pstrf on positive semi-definite matrices

[fKunstner]

🐛 Bug

pstrf does not work on Positive Semi-Definite matrices while the documentation says it should.

The issue might be a misunderstanding on my part of the documentation.
Any correction to this misunderstanding is welcome :)

I am assuming that for M to be a Positive SemiDefinite matrix, it needs to hold that for any real vector x, x.t() @ M @ x >= 0,
whereas to be Positive Definite, it needs to hold that x.t() @ M @ x > 0.

To Reproduce

Using Pytorch 0.4.1, on CPU

import torch 
Zero = torch.tensor([0.0]).view(1,1)
print(torch.pstrf(Zero))

throws

RuntimeError: 
Lapack Error pstrf : 
matrix is rank deficient or not positive semidefinite at 
c:\programdata\miniconda3\conda-bld\pytorch_1532505617613\work\aten\src\th\generic/THTensorLapack.cpp:744

Expected behavior

pstrf is supposed to

Computes the pivoted Cholesky decomposition of a positive semidefinite matrix a. returns matrices u and piv.

In the example provided above, I would expected u should be [[0]] and piv should be [0]

Additional Details

I don't understand the error message,

matrix is rank deficient or not positive semidefinite

From my understanding, Positive SemiDefinite matrices that are not Positive Definite are by definition rank deficient.

cc @jianyuh @nikitaved @pearu @mruberry @heitorschueroff @walterddr @IvanYashchuk

[vishwakftw]

I think pstrf requires a matrix with rank at least 1. I don’t think this is mentioned in the docs, will send in a patch.

[fKunstner]

Interesting. However, it does not seem to work with a Rank-1, 2x2 matrix, unless I misunderstood your point.

import torch 
RankOne = torch.tensor([[1.0,0.0],[0.0,0.0]])
print(RankOne)
print(torch.pstrf(RankOne))

tensor([[1., 0.],
        [0., 0.]])
[...]
RuntimeError: 
Lapack Error pstrf : 
matrix is rank deficient or not positive semidefinite at
c:\programdata\miniconda3\conda-bld\pytorch_1532505617613\work\aten\src\th\generic/THTensorLapack.cpp:744

[vishwakftw]

Oh, my bad. I thought that was the case. I will look into it in more detail.

[vishwakftw]

I looked into it further, and there's nothing interfering with the LAPACK calls for pstrf. This probably means that there's some inaccuracy in the LAPACK implementation creeping in and playing spoilsport.

I also found out that pstrf doesn't have a CUDA implementation, which is an asymmetry.

I think we should just remove it. cc: @apaszke @soumith

[zou3519]

The matrix is rank deficient, which is correct, but the error message is indeed confusing. Maybe we should just make it say that the matrix is not positive definite (this is what scipy.linalg.cholesky says)

[vishwakftw]

pstrf is meant for positive semi-definite matrices, meaning that there is something wrong going on with the errors. For positive definite we have potrf, which works fine I suppose.

[vishwakftw]

Further more, do we need to have a function with only an implementation for the CPU. I think that would be inconvenient.

[ngimel]

Pytorch no longer has pstrf function. Labeling this for linalg in case we need to add support.

[lezcano]

Same as with the other issue, I think LDL provides a good enough solution for this. cc @IvanYashchuk