spd_learn.models.GeodesicCCALoss#

class spd_learn.models.GeodesicCCALoss(outdim_size: int = 10, eps: float = 0.0001)[source]#

Bases: Module

Geodesic Canonical Correlation Analysis (GeodesicCCA) Loss.

Computes the negative geodesic canonical correlation loss between two batches of Symmetric Positive Definite (SPD) matrices [Ju et al., 2024].

Each batch of SPD matrices \(Y_1, Y_2 \in \mathbb{P}_d\) is mapped to the tangent space at their empirical Fréchet mean via the matrix logarithm:

\[L_1 = \log(Y_1), \quad L_2 = \log(Y_2)\]

The tangent matrices are centered across the batch and vectorized into Euclidean representations \(H_1, H_2 \in \mathbb{R}^{B \times m}\) where \(m = d(d+1)/2\). Regularized covariance matrices are then computed:

\[\Sigma_{11} = \frac{1}{N} H_1^\top H_1 + \epsilon I_m, \quad \Sigma_{22} = \frac{1}{N} H_2^\top H_2 + \epsilon I_m, \quad \Sigma_{12} = \frac{1}{N} H_1^\top H_2\]

where \(N = \max(B - 1, 1)\). The whitened cross-correlation matrix \(T = \Sigma_{11}^{-1/2} \Sigma_{12} \Sigma_{22}^{-1/2}\) is formed, and its top-\(k\) singular values \(\rho_i\) (\(k = \min(\text{outdim\_size}, m)\)) are extracted via the eigendecomposition of \(T^\top T + \epsilon I_m\). The loss returned is the negative sum of these canonical correlations:

\[\mathcal{L}_{\text{GeoCCA}} = -\sum_{i=1}^k \rho_i\]
Parameters:

  • outdim_size (int, default=10) – Number of top canonical correlation components to extract and maximize.

  • eps (float, default=1e-4) – Regularization parameter added to covariance matrices and eigenvalue clamping for numerical stability.

forward(Y1: Tensor, Y2: Tensor) → Tensor[source]#

Compute the negative geodesic canonical correlation loss.

Parameters:

  • Y1 (torch.Tensor) – Batch of SPD matrices from view 1 of shape (batch_size, d, d).

  • Y2 (torch.Tensor) – Batch of SPD matrices from view 2 of shape (batch_size, d, d).

Returns:

Scalar negative total geodesic correlation.

Return type:

torch.Tensor