I'm trying to analyze convergence of $x\in \mathbb{R}^{+d}$ which follows the following recurrence

$$\mathbf{x}\leftarrow (\mathbf{1}-\mathbf{h})^2 \mathbf{x} + \mathbf{h}\langle \mathbf{x}, \mathbf{h}\rangle$$

Here $\mathbf{1}$ indicates a vector of $1$'s, vector multiplication, addition, squaring are applied pointwise and $\mathbf{h}\propto (1^{-p},2^{-p},\ldots ,d^{-p})$ for some $p\in (1,2)$.

I care about L1 norm after $t$ steps which is well approximated by the following:

$$f(t)=\|\mathbf{x}_t\|_1\approx e^{t(\operatorname{diag}[(\mathbf{1}-\mathbf{h})^2]+\mathbf{h}\mathbf{h}^T)}\mathbf{h}$$

For a given $p$, how do I get a nice upper bound on $f(t)$ which holds when $d\to \infty$?

Things are easy if we didn't have the the $\mathbf{h}\mathbf{h}^T$ term: approximating $\|\mathbf{x}_t\|_1$ with an integral I get formulas which match observed behavior very well:

[![enter image description here][1]][1]
<sub><sup>[Notebook](https://www.wolframcloud.com/obj/yaroslavvb/nn-linear/mathoverflow-rank1-recurrence-simple.nb)</sup></sub>


Keeping $\mathbf{h}\mathbf{h}^T$ term makes things much harder to handle. Straightforward approach [gives formula](https://math.stackexchange.com/a/4668173/998) in terms of roots of diagonal + rank-1 matrix, but not practical for large $d$. There's also a numeric [approach](https://mathoverflow.net/a/443030/7655) which works but also leaves unclear the dependence on $p$.


Motivation: this equation models expected value of iterated Gaussian linear system [like this](https://mathoverflow.net/questions/443143/when-is-prod-i-0-infty-i-x-i-x-it-0-for-isotropic-gaussian-x-i) for non-isotropic Gaussian case. Used to model training curve of neural network training.


  [1]: https://i.sstatic.net/6bjVX.png
  [2]: https://i.sstatic.net/3a3jd.png