When is full SVD used, if at all?

Question

Whenever I see SVD, it's first introduced in a full form (i.e. for $X_{n \times m}$, where $n > m$, matrix $U$ has $n$ columns) and then truncated (so $U$ has $m$ columns). It's always also stated that we "almost never" use full SVD. So when would we use it, apart from theory (e.g. having orthgonal $U$ for proofs)? Or even in theory, where would we use full SVD instead of reduced?

Answer 1

The only setting in which the full SVD would be preferred is in the "theoretical" setting. In this setting, the full SVD is indeed strongly preferred. For example, if you look up proofs of the EYM theorem, pretty much all of them use the full SVD because it is much easier to work with an orthogonal $U$ that with a $U$ for which we only have $U^TU = I$ (for the $n>m$ case)