Given a target matrix $\mathbf{T}\in \mathbb{C}^{M\times M}$ and a basis (not necessarily orthonormal) $\mathbf{V}\in \mathbb{C}^{M\times K}$, how would I find the optimal reconstruction of $\mathbf{T}$
$$\min_{\mathbf{D}} \left\| \mathbf{T-VDV}^{H} \right\|_F \quad s.t. \mathbf{D}\: \mathrm{diagonal}$$
So far I recognize this is equivalent to finding the optimal $\sigma$ to minimize:
$$\min_{\mathbf{\sigma}} \left\| \mathbf{T}-\Big(\sum_{i=1}^K \sigma_i \mathbf{v}_i\mathbf{v}_i^H\Big) \right\|_F$$
where $\mathbf{v}_i$ is column $i$ of $\mathbf{V}$. I suspect the optimal $\sigma$ values are related to the magnitude of $\mathbf{T}$'s projection onto each $\mathbf{v}_i$, but have not managed to make anything work so far.