left_svd
========

.. py:function:: t3toolbox.backend.linalg.left_svd(G0_i_a_j, min_rank = None, max_rank = None, rtol = None, atol = None)

   .. code-block:: python

      def left_svd(
              G0_i_a_j: NDArray, # shape=(..., ni, na, nj)
              min_rank: int = None, # 1 <= min_rank <= max_rank <= minimum(ni*na, nj)
              max_rank: int = None, # 1 <= min_rank <= max_rank <= minimum(ni*na, nj)
              rtol: float = None, # removes singular values satisfying sigma < maximum(atol, rtol*sigma1)
              atol: float = None, # removes singular values satisfying sigma < maximum(atol, rtol*sigma1)
      ) -> typ.Tuple[
          NDArray, # U_i_a_x, shape=(..., ni, na, r)
          NDArray, # ss_x,    shape=(.., r)
          NDArray, # Vt_x_j,  shape=(..., r, nj)
      ]:


   Compute (truncated) singular value decomposition of 3-tensor left unfolding.

   First two indices of the tensor are grouped for the SVD: ``G[i,a,j] = sum_x U[i,a,x] ss[x] Vt[x,j]``,
   with ``U`` orthonormal in its grouped ``(i,a)`` rows. Truncation args behave as in
   :py:func:`truncated_svd`.

   .. rubric:: Examples

   >>> import numpy as np
   >>> import t3toolbox.backend.linalg as linalg
   >>> np.random.seed(0)
   >>> G = np.random.randn(4, 5, 6)                      # (ni, na, nj)
   >>> U, ss, Vt = linalg.left_svd(G)
   >>> print(U.shape, ss.shape, Vt.shape)               # U keeps (i,a); Vt is the 2d right factor
   (4, 5, 6) (6,) (6, 6)
   >>> print(np.allclose(np.einsum('iax,x,xj->iaj', U, ss, Vt), G))   # reconstructs G
   True
   >>> Um = U.reshape(4 * 5, -1)
   >>> print(np.allclose(Um.T @ Um, np.eye(Um.shape[1])))   # left unfolding of U is orthonormal
   True