def _iterate_to_update_x_transforms(self):
        """Perform the inner iteration."""
        self._inner_iters = 0
        self._last_inner_error = float('inf')
        while self._inner_error_is_decreasing():
            print('  Starting inner iteration {0:03d}. Current err = {1:12.5E}'
                  ''.format(self._inner_iters, self._last_inner_error))
            self._update_x_transforms()
            self._inner_iters += 1
Perform the inner iteration.