spd_learn.functional.exp_map_airm#

spd_learn.functional.exp_map_airm(P, V, t=1.0)[source]#

Riemannian exponential map under the Affine-Invariant metric.

Maps a tangent vector \(V\) at base point \(P\) to a point on the SPD manifold by shooting along the geodesic in direction \(V\).

Under the Affine-Invariant Riemannian Metric (AIRM), the exponential map is:

\[\text{Exp}_P(tV) = P^{1/2} \exp(t P^{-1/2} V P^{-1/2}) P^{1/2}\]

where \(\exp\) denotes the matrix exponential and \(V\) is a symmetric matrix representing a tangent vector at \(P\).

Parameters:

P (torch.Tensor) – Base point on the SPD manifold with shape (…, n, n).
V (torch.Tensor) – Tangent vector at P (symmetric matrix) with shape (…, n, n).
t (float, optional) – Scaling factor for the tangent vector. Default is 1.0.

Returns:

Point on the SPD manifold with shape (…, n, n).

Return type:

torch.Tensor

Notes

The AIRM exponential map is a diffeomorphism from the tangent space at \(P\) to the entire SPD manifold, because the SPD manifold with AIRM forms a Hadamard manifold (complete, simply connected, nonpositive sectional curvature).

Examples

>>> import torch
>>> from spd_learn.functional.metrics import exp_map_airm, log_map_airm
>>> P = torch.eye(3)
>>> V = 0.1 * torch.randn(3, 3)
>>> V = (V + V.T) / 2  # Symmetrize
>>> Q = exp_map_airm(P, V)  # Shoot from P in direction V
>>> V_back = log_map_airm(P, Q)  # Should recover V
>>> torch.allclose(V, V_back, atol=1e-5)
True