_lobpcg.py

"""Locally Optimal Block Preconditioned Conjugate Gradient methods.
"""
# Author: Pearu Peterson
# Created: February 2020

import torch
from . import _linalg_utils as _utils
from ._overrides import has_torch_function, handle_torch_function


__all__ = ['lobpcg']


def lobpcg(A,                   # type: Tensor
           k=None,              # type: Optional[int]
           B=None,              # type: Optional[Tensor]
           X=None,              # type: Optional[Tensor]
           n=None,              # type: Optional[int]
           iK=None,             # type: Optional[Tensor]
           niter=None,          # type: Optional[int]
           tol=None,            # type: Optional[float]
           largest=None,        # type: Optional[bool]
           method=None,         # type: Optional[str]
           tracker=None,        # type: Optional[None]
           ortho_iparams=None,  # type: Optional[Dict[str, int]]
           ortho_fparams=None,  # type: Optional[Dict[str, float]]
           ortho_bparams=None,  # type: Optional[Dict[str, bool]]
           ):
    # type: (...) -> Tuple[Tensor, Tensor]

    """Find the k largest (or smallest) eigenvalues and the corresponding
    eigenvectors of a symmetric positive defined generalized
    eigenvalue problem using matrix-free LOBPCG methods.

    This function is a front-end to the following LOBPCG algorithms
    selectable via `method` argument:

      `method="basic"` - the LOBPCG method introduced by Andrew
      Knyazev, see [Knyazev2001]. A less robust method, may fail when
      Cholesky is applied to singular input.

      `method="ortho"` - the LOBPCG method with orthogonal basis
      selection [StathopoulosEtal2002]. A robust method.

    Supported inputs are dense, sparse, and batches of dense matrices.

    .. note:: In general, the basic method spends least time per
      iteration. However, the robust methods converge much faster and
      are more stable. So, the usage of the basic method is generally
      not recommended but there exist cases where the usage of the
      basic method may be preferred.

    Arguments:

      A (Tensor): the input tensor of size :math:`(*, m, m)`

      B (Tensor, optional): the input tensor of size :math:`(*, m,
                  m)`. When not specified, `B` is interpereted as
                  identity matrix.

      X (tensor, optional): the input tensor of size :math:`(*, m, n)`
                  where `k <= n <= m`. When specified, it is used as
                  initial approximation of eigenvectors. X must be a
                  dense tensor.

      iK (tensor, optional): the input tensor of size :math:`(*, m,
                  m)`. When specified, it will be used as preconditioner.

      k (integer, optional): the number of requested
                  eigenpairs. Default is the number of :math:`X`
                  columns (when specified) or `1`.

      n (integer, optional): if :math:`X` is not specified then `n`
                  specifies the size of the generated random
                  approximation of eigenvectors. Default value for `n`
                  is `k`. If :math:`X` is specifed, the value of `n`
                  (when specified) must be the number of :math:`X`
                  columns.

      tol (float, optional): residual tolerance for stopping
                 criterion. Default is `feps ** 0.5` where `feps` is
                 smallest non-zero floating-point number of the given
                 input tensor `A` data type.

      largest (bool, optional): when True, solve the eigenproblem for
                 the largest eigenvalues. Otherwise, solve the
                 eigenproblem for smallest eigenvalues. Default is
                 `True`.

      method (str, optional): select LOBPCG method. See the
                 description of the function above. Default is
                 "ortho".

      niter (int, optional): maximum number of iterations. When
                 reached, the iteration process is hard-stopped and
                 the current approximation of eigenpairs is returned.
                 For infinite iteration but until convergence criteria
                 is met, use `-1`.

      tracker (callable, optional) : a function for tracing the
                 iteration process. When specified, it is called at
                 each iteration step with LOBPCG instance as an
                 argument. The LOBPCG instance holds the full state of
                 the iteration process in the following attributes:

                   `iparams`, `fparams`, `bparams` - dictionaries of
                   integer, float, and boolean valued input
                   parameters, respectively

                   `ivars`, `fvars`, `bvars`, `tvars` - dictionaries
                   of integer, float, boolean, and Tensor valued
                   iteration variables, respectively.

