# Is low rank finite-iteration manifold identification possible?

In sparse optimization, I am trying to solve the problem $$\min_{x\in \mathbb R^{n}} \quad f(x) + \|x\|_1$$ and at optimality, $$x^*$$ may be sparse. If I define the sparse manifold as $$\mathcal M = \{z : z_i = 0 \text{ whenever } x_i^* = 0\}$$, there exist methods in which, for certain cases of $$f$$, can identify the sparse manifold; e.g. $$x^{(k)} \to x^*$$, there exist no finite $$k_0$$ for which $$x^{(k)} = x^*$$, but there does exist a finite $$k_0$$ for which, for all $$k > k_0$$, $$x^{(k)} \in \mathcal M$$.

Now is it possible to extend this to low-rank matrices? Specifically, now I am trying to solve some problem $$\min_{X\in \mathbb R^{m\times n}} \quad f(X) + \|X\|_*$$ where $$\|X\|_* =$$ the sum of the singular values of $$X$$, and at optimality, $$\text{rank}(X) = r< \min\{m,n\}$$. Then I define the low rank manifold as $$\mathcal M = \left\{\sum_{i=1}^r s_i u_iv_i^T: s_i \in \mathbb R\right\}$$ where $$X^* = \sum_{i=1}^r \sigma_i u_i v_i^T$$ is the economic" SVD of $$X^*$$.

Are there any existing methods in which $$X^{(k)} \to X^*$$, and for some finite $$k_0$$, for all $$k \geq k_0$$, $$X^{(k)}\in \mathcal M$$?