SemidefiniteMean

SPDEuclideanMean

class hypercoil.init.semidefinite.SPDEuclideanMean(axis: int | Sequence[int] = (0,))

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

This is the familiar arithmetic mean:

\(\frac{1}{N}\sum_{i=1}^N X_{i}\)

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:

axisint

Axis or axes over which the mean is computed.

Methods

__call__(input)

Call self as a function.

SPDHarmonicMean

class hypercoil.init.semidefinite.SPDHarmonicMean(axis: int | Sequence[int] = (0,))

The harmonic mean is computed as the matrix inverse of the Euclidean mean of matrix inverses:

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

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:

axisint

Axis or axes over which the mean is computed.

Methods

__call__(input)

Call self as a function.

SPDLogEuclideanMean

class hypercoil.init.semidefinite.SPDLogEuclideanMean(axis: int | Sequence[int] = (0,), psi: float = 0.0)

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:

axisint

Axis or axes over which the mean is computed. {semidefinite_psi_spec}

Methods

__call__(input)

Call self as a function.

SPDGeometricMean

class hypercoil.init.semidefinite.SPDGeometricMean(axis: int | Sequence[int] = (0,), psi: float = 0.0, eps: float = 1e-05, max_iter: int = 10)

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

The geometric mean is computed via gradient descent along the geodesic on the manifold. In brief:

Initialisation :

• The estimate of the mean is initialised to the Euclidean mean.

Iteration :

• Using the working estimate of the mean as the point of tangency, the tensors are projected into a tangent space.

• The arithmetic mean of the tensors is computed in tangent space.

• This mean is projected back into the positive semidefinite cone using the same point of tangency. It now becomes a new working estimate of the mean and thus a new point of tangency.

Termination / convergence :

• The algorithm terminates either when the Frobenius norm of the difference between the new estimate and the previous estimate is less than a specified threshold, or when a maximum number of iterations has been attained.

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:

axisint

Axis or axes over which the mean is computed.

psifloat in [0, 1]

Conditioning factor to promote positive definiteness. If this is in (0, 1], the original input will be replaced with a convex combination of the input and an identity matrix.

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

A suitable value can be used to ensure that all eigenvalues are positive and therefore guarantee that the matrix is in the domain of projection operations.

max_iternonnegative int

The maximum number of iterations of gradient descent to run before termination.

Methods

__call__(input)

Call self as a function.