                   `A`, `B`, `iK` - input Tensor arguments.

                   `E`, `X`, `S`, `R` - iteration Tensor variables.

                 For instance:

                   `ivars["istep"]` - the current iteration step
                   `X` - the current approximation of eigenvectors
                   `E` - the current approximation of eigenvalues
                   `R` - the current residual
                   `ivars["converged_count"]` - the current number of converged eigenpairs
                   `tvars["rerr"]` - the current state of convergence criteria

                 Note that when `tracker` stores Tensor objects from
                 the LOBPCG instance, it must make copies of these.

                 If `tracker` sets `bvars["force_stop"] = True`, the
                 iteration process will be hard-stopped.

      ortho_iparams, ortho_fparams, ortho_bparams (dict, optional):
                 various parameters to LOBPCG algorithm when using
                 `method="ortho"`.

    Returns:

      E (Tensor): tensor of eigenvalues of size :math:`(*, k)`

      X (Tensor): tensor of eigenvectors of size :math:`(*, m, k)`

    References:

      [Knyazev2001] Andrew V. Knyazev. (2001) Toward the Optimal
      Preconditioned Eigensolver: Locally Optimal Block Preconditioned
      Conjugate Gradient Method. SIAM J. Sci. Comput., 23(2),
      517-541. (25 pages)
      `https://epubs.siam.org/doi/abs/10.1137/S1064827500366124`_

      [StathopoulosEtal2002] Andreas Stathopoulos and Kesheng
      Wu. (2002) A Block Orthogonalization Procedure with Constant
      Synchronization Requirements. SIAM J. Sci. Comput., 23(6),
      2165-2182. (18 pages)
      `https://epubs.siam.org/doi/10.1137/S1064827500370883`_

      [DuerschEtal2018] Jed A. Duersch, Meiyue Shao, Chao Yang, Ming
      Gu. (2018) A Robust and Efficient Implementation of LOBPCG.
      SIAM J. Sci. Comput., 40(5), C655-C676. (22 pages)
      `https://epubs.siam.org/doi/abs/10.1137/17M1129830`_

    """

    if not torch.jit.is_scripting():
        tensor_ops = (A, B, X, iK)
        if (not set(map(type, tensor_ops)).issubset((torch.Tensor, type(None))) and has_torch_function(tensor_ops)):
            return handle_torch_function(
                lobpcg, tensor_ops, A, k=k,
                B=B, X=X, n=n, iK=iK, niter=niter, tol=tol,
                largest=largest, method=method, tracker=tracker,
                ortho_iparams=ortho_iparams,
                ortho_fparams=ortho_fparams,
                ortho_bparams=ortho_bparams)

    # A must be square:
    assert A.shape[-2] == A.shape[-1], A.shape
    if B is not None:
        # A and B must have the same shapes:
        assert A.shape == B.shape, (A.shape, B.shape)

    dtype = _utils.get_floating_dtype(A)
    device = A.device
    if tol is None:
        feps = {torch.float32: 1.2e-07,
                torch.float64: 2.23e-16}[dtype]
        tol = feps ** 0.5

    m = A.shape[-1]
    k = (1 if X is None else X.shape[-1]) if k is None else k
    n = (k if n is None else n) if X is None else X.shape[-1]

    if (m < 3 * n):
        raise ValueError(
            'LPBPCG algorithm is not applicable when the number of A rows (={})'
            ' is smaller than 3 x the number of requested eigenpairs (={})'
            .format(m, n))

    method = 'ortho' if method is None else method

    iparams = {
        'm': m,
        'n': n,
        'k': k,
        'niter': 1000 if niter is None else niter,
    }

