A measure concentration effect for matrices of high, higher, and even higher dimension

Abstract

Let $n>m$, and let $A$ be an $(m\times n)$-matrix of full rank. Then obviously the estimate $\Vert Ax\Vert \le \Vert A\Vert \Vert x\Vert$ holds for the euclidean norm of $x$ and $Ax$ and the spectral norm as the assigned matrix norm. We study the sets of all $x$ for which, for fixed $\delta <1$, conversely $\Vert Ax\Vert \ge \delta \Vert A\Vert \Vert x\Vert$ holds. It turns out that these sets fill, in the high-dimensional case, almost the complete space once $\delta$ falls below a bound that depends on the extremal singular values of $A$ and on the ratio of the dimensions. This effect has much to do with the random projection theorem, which plays an important role in the data sciences. As a byproduct, we calculate the probabilities this theorem deals with exactly.

to appear in SIAM Journal on Matrix Analysis and Applications

Cited by in corpus (1)

• An iterative method for the solution of Laplace-like equations in high and very high space dimensions