irrep_radii

irrep_radii(x, rep)

Compute Euclidean radii for all irreducible-subspace features.

Let \(\rho_{\mathcal{X}}\) be the (possibly decomposable) representation of a vector space \(\mathcal{X}\):

\[\rho_{\mathcal{X}} = \mathbf{Q}\left( \bigoplus_{k\in[1,n_{\text{iso}}]} \bigoplus_{i\in[1,n_k]} \hat{\rho}_k \right)\mathbf{Q}^T.\]

We first change to the irrep-spectral basis induced by this isotypic decomposition (as returned by isotypic_decomp_rep()), \(\hat{\mathbf{x}}=\mathbf{Q}^T\mathbf{x}\), and then compute the radius of each irrep copy:

\[r_{k,i} = \lVert \hat{\mathbf{x}}_{k,i} \rVert_2.\]

Parameters:

• x (Tensor) – (Tensor) of shape \((..., D)\) describing vectors transforming according to rep.

• rep (Representation) – (Representation) acting on the last dimension of x.

Returns:

Radii of shape \((..., N)\) where \(N=\texttt{len(rep.irreps)}\). The output order follows rep.irreps (one radius per irreducible copy in the decomposition).

Return type:

(Tensor)

Shape:

• Input x: \((..., D)\) with \(D=\dim(\rho_{\mathcal{X}})\).

• Output: \((..., N)\) containing the per-irrep Euclidean norms.

Note

For repeated calls with the same representation object rep, the matrix \(\mathbf{Q}^{-1}\) is cached in rep.attributes["Q_inv"] and reused.