torch.cholesky

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

Computes the Cholesky decomposition of a symmetric positive-definite matrix $A$ 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:

$$A = U^TU$$

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

$$A = LL^T$$

If upper is True, and $A$ 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.

Note

torch.linalg.cholesky() should be used over torch.cholesky when possible. Note however that torch.linalg.cholesky() does not yet support the upper parameter and instead always returns the lower triangular matrix.

Parameters

• input (Tensor) – the input tensor $A$ of size $(*, n, n)$ 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 = torch.mm(a, a.t()) # 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]])
>>> torch.mm(l, l.t())
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)
>>> a = torch.matmul(a, a.transpose(-1, -2)) + 1e-03 # make symmetric positive-definite
>>> l = torch.cholesky(a)
>>> z = torch.matmul(l, l.transpose(-1, -2))
>>> torch.max(torch.abs(z - a)) # Max non-zero
tensor(2.3842e-07)