symsqrt

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

Matrix square root of a batch of symmetric, positive definite matrices.

Computed by diagonalising the matrix \(X = Q_X \Lambda_X Q_X^\intercal\), computing the square root of the eigenvalues, and recomposing.

\(\sqrt{X} = Q_X \sqrt{\Lambda_X} Q_X^\intercal\)

Note that this will be infeasible for matrices with negative eigenvalues, and potentially singular matrices due to numerical rounding errors. To guard against the infeasible case, consider specifying a recondition parameter.

Note

This approach is in principle faster than the matrix square root in JAX, but it is not as robust or general as the JAX implementation (jax.linalg.sqrtm).

Dimension:

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

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

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

As above.

Parameters:

inputTensor

Batch of symmetric tensors to transform using the matrix square root.

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.

Returns:

outputTensor

Square root of each matrix in the input batch.