I have a matrix $M\in\mathbb{R}^{n\times k}$, with $k<n$ whose columns $m_i$ are linearly independent.
So we have $M := [m_1|..|m_k]$.
From the columns of $M$ I can define the following matrix $$ C = \sum_{i=1}^{k}m_i \otimes m_i $$ that has rank $k$ and the following family of tensors $$ T_h= \sum_{i=1}^{k}m_i\otimes...\otimes m_i = \sum_{i=1}^{k}m_i^{\otimes h} $$
Let now, for a $l<k$ $$ \tilde{M} = [\tilde{m}_1|..|\tilde{m}_l] \in\mathbb{R}^{n\times l} $$ be the rank $l$ matrix that minimizes the following distance according to the Frobenius norm: $$ D(\tilde{M}) = ||C - \sum_{i=1}^{l}\tilde{m}_i \otimes \tilde{m}_i||_F $$ I have the impression that if $\tilde{M}$ minimizes $D(\tilde{M})$, then it also minimizes the following distance, for any $h>2$ $$ D_h(\tilde{M}) = ||T_h - \sum_{i=1}^{l}\tilde{m}_i^{\otimes h} ||_F $$ Is that true? There exist any result in this direction?
