A Different Approach to Low-Rank Matrix Completion

In space of all matrices, the set of rank one matrices is called a Segre variety[2]. Segre variety is cut out by 2x2 minors of a generic matrix, which is what the author's post is all about. What he's effectively trying to do is given a plane (which is the set of possible matrices having some given values), finding a point in the intersection of the Segre variety and that plane. The way he does it is by intersecting that plane one by one with large varieties cut out by a single minor, and noting that all points in the intersection must lie in a hyperplane not already containing the plane of possible matrices. This reduces the dimension of the plane of possible matrices, as they must lie in both the original plane and the constraint hyperplane. After enough steps we're left with a single point. The clever thing here is finding a good order of intersections, so that at each step we can find that hyperplane.

Now, rank k matrices form a k-th secant variety to the Segre variety, and they are cut out by k x k minors, and so for their part 2, I expect using larger minors.

[1] - https://en.wikipedia.org/wiki/Secant_variety [2] - https://en.wikipedia.org/wiki/Segre_embedding

https://en.wikipedia.org/wiki/Compressed_sensing

Nice to see that similar mathematical tools are being developed for similar problems.

Edit: typo

- if the results of the two methods are the same?

- how about the numerical stability? (many times tricky matrix algorithms have poor stability)