Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$:

$$\frac{v_1}{\|v_1\|},\frac{v_2}{\|v_2\|}, \ldots ,\frac{v_k}{\|v_k\|}$$

We apply Gram-Schmidt orthogonalization to this list to get a new set of vectors

$$u_1,u_2,u_3,\ldots,u_k$$

Can we say how the expected value of $\|u_i\|^2$ decays with $i$? For isotropic Gaussian, it decays [linearly](https://stats.stackexchange.com/a/583564/511), while simulation [suggests](https://www.wolframcloud.com/obj/yaroslavvb/newton/forum-decay-of-orthogonal-contributions.nb) there is a simple correspondence for the case at hand, can we say what it is?


[![enter image description here][1]][1]


**Motivation** some optimization tasks in ML [exhibit](https://arxiv.org/abs/1907.04164) $1/k$ decay of eigenvalues, knowing this rate would let us bound the regret from using batch size $<d$ in each optimization step

  [1]: https://i.sstatic.net/xrVRu.png