Let $X \in \mathbb{R}^{m \times n}$ with rank at most $r$. Let $\mathcal{M}$ be the set of matrices 2-tuples $(A,B) \in \mathbb{R}^{m \times r} \times \mathbb{R}^{n \times r}$ such that $X = AB^T$. Do you have any idea whether $\mathbb{M}$ can be viewed as a (Riemannian) manifold?
N.B.: I don't have a lot of knowlegde about manifolds. I only know it possible to make some optimization on manifolds and in this case I would like to optimize $(A,B)$ so that they are as sparse as possible.
Thank you.