torch.cholesky (original) (raw)

torch.cholesky(input, upper=False, *, out=None) → Tensor#

Computes the Cholesky decomposition of a symmetric positive-definite matrix AA or for batches of symmetric positive-definite matrices.

If upper is True, the returned matrix U is upper-triangular, and the decomposition has the form:

A=UTUA = U^TU

If upper is False, the returned matrix L is lower-triangular, and the decomposition has the form:

A=LLTA = LL^T

If upper is True, and AA is a batch of symmetric positive-definite matrices, then the returned tensor will be composed of upper-triangular Cholesky factors of each of the individual matrices. Similarly, when upper is False, the returned tensor will be composed of lower-triangular Cholesky factors of each of the individual matrices.

Warning

torch.cholesky() is deprecated in favor of torch.linalg.cholesky()and will be removed in a future PyTorch release.

L = torch.cholesky(A) should be replaced with

L = torch.linalg.cholesky(A)

U = torch.cholesky(A, upper=True) should be replaced with

U = torch.linalg.cholesky(A).mH

This transform will produce equivalent results for all valid (symmetric positive definite) inputs.

Parameters

• input (Tensor) – the input tensor AA of size (∗,n,n)(*, n, n) where * is zero or more batch dimensions consisting of symmetric positive-definite matrices.
• upper (bool, optional) – flag that indicates whether to return a upper or lower triangular matrix. Default: False

Keyword Arguments

out (Tensor, optional) – the output matrix

Example:

a = torch.randn(3, 3) a = a @ a.mT + 1e-3 # make symmetric positive-definite l = torch.cholesky(a) a tensor([[ 2.4112, -0.7486, 1.4551], [-0.7486, 1.3544, 0.1294], [ 1.4551, 0.1294, 1.6724]]) l tensor([[ 1.5528, 0.0000, 0.0000], [-0.4821, 1.0592, 0.0000], [ 0.9371, 0.5487, 0.7023]]) l @ l.mT tensor([[ 2.4112, -0.7486, 1.4551], [-0.7486, 1.3544, 0.1294], [ 1.4551, 0.1294, 1.6724]]) a = torch.randn(3, 2, 2) # Example for batched input a = a @ a.mT + 1e-03 # make symmetric positive-definite l = torch.cholesky(a) z = l @ l.mT torch.dist(z, a) tensor(2.3842e-07)