    fparams = {
        'tol': tol,
    }

    bparams = {
        'largest': True if largest is None else largest
    }

    if method == 'ortho':
        if ortho_iparams is not None:
            iparams.update(ortho_iparams)
        if ortho_fparams is not None:
            fparams.update(ortho_fparams)
        if ortho_bparams is not None:
            bparams.update(ortho_bparams)
        iparams['ortho_i_max'] = iparams.get('ortho_i_max', 3)
        iparams['ortho_j_max'] = iparams.get('ortho_j_max', 3)
        fparams['ortho_tol'] = fparams.get('ortho_tol', tol)
        fparams['ortho_tol_drop'] = fparams.get('ortho_tol_drop', tol)
        fparams['ortho_tol_replace'] = fparams.get('ortho_tol_replace', tol)
        bparams['ortho_use_drop'] = bparams.get('ortho_use_drop', False)

    if not torch.jit.is_scripting():
        LOBPCG.call_tracker = LOBPCG_call_tracker

    if len(A.shape) > 2:
        N = int(torch.prod(torch.tensor(A.shape[:-2])))
        bA = A.reshape((N,) + A.shape[-2:])
        bB = B.reshape((N,) + A.shape[-2:]) if B is not None else None
        bX = X.reshape((N,) + X.shape[-2:]) if X is not None else None
        bE = torch.empty((N, k), dtype=dtype, device=device)
        bXret = torch.empty((N, m, k), dtype=dtype, device=device)

        for i in range(N):
            A_ = bA[i]
            B_ = bB[i] if bB is not None else None
            X_ = torch.randn((m, n), dtype=dtype, device=device) if bX is None else bX[i]
            assert len(X_.shape) == 2 and X_.shape == (m, n), (X_.shape, (m, n))
            iparams['batch_index'] = i
            worker = LOBPCG(A_, B_, X_, iK, iparams, fparams, bparams, method, tracker)
            worker.run()
            bE[i] = worker.E[:k]
            bXret[i] = worker.X[:, :k]

        if not torch.jit.is_scripting():
            LOBPCG.call_tracker = LOBPCG_call_tracker_orig

        return bE.reshape(A.shape[:-2] + (k,)), bXret.reshape(A.shape[:-2] + (m, k))

    X = torch.randn((m, n), dtype=dtype, device=device) if X is None else X
    assert len(X.shape) == 2 and X.shape == (m, n), (X.shape, (m, n))

    worker = LOBPCG(A, B, X, iK, iparams, fparams, bparams, method, tracker)

    worker.run()

    if not torch.jit.is_scripting():
        LOBPCG.call_tracker = LOBPCG_call_tracker_orig

    return worker.E[:k], worker.X[:, :k]


class LOBPCG(object):
    """Worker class of LOBPCG methods.
    """

    def __init__(self,
                 A,        # type: Optional[Tensor]
                 B,        # type: Optional[Tensor]
                 X,        # type: Tensor
                 iK,       # type: Optional[Tensor]
                 iparams,  # type: Dict[str, int]
                 fparams,  # type: Dict[str, float]
                 bparams,  # type: Dict[str, bool]
                 method,   # type: str
                 tracker   # type: Optional[None]
                 ):
        # type: (...) -> None

        # constant parameters
        self.A = A
        self.B = B
        self.iK = iK
        self.iparams = iparams
        self.fparams = fparams
        self.bparams = bparams
        self.method = method
        self.tracker = tracker
        m = iparams['m']
        n = iparams['n']

        # variable parameters
        self.X = X
        self.E = torch.zeros((n, ), dtype=X.dtype, device=X.device)
        self.R = torch.zeros((m, n), dtype=X.dtype, device=X.device)
        self.S = torch.zeros((m, 3 * n), dtype=X.dtype, device=X.device)
        self.tvars = {}               # type: Dict[str, Tensor]
        self.ivars = {'istep': 0}     # type: Dict[str, int]
        self.fvars = {'_': 0.0}       # type: Dict[str, float]
        self.bvars = {'_': False}     # type: Dict[str, bool]

