spd_learn.functional.orthogonal_polar_factor#

spd_learn.functional.orthogonal_polar_factor(W: Tensor) → Tensor[source]#

Compute the orthogonal polar factor of a matrix.

Projects a matrix onto the Stiefel manifold \(\text{St}(n, k)\) via polar decomposition. The orthogonal polar factor is:

\[W_{\perp} = W (W^\top W)^{-1/2}\]

This is the unique matrix with orthonormal columns that is closest to \(W\) in Frobenius norm.

Parameters:: W (torch.Tensor) – Input matrix with shape (…, n, k) where n >= k.
Returns:: Orthogonal polar factor with shape (…, n, k) satisfying \(W_{\perp}^\top W_{\perp} = I_k\).
Return type:: torch.Tensor

Notes

This function uses the matrix inverse square root computed via eigendecomposition for numerical stability.

Examples

>>> import torch
>>> from spd_learn.functional import orthogonal_polar_factor
>>> W = torch.randn(8, 4)
>>> W_orth = orthogonal_polar_factor(W)
>>> # Verify orthonormality
>>> torch.allclose(W_orth.T @ W_orth, torch.eye(4), atol=1e-5)
True