Implement torch.linalg.qr (#47764)
Summary:
I am opening this PR early to have a place to discuss design issues.
The biggest difference between `torch.qr` and `numpy.linalg.qr` is that the former `torch.qr` takes a boolean parameter `some=True`, while the latter takes a string parameter `mode='reduced'` which can be one of the following:
`reduced`
this is completely equivalent to `some=True`, and both are the default.
`complete`
this is completely equivalent to `some=False`.
`r`
this returns only `r` instead of a tuple `(r, q)`. We have already decided that we don't want different return types depending on the parameters, so I propose to return `(r, empty_tensor)` instead. I **think** that in this mode it will be impossible to implement the backward pass, so we should raise an appropriate error in that case.
`raw`
in this mode, it returns `(h, tau)` instead of `(q, r)`. Internally, `h` and `tau` are obtained by calling lapack's `dgeqrf` and are later used to compute the actual values of `(q, r)`. The numpy docs suggest that these might be useful to call other lapack functions, but at the moment none of them is exposed by numpy and I don't know how often it is used in the real world.
I suppose the implementing the backward pass need attention to: the most straightforward solution is to use `(h, tau)` to compute `(q, r)` and then use the normal logic for `qr_backward`, but there might be faster alternatives.
`full`, `f`
alias for `reduced`, deprecated since numpy 1.8.0
`economic`, `e`
similar to `raw but it returns only `h` instead of `(h, tau). Deprecated since numpy 1.8.0
To summarize:
* `reduce`, `complete` and `r` are straightforward to implement.
* `raw` needs a bit of extra care, but I don't know how much high priority it is: since it is used rarely, we might want to not support it right now and maybe implement it in the future?
* I think we should just leave `full` and `economic` out, and possibly add a note to the docs explaining what you need to use instead
/cc mruberry
Pull Request resolved: https://github.com/pytorch/pytorch/pull/47764
Reviewed By: ngimel
Differential Revision: D25708870
Pulled By: mruberry
fbshipit-source-id: c25c70a23a02ec4322430d636542041e766ebe1b
