I would like to ask the following question:

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 $$\mathbf{E}_{\eta\sim\mathcal{N}}\left[\frac{\lVert M(x+\lVert x\rVert \eta)-Mx\rVert}{\lVert Mx \rVert}\right]=\lVert x\rVert\mathbf{E}_{\eta\sim\mathcal{N}}\left[\frac{\lVert M \eta\rVert}{\lVert Mx \rVert}\right]$$

Should I ask here or on Mathematics?

  • 2
    It looks on topic; but given this is not the actual question, it's not clear the extent to which you're better off asking here or there. – Glen_b Oct 5 at 12:51
  • @Glen_b ok: I'll write the full question, so that it's more clear. Give me an hour, top. – DeltaIV Oct 5 at 12:57
  • @Glen_b done, let me know what you think about it now. – DeltaIV Oct 5 at 13:32
  • 8
    It's on topic on either site. I would guess it's a better fit on Mathematics, but as @Tim notes, you are likely to get different kinds / styles of answers between the two sites, so you should probably choose based on your preference for the style of answer that would most benefit you. – gung Oct 5 at 13:52
  • @gung question asked here. – DeltaIV Oct 7 at 8:10
up vote 10 down vote accepted

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