Suppose $x\sim \text{Normal}(0,\Sigma)$ with diagonal $\Sigma$.

Is there a closed form solution or a good approximation to the following quantity where $y=x/\|x\|$:


Target is to understand the growth rate of $g(k)=\operatorname{Tr}(\underbrace{f\circ f \circ \cdots \circ f}_{k} \circ I)$ in terms of $k$.

For unnormalized $x$, I can use Wick's theorem, and this iteration nicely factors, giving nice formulas for $g(k)$.

Wondering if there's a trick I can use to make it work for $y$.

Motivation: factoring density of $x$ is used in Bordelon paper to give loss curve of SGD. Extending this to $y$ will transfer this analysis to the Kaczmarz method.

  • $\begingroup$ What do you know about $C$? $\endgroup$ Mar 11 at 14:12
  • $\begingroup$ C0 is identity matrix and C1=f(C0). (Diagonal perhaps?). So these are two main cases I need to solve $\endgroup$ Mar 11 at 15:12


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