equiv_orthogonal_projection_coefficients#

equiv_orthogonal_projection_coefficients(W, rep_x, rep_y, tensor_cache=None)[source]#

Return the flattened homomorphism-basis coefficients of \(\Pi_{\mathrm{Hom}_{\mathbb{G}}}(\mathbf{W})\).

Let \(\mathbf{W}\in\mathbb{R}^{D_y\times D_x}\) be any dense linear map between \((\mathcal{X}, \rho_{\mathcal{X}})\) and \((\mathcal{Y}, \rho_{\mathcal{Y}})\). This function computes the orthogonal projection

\[\Pi_{\mathrm{Hom}_{\mathbb{G}}}(\mathbf{W}) \in \mathrm{Hom}_{\mathbb{G}}(\rho_{\mathcal{X}}, \rho_{\mathcal{Y}})\]

and returns its coefficients in the blockwise isotypic basis described in Leveraging the structure of Equivariant Linear maps.

In isotypic coordinates, the projected operator decomposes as

\[\Pi_{\mathrm{Hom}_{\mathbb{G}}}(\mathbf{W}) = \mathbf{Q}_{\mathcal{Y}} \left( \bigoplus_{k\in[1,n_{\text{iso}}]} \mathbf{W}^{(k)} \right) \mathbf{Q}_{\mathcal{X}}^T.\]

For each common irrep type \(k\), the corresponding block is written as

\[\mathbf{W}^{(k)} = \sum_{s=1}^{S_k}\mathbf{\Theta}^{(k)}_s \otimes \mathbf{\Psi}^{(k)}_s,\]

where \(\mathbf{\Theta}^{(k)}_s \in \mathbb{R}^{m_k^{\mathcal{Y}} \times m_k^{\mathcal{X}}}\), and \(\{\mathbf{\Psi}^{(k)}_s\}_{s=1}^{S_k}\) is a basis of \(\mathrm{End}_{\mathbb{G}}(\hat{\rho}_k)\). The output concatenates all coefficient blocks after flattening each tensor of shape \((m_k^{\mathcal{Y}} m_k^{\mathcal{X}}, S_k)\), with the endomorphism-basis index varying fastest.

Parameters:

  • W (Tensor) – Dense map (or batch of maps) of shape \((..., D_y, D_x)\).

  • rep_x (Representation) – Input representation \(\rho_{\mathcal{X}}\).

  • rep_y (Representation) – Output representation \(\rho_{\mathcal{Y}}\).

  • tensor_cache (IsotypicTensorCache, optional) – Optional override containing the tensor cache required by project_in_isobasis(). When provided, all required tensors must be present.

Returns:

Flattened coefficient vector(s) of shape \((..., \dim(\mathrm{Hom}_{\mathbb{G}}(\rho_{\mathcal{X}}, \rho_{\mathcal{Y}})))\).

Return type:

Tensor

Shape:

  • W: \((..., D_y, D_x)\).

  • Output: \((..., |\mathrm{Hom}_{\mathbb{G}}(\rho_{\mathcal{X}}, \rho_{\mathcal{Y}})|)\).