Algorithm to minimize the 2-norm between two matrices

Question

What algorithm can you use to minimize the distance between two matrices? For example:

$$\min_{X \in \Gamma} \| A - X\|_2$$

$\Gamma$ all m $\times$ m rank $k$ matrices.

How can you think of this geometrically? If for least squares we minimize $Ax-b$ and look for $x$, but what vector do we look for in the difference between two matrices? I wish I could contribute more but I'm not really sure how to intrepret this.

You can rewrite this assuming a Singular Value Decomposition (SVD) on $X = U\Sigma V^*$ and another one on $A$. We can without lack of generality assume $\Sigma$ sorted along diagonal if we wish. How does rank translate into constraints on $\Sigma$? — mathreadler, Oct 30 '19 at 13:29
SVD is how I was approaching it. Also I don't know the rank question. — Math Stout, Oct 30 '19 at 17:57

score 1 · Answer 1 · answered Oct 30 '19 at 13:49

1

This corresponds to optimisation the cost function on the Grassmanian manifold. One can use the theory of optimisation on manifolds. In particular, reimannian gradient descent on Grassmanian manifolds provides an iterative algorithm.

answered Oct 30 '19 at 13:49

Siddharth Bhat

8,452

Algorithm to minimize the 2-norm between two matrices

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