    def __str__(self):
        lines = ['LOPBCG:']
        lines += ['  iparams={}'.format(self.iparams)]
        lines += ['  fparams={}'.format(self.fparams)]
        lines += ['  bparams={}'.format(self.bparams)]
        lines += ['  ivars={}'.format(self.ivars)]
        lines += ['  fvars={}'.format(self.fvars)]
        lines += ['  bvars={}'.format(self.bvars)]
        lines += ['  tvars={}'.format(self.tvars)]
        lines += ['  A={}'.format(self.A)]
        lines += ['  B={}'.format(self.B)]
        lines += ['  iK={}'.format(self.iK)]
        lines += ['  X={}'.format(self.X)]
        lines += ['  E={}'.format(self.E)]
        r = ''
        for line in lines:
            r += line + '\n'
        return r

    def update(self):
        """Set and update iteration variables.
        """
        if self.ivars['istep'] == 0:
            X_norm = float(torch.norm(self.X))
            iX_norm = X_norm ** -1
            A_norm = float(torch.norm(_utils.matmul(self.A, self.X))) * iX_norm
            B_norm = float(torch.norm(_utils.matmul(self.B, self.X))) * iX_norm
            self.fvars['X_norm'] = X_norm
            self.fvars['A_norm'] = A_norm
            self.fvars['B_norm'] = B_norm
            self.ivars['iterations_left'] = self.iparams['niter']
            self.ivars['converged_count'] = 0
            self.ivars['converged_end'] = 0

        if self.method == 'ortho':
            self._update_ortho()
        else:
            self._update_basic()

        self.ivars['iterations_left'] = self.ivars['iterations_left'] - 1
        self.ivars['istep'] = self.ivars['istep'] + 1

    def update_residual(self):
        """Update residual R from A, B, X, E.
        """
        mm = _utils.matmul
        self.R = mm(self.A, self.X) - mm(self.B, self.X) * self.E

    def update_converged_count(self):
        """Determine the number of converged eigenpairs using backward stable
        convergence criterion, see discussion in Sec 4.3 of [DuerschEtal2018].

        Users may redefine this method for custom convergence criteria.
        """
        # (...) -> int
        prev_count = self.ivars['converged_count']
        tol = self.fparams['tol']
        A_norm = self.fvars['A_norm']
        B_norm = self.fvars['B_norm']
        E, X, R = self.E, self.X, self.R
        rerr = torch.norm(R, 2, (0, )) * (torch.norm(X, 2, (0, )) * (A_norm + E[:X.shape[-1]] * B_norm)) ** -1
        converged = rerr < tol
        count = 0
        for b in converged:
            if not b:
                # ignore convergence of following pairs to ensure
                # strict ordering of eigenpairs
                break
            count += 1
        assert count >= prev_count, (
            'the number of converged eigenpairs '
            '(was %s, got %s) cannot decrease' % (prev_count, count))
        self.ivars['converged_count'] = count
        self.tvars['rerr'] = rerr
        return count

    def stop_iteration(self):
        """Return True to stop iterations.

        Note that tracker (if defined) can force-stop iterations by
        setting ``worker.bvars['force_stop'] = True``.
        """
        return (self.bvars.get('force_stop', False)
                or self.ivars['iterations_left'] == 0
                or self.ivars['converged_count'] >= self.iparams['k'])

    def run(self):
        """Run LOBPCG iterations.

        Use this method as a template for implementing LOBPCG
        iteration scheme with custom tracker that is compatible with
        TorchScript.
        """
        self.update()

        if not torch.jit.is_scripting() and self.tracker is not None:
            self.call_tracker()

        while not self.stop_iteration():

            self.update()

            if not torch.jit.is_scripting() and self.tracker is not None:
                self.call_tracker()

    @torch.jit.unused
    def call_tracker(self):
        """Interface for tracking iteration process in Python mode.

        Tracking the iteration process is disabled in TorchScript
        mode. In fact, one should specify tracker=None when JIT
        compiling functions using lobpcg.
        """
        # do nothing when in TorchScript mode
        pass

    # Internal methods

    def _update_basic(self):
        """
        Update or initialize iteration variables when `method == "basic"`.
        """
        mm = torch.matmul
        ns = self.ivars['converged_end']
        nc = self.ivars['converged_count']
        n = self.iparams['n']
        largest = self.bparams['largest']

        if self.ivars['istep'] == 0:
            Ri = self._get_rayleigh_ritz_transform(self.X)
            M = _utils.qform(_utils.qform(self.A, self.X), Ri)
            E, Z = _utils.symeig(M, largest)
            self.X[:] = mm(self.X, mm(Ri, Z))
            self.E[:] = E
            np = 0
            self.update_residual()
            nc = self.update_converged_count()
            self.S[..., :n] = self.X

            W = _utils.matmul(self.iK, self.R)
            self.ivars['converged_end'] = ns = n + np + W.shape[-1]
            self.S[:, n + np:ns] = W
        else:
            S_ = self.S[:, nc:ns]
            Ri = self._get_rayleigh_ritz_transform(S_)
            M = _utils.qform(_utils.qform(self.A, S_), Ri)
            E_, Z = _utils.symeig(M, largest)
            self.X[:, nc:] = mm(S_, mm(Ri, Z[:, :n - nc]))
            self.E[nc:] = E_[:n - nc]
            P = mm(S_, mm(Ri, Z[:, n:2 * n - nc]))
            np = P.shape[-1]

            self.update_residual()
            nc = self.update_converged_count()
            self.S[..., :n] = self.X
            self.S[:, n:n + np] = P
            W = _utils.matmul(self.iK, self.R[:, nc:])

            self.ivars['converged_end'] = ns = n + np + W.shape[-1]
            self.S[:, n + np:ns] = W

    def _update_ortho(self):
        """
        Update or initialize iteration variables when `method == "ortho"`.
        """
        mm = torch.matmul
        ns = self.ivars['converged_end']
        nc = self.ivars['converged_count']
        n = self.iparams['n']
        largest = self.bparams['largest']

        if self.ivars['istep'] == 0:
            Ri = self._get_rayleigh_ritz_transform(self.X)
            M = _utils.qform(_utils.qform(self.A, self.X), Ri)
            E, Z = _utils.symeig(M, largest)
            self.X = mm(self.X, mm(Ri, Z))
            self.update_residual()
            np = 0
            nc = self.update_converged_count()
            self.S[:, :n] = self.X
            W = self._get_ortho(self.R, self.X)
            ns = self.ivars['converged_end'] = n + np + W.shape[-1]
            self.S[:, n + np:ns] = W

        else:
            S_ = self.S[:, nc:ns]
            # Rayleigh-Ritz procedure
            E_, Z = _utils.symeig(_utils.qform(self.A, S_), largest)

            # Update E, X, P
            self.X[:, nc:] = mm(S_, Z[:, :n - nc])
            self.E[nc:] = E_[:n - nc]
            P = mm(S_, mm(Z[:, n - nc:], _utils.basis(_utils.transpose(Z[:n - nc, n - nc:]))))
            np = P.shape[-1]

            # check convergence
            self.update_residual()
            nc = self.update_converged_count()

            # update S
            self.S[:, :n] = self.X
            self.S[:, n:n + np] = P
            W = self._get_ortho(self.R[:, nc:], self.S[:, :n + np])
            ns = self.ivars['converged_end'] = n + np + W.shape[-1]
            self.S[:, n + np:ns] = W

    def _get_rayleigh_ritz_transform(self, S):
        """Return a transformation matrix that is used in Rayleigh-Ritz
        procedure for reducing a general eigenvalue problem :math:`(S^TAS)
        C = (S^TBS) C E` to a standard eigenvalue problem :math: `(Ri^T
        S^TAS Ri) Z = Z E` where `C = Ri Z`.

        .. note:: In the original Rayleight-Ritz procedure in
          [DuerschEtal2018], the problem is formulated as follows::

            SAS = S^T A S
            SBS = S^T B S
            D = (<diagonal matrix of SBS>) ** -1/2
            R^T R = Cholesky(D SBS D)
            Ri = D R^-1
            solve symeig problem Ri^T SAS Ri Z = Theta Z
            C = Ri Z

          To reduce the number of matrix products (denoted by empty
          space between matrices), here we introduce element-wise
          products (denoted by symbol `*`) so that the Rayleight-Ritz
          procedure becomes::

            SAS = S^T A S
            SBS = S^T B S
            d = (<diagonal of SBS>) ** -1/2    # this is 1-d column vector
            dd = d d^T                         # this is 2-d matrix
            R^T R = Cholesky(dd * SBS)
            Ri = R^-1 * d                      # broadcasting
            solve symeig problem Ri^T SAS Ri Z = Theta Z
            C = Ri Z

          where `dd` is 2-d matrix that replaces matrix products `D M
          D` with one element-wise product `M * dd`; and `d` replaces
          matrix product `D M` with element-wise product `M *
          d`. Also, creating the diagonal matrix `D` is avoided.

        Arguments:
        S (Tensor): the matrix basis for the search subspace, size is
                    :math:`(m, n)`.

        Returns:
        Ri (tensor): upper-triangular transformation matrix of size
                     :math:`(n, n)`.

        """
        B = self.B
        mm = torch.matmul
        SBS = _utils.qform(B, S)
        d_row = SBS.diagonal(0, -2, -1) ** -0.5
        d_col = d_row.reshape(d_row.shape[0], 1)
        R = torch.cholesky((SBS * d_row) * d_col, upper=True)
        # TODO: could use LAPACK ?trtri as R is upper-triangular
        Rinv = torch.inverse(R)
        return Rinv * d_col

    def _get_svqb(self,
                  U,     # Tensor
                  drop,  # bool
                  tau    # float
                  ):
        # type: (Tensor, bool, float) -> Tensor
        """Return B-orthonormal U.

        .. note:: When `drop` is `False` then `svqb` is based on the
                  Algorithm 4 from [DuerschPhD2015] that is a slight
                  modification of the corresponding algorithm
                  introduced in [StathopolousWu2002].

        Arguments:

          U (Tensor) : initial approximation, size is (m, n)
          drop (bool) : when True, drop columns that
                     contribution to the `span([U])` is small.
          tau (float) : positive tolerance

        Returns:

          U (Tensor) : B-orthonormal columns (:math:`U^T B U = I`), size
                       is (m, n1), where `n1 = n` if `drop` is `False,
                       otherwise `n1 <= n`.

        """
        if torch.numel(U) == 0:
            return U
        UBU = _utils.qform(self.B, U)
        d = UBU.diagonal(0, -2, -1)

        # Detect and drop exact zero columns from U. While the test
        # `abs(d) == 0` is unlikely to be True for random data, it is
        # possible to construct input data to lobpcg where it will be
        # True leading to a failure (notice the `d ** -0.5` operation
        # in the original algorithm). To prevent the failure, we drop
        # the exact zero columns here and then continue with the
        # original algorithm below.
        nz = torch.where(abs(d) != 0.0)
        assert len(nz) == 1, nz
        if len(nz[0]) < len(d):
            U = U[:, nz[0]]
            if torch.numel(U) == 0:
                return U
            UBU = _utils.qform(self.B, U)
            d = UBU.diagonal(0, -2, -1)
            nz = torch.where(abs(d) != 0.0)
            assert len(nz[0]) == len(d)

        # The original algorithm 4 from [DuerschPhD2015].
        d_col = (d ** -0.5).reshape(d.shape[0], 1)
        DUBUD = (UBU * d_col) * _utils.transpose(d_col)
        E, Z = _utils.symeig(DUBUD, eigenvectors=True)
        t = tau * abs(E).max()
        if drop:
            keep = torch.where(E > t)
            assert len(keep) == 1, keep
            E = E[keep[0]]
            Z = Z[:, keep[0]]
            d_col = d_col[keep[0]]
        else:
            E[(torch.where(E < t))[0]] = t

        return torch.matmul(U * _utils.transpose(d_col), Z * E ** -0.5)

    def _get_ortho(self, U, V):
        """Return B-orthonormal U with columns are B-orthogonal to V.

