implement `linalg.cholesky` - Githubissues

helmholtz-analytics / heat

Distributed tensors and Machine Learning framework with GPU and MPI acceleration in Python

https://heat.readthedocs.io/

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implement `linalg.cholesky` #886

Open mtar opened 3 years ago

mtar commented 3 years ago

Feature functionality computes the cholesky decomposition Intended scope:

real/complex matrices
split $\in${ 0,1,None}
of course, appropriate scalability of the routine is desired

Additional context

https://numpy.org/doc/stable/reference/generated/numpy.linalg.cholesky.html https://pytorch.org/docs/stable/generated/torch.linalg.cholesky.html https://data-apis.org/array-api/latest/extensions/linear_algebra_functions.html?#linalg-cholesky-x-upper-false

After obtaining a triangular matrix by Cholesky, one often uses triangular solvers #1096

mrfh92 commented 10 months ago

My suggestion: implement "standard" Cholesky (see, e.g., https://en.wikipedia.org/wiki/Cholesky_decomposition) but block-wise using PyTorch for the blocks; in this sense, the formulas from wikipedia can be used but the $L{i,j}$'s and $A{i,j}$'s are blocks (i.e. matrices) instead of numbers. Of course, dividing by some $L{i,i}$ has now to be replaced by applying the matrix-inverse of $L{i,i}$ (by triangular solve in PyTorch) etc. ...