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Singular value decomposition (SVD) of a matrix $\mathbf{A}$ is given by $\mathbf{A} = \mathbf{USV}^\top$ where $\mathbf{U}$ and $\mathbf{V}$ are orthogonal matrices and $\mathbf{S}$ is a diagonal matrix.

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Is low rank finite-iteration manifold identification possible?

Yes. You can see results like this in the thesis of Jingwei Liang titled "Convergence Rates of First-Order Operator Splitting Methods" where they are called finite identification results (ch. 6, 7, 8) …
giorgi nguyen's user avatar