I am trying to solve the following rank-k decomposition $$\Sigma \approx W\Lambda W^T$$ where $\Sigma \in R^{n \times n}$ is a positive definite symmetric matrix having diagonal elements 1, $\Lambda \in R^{k \times k}$ is a diagonal matrix having positive element. Let the columns of $W$ matrix be $w^l$ then $\| w^l\|_1 \leq \alpha$ and $\| w^l\|_\infty \leq 1$. I am trying to show that for a given $\alpha$, all the solutions of the below equation are similar $$\text{arg min}_W \|\Sigma - W\Lambda W^T \|_F$$ Let $W_1$ and $W_2$ be two solutions then the average correlation between columns of $W_1$ and $W_2$ is very high i.e. the solutions are similar. The idea here is that if I constrain $W$ then the solution of the above minimization problem becomes constrain such that they lie in a similar landscape. But, I don't know how to prove this.
Edit 1: Let $\Sigma \approx W_1\Lambda_1 W_1^T \approx W_2\Lambda_2 W_2^T$ where the elements in $\Lambda_1$ and $\Lambda_2$ are sorted from high to low. Now the similarity is defined as $\frac{1}{k}\sum_{l=1}^{k}corr(w_1^l,w_2^l)$ where $corr$ is the correlation.