reduction and random matrix theory.
The paper tackles the problem of understanding how random projections affect the norms of random vectors with i.i.d. entries, proving that the distribution of the norm is preserved under such projections, specifically showing convergence to a normal distribution with given parameters as dimensions increase.
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 $X \in \mathbb{R}^n$, whose entries are i.i.d. random variables, is preserved by random projection $S:\mathbb{R}^n \to \mathbb{R}^m$. More precisely, \[ \frac{X^TS^TSX - mn}{\sqrt{σ^2 m^2n+2mn^2}} \xrightarrow[\quad m/n\to 0 \quad ]{ m,n\to \infty } \mathcal{N}(0,1) \] 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.