mean_logeuc_spd

hypercoil.functional.semidefinite.mean_logeuc_spd(input: Tensor, axis: int | Sequence[int] = 0, psi: float = 0, key: Tensor | None = None, recondition: Literal['eigenspaces', 'convexcombination'] = 'eigenspaces', fill_nans: bool = True, truncate_eigenvalues: bool = False) → Tensor

Batch-wise log-Euclidean mean of tensors in the positive semidefinite cone.

The log-Euclidean mean is computed as the matrix exponential of the mean of matrix logarithms.

\(\bar{X} = \exp \left(\frac{1}{N}\sum_{i=1}^N \log X_{i}\right)\)

Dimension:

Input : \((N, *, D, D)\)

N denotes batch size, * denotes any number of intervening dimensions, D denotes matrix row and column dimension.

Output : \((*, D, D)\)

As above.

Parameters:

inputTensor

Batch of matrices over which the Euclidean mean is to be computed.

axisint

Axis or axes over which the mean is computed.

psifloat in [0, 1]

Conditioning factor to promote positive definiteness.

key: Tensor or None (default None)

Key for pseudo-random number generation. Required if recondition is set to 'eigenspaces' and psi is in (0, 1].

recondition'convexcombination' or 'eigenspaces' (default 'eigenspaces')

Method for reconditioning.

• 'convexcombination' denotes that the original input will be replaced with a convex combination of the input and an identity matrix.

  \(\widetilde{X} = (1 - \psi) X + \psi I\)

  A suitable \(\psi\) can be used to ensure that all eigenvalues are positive.

• 'eigenspaces' denotes that noise will be added to the original input along the diagonal.

  \(\widetilde{X} = X + \psi I - \xi I\)

  where each element of \(\xi\) is independently sampled uniformly from \((0, \psi)\). In addition to promoting positive definiteness, this method promotes eigenspaces with dimension 1 (no degenerate/repeated eigenvalues). Nondegeneracy of eigenvalues is required for differentiation through SVD.

fill_nansbool (default True)

Indicates that any NaNs among the transformed eigenvalues should be replaced with zeros.

truncate_eigenvaluesbool (default False)

Indicates that very small eigenvalues, which might for instance occur due to numerical errors in the decomposition, should be truncated to zero. Note that you should not do this if you wish to differentiate through this operation, or if you require the input to be positive definite. For these use cases, consider using the psi and recondition parameters.

require_nonsingularbool (default True)

Indicates that the input matrix must be nonsingular. If this is False, then the Moore-Penrose pseudoinverse is computed instead of the inverse.

Returns:

outputTensor

Log-Euclidean mean of the input batch.