Refactor orthogonalize_matrix_nearest function to include center para…

…meter and improve code readability

bnpm/similarity.py

@@ -237,11 +237,15 @@ def orthogonalize(v1, v2, method='OLS', device='cpu', thresh_EVR_PCA=1e-15):
 
 def orthogonalize_matrix_nearest(X: Union[np.ndarray, torch.Tensor], center=True):
     """
-    Orthogonalizes a matrix by finding the nearest orthogonal matrix. Nearest is
-    defined as solving the Procrustes problem via minimizing the Frobenius norm
-    of the difference between the original input matrix and the orthogonal
-    matrix. \n
-    X_orth = argmin ||X - X_orth||_F
+    Orthogonalizes a matrix by finding the nearest orthonormal matrix to an
+    input ``X``. 'Nearest' is defined as solving the orthogonal Procrustes
+    problem via minimizing the Frobenius norm of the difference between the
+    original input matrix and a orthonormal matrix that spans the same
+    dimensions as ``X``. For the initial orthonormal matrix, we use a centered
+    ``Q`` from a QR decomposition of ``X``. \n
+    ``X_orth = argmin: X_orth for ||X - X_orth||_F`` , where ``X_orth = Q @ R`` , where
+    ``R`` is a rotation matrix derived from solving the orthogonal Procrustes
+    problem for ``R = argmin: R for ||X - Q @ R||_F``. \n
 
     RH 2024
 
@@ -256,19 +260,21 @@ def orthogonalize_matrix_nearest(X: Union[np.ndarray, torch.Tensor], center=True
 
     Returns:
         (ndarray):
-            X_orth: orthogonalized matrix. shape: (n_samples, n_features)
+            X_orth: 
+                Orthogonalized matrix. shape: (n_samples, n_features)
     """
     if isinstance(X, np.ndarray):
-        op, qr = scipy.linalg.orthogonal_procrustes, np.linalg.qr
+        op, qr, svd = scipy.linalg.orthogonal_procrustes, np.linalg.qr, np.linalg.svd
     elif isinstance(X, torch.Tensor):
-        op, qr = torch_helpers.orthogonal_procrustes, torch.linalg.qr
+        op, qr, svd = torch_helpers.orthogonal_procrustes, torch.linalg.qr, torch.linalg.svd
 
     if center:
         X = X - X.mean(axis=0, keepdims=True)
 
     Q = qr(X)[0]
-    w, scale = op(Q, X)
-    return Q @ w
+    Q -= Q.mean(axis=0, keepdims=True)
+    R, scale = op(Q, X)
+    return Q @ R
 
 
 @njit