spd_learn.functional.tril_embed

spd_learn.functional.tril_embed(X: Tensor) → Tensor

Extract strictly lower-triangular entries scaled by \(\sqrt{2}\).

Provides an isometric embedding of hollow symmetric matrices \(\text{Hol}(\mathcal{S}^n)\) into Euclidean space \(\mathbb{R}^{n(n-1)/2}\) under the Frobenius metric:

\[\langle X, Y \rangle_F = \text{tr}(X Y) = \langle \text{tril\_embed}(X), \text{tril\_embed}(Y) \rangle\]

Parameters:

X (torch.Tensor) – Hollow symmetric matrices of shape (…, n, n).

Returns:

Vectorized representation of shape (…, n*(n-1)//2).

Return type:

torch.Tensor