Suppose $x_i$ come from 2-d standard Normal centered at 0. What is the range of $a$ for which the following iteration converges almost surely?

$$w_{i+1} = w_i-a x_i \langle w_i, x_i \rangle$$

For 1-d, one can write $w_{i+1}$ as $w_i=w_0\prod_i (1-\alpha x_i^2)$, take log and apply central limit theorem. For 2-D, one could use a similar approach, but using matrix logarithm instead of regular logarithm. However, this side-steps some technical details (we have a product of rank-1 matrices, how can you take matrix logarithm of that?) so I'm wondering if the resulting estimate is valid.

In particular, it predicts that iteration converges iff $\alpha \in (0.12131, 0.91861)$, while in simulations, I'm seeing convergence for $\alpha=0.93$ (but divergence for $\alpha=0.94$)

The question is equivalent to finding $\alpha$ such that $C_i$ converges almost surely where

$$C_i=w_i w_i^T$$ $$C_{i+1}=(1-\alpha x_i x_i^T)C_i(1-\alpha x_i x_i^T)$$

Suppose $x_i$ come from 2-d standard Normal centered at 0. What is the range of $a$ for which the following iteration converges almost surely?

$$w_{i+1} = w_i-a x_i \langle w_i, x_i \rangle$$

For 1-d, one can write $w_{i+1}^2$ as $w_i^2=w_0^2\prod_i (1-\alpha x_i^2)$, take log and apply central limit theorem. For 2-D, one could use a similar approach, but using matrix logarithm instead of regular logarithm. However, this side-steps some technical details (we have a product of rank-1 matrices, how can you take matrix logarithm of that?) so I'm wondering if the resulting estimate is valid.

In particular, it predicts that iteration converges iff $\alpha \in (0.12131, 0.91861)$, while in simulations, I'm seeing convergence for $\alpha=0.93$ (but divergence for $\alpha=0.94$)

The question is equivalent to finding $\alpha$ such that $C_i$ converges almost surely where

$$C_i=w_i w_i^T$$ $$C_{i+1}=(1-\alpha x_i x_i^T)C_i(1-\alpha x_i x_i^T)$$