Compute the mean of normalized norms of linear transformations of Gaussian random vectors

by DeltaIV   Last Updated October 07, 2018 08:19 AM

if $M$ is a $m\times n$ constant matrix and $\eta\sim\mathcal{N}(0,I)$, then does $$\mathbf{E}_{\eta\sim\mathcal{N}}\left[\frac{\lVert M\eta\rVert}{\lVert\eta\rVert}\right]$$ exist? Also, let $x\in \mathbb{R}^n_{\ne 0}$ be an arbitrary non-zero vector. Is it possible to compute the maximum (or at least to find a tight upper-bound) over all $x$, of the quantity $$\mathbf{E}_{\eta\sim\mathcal{N}}\left[\frac{\lVert M(x+\lVert x\rVert \eta)-Mx\rVert}{\lVert Mx \rVert}\right]=\lVert x\rVert\mathbf{E}_{\eta\sim\mathcal{N}}\left[\frac{\lVert M \eta\rVert}{\lVert Mx \rVert}\right]$$

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