        .. note:: When `bparams["ortho_use_drop"] == False` then
                  `_get_ortho` is based on the Algorithm 3 from
                  [DuerschPhD2015] that is a slight modification of
                  the corresponding algorithm introduced in
                  [StathopolousWu2002]. Otherwise, the method
                  implements Algorithm 6 from [DuerschPhD2015]

        .. note:: If all U columns are B-collinear to V then the
                  returned tensor U will be empty.

        Arguments:

          U (Tensor) : initial approximation, size is (m, n)
          V (Tensor) : B-orthogonal external basis, size is (m, k)

        Returns:

          U (Tensor) : B-orthonormal columns (:math:`U^T B U = I`)
                       such that :math:`V^T B U=0`, size is (m, n1),
                       where `n1 = n` if `drop` is `False, otherwise
                       `n1 <= n`.
        """
        mm = torch.matmul
        mm_B = _utils.matmul
        m = self.iparams['m']
        tau_ortho = self.fparams['ortho_tol']
        tau_drop = self.fparams['ortho_tol_drop']
        tau_replace = self.fparams['ortho_tol_replace']
        i_max = self.iparams['ortho_i_max']
        j_max = self.iparams['ortho_j_max']
        # when use_drop==True, enable dropping U columns that have
        # small contribution to the `span([U, V])`.
        use_drop = self.bparams['ortho_use_drop']

        # clean up variables from the previous call
        for vkey in list(self.fvars.keys()):
            if vkey.startswith('ortho_') and vkey.endswith('_rerr'):
                self.fvars.pop(vkey)
        self.ivars.pop('ortho_i', 0)
        self.ivars.pop('ortho_j', 0)

        BV_norm = torch.norm(mm_B(self.B, V))
        BU = mm_B(self.B, U)
        VBU = mm(_utils.transpose(V), BU)
        i = j = 0
        stats = ''
        for i in range(i_max):
            U = U - mm(V, VBU)
            drop = False
            tau_svqb = tau_drop
            for j in range(j_max):
                if use_drop:
                    U = self._get_svqb(U, drop, tau_svqb)
                    drop = True
                    tau_svqb = tau_replace
                else:
                    U = self._get_svqb(U, False, tau_replace)
                if torch.numel(U) == 0:
                    # all initial U columns are B-collinear to V
                    self.ivars['ortho_i'] = i
                    self.ivars['ortho_j'] = j
                    return U
                BU = mm_B(self.B, U)
                UBU = mm(_utils.transpose(U), BU)
                U_norm = torch.norm(U)
                BU_norm = torch.norm(BU)
                R = UBU - torch.eye(UBU.shape[-1],
                                    device=UBU.device,
                                    dtype=UBU.dtype)
                R_norm = torch.norm(R)
                # https://github.com/pytorch/pytorch/issues/33810 workaround:
                rerr = float(R_norm) * float(BU_norm * U_norm) ** -1
                vkey = 'ortho_UBUmI_rerr[{}, {}]'.format(i, j)
                self.fvars[vkey] = rerr
                if rerr < tau_ortho:
                    break
            VBU = mm(_utils.transpose(V), BU)
            VBU_norm = torch.norm(VBU)
            U_norm = torch.norm(U)
            rerr = float(VBU_norm) * float(BV_norm * U_norm) ** -1
            vkey = 'ortho_VBU_rerr[{}]'.format(i)
            self.fvars[vkey] = rerr
            if rerr < tau_ortho:
                break
            if m < U.shape[-1] + V.shape[-1]:
                raise ValueError(
                    'Overdetermined shape of U:'
                    ' #B-cols(={}) >= #U-cols(={}) + #V-cols(={}) must hold'
                    .format(self.B.shape[-1], U.shape[-1], V.shape[-1]))
        self.ivars['ortho_i'] = i
        self.ivars['ortho_j'] = j
        return U


# Calling tracker is separated from LOBPCG definitions because
# TorchScript does not support user-defined callback arguments:
LOBPCG_call_tracker_orig = LOBPCG.call_tracker
def LOBPCG_call_tracker(self):
    self.tracker(self)