def symeig(A: LinearOperator,
neig: Optional[int] = None,
mode: str = "lowest",
M: Optional[LinearOperator] = None,
bck_options: Mapping[str, Any] = {},
method: Union[str, Callable, None] = None,
**fwd_options) -> Tuple[torch.Tensor, torch.Tensor]:
r"""
Obtain ``neig`` lowest eigenvalues and eigenvectors of a linear operator,
.. math::
\mathbf{AX = MXE}
where :math:`\mathbf{A}, \mathbf{M}` are linear operators,
:math:`\mathbf{E}` is a diagonal matrix containing the eigenvalues, and
:math:`\mathbf{X}` is a matrix containing the eigenvectors.
Arguments
---------
A: xitorch.LinearOperator
The linear operator object on which the eigenpairs are constructed.
It must be a Hermitian linear operator with shape ``(*BA, q, q)``
neig: int or None
The number of eigenpairs to be retrieved. If ``None``, all eigenpairs are
retrieved
mode: str
``"lowest"`` or ``"uppermost"``/``"uppest"``. If ``"lowest"``,
it will take the lowest ``neig`` eigenpairs.
If ``"uppest"``, it will take the uppermost ``neig``.
M: xitorch.LinearOperator
The transformation on the right hand side. If ``None``, then ``M=I``.
If specified, it must be a Hermitian with shape ``(*BM, q, q)``.
bck_options: dict
Method-specific options for :func:`solve` which used in backpropagation
calculation.
method: str or callable or None
Method for the eigendecomposition. If ``None``, it will choose
``"exacteig"``.
**fwd_options
Method-specific options (see method section below).
Returns
-------
tuple of tensors (eigenvalues, eigenvectors)
It will return eigenvalues and eigenvectors with shapes respectively
``(*BAM, neig)`` and ``(*BAM, na, neig)``, where ``*BAM`` is the
broadcasted shape of ``*BA`` and ``*BM``.
"""
assert_runtime(A.is_hermitian, "The linear operator A must be Hermitian")
assert_runtime(
not torch.is_grad_enabled() or A.is_getparamnames_implemented,
"The _getparamnames(self, prefix) of linear operator A must be "
"implemented if using symeig with grad enabled")
if M is not None:
assert_runtime(M.is_hermitian,
"The linear operator M must be Hermitian")
assert_runtime(
M.shape[-1] == A.shape[-1],
"The shape of A & M must match (A: %s, M: %s)" %
(A.shape, M.shape))
assert_runtime(
not torch.is_grad_enabled() or M.is_getparamnames_implemented,
"The _getparamnames(self, prefix) of linear operator M must be "
"implemented if using symeig with grad enabled")
mode = mode.lower()
if mode == "uppermost":
mode = "uppest"
if method is None:
if isinstance(A, MatrixLinearOperator) and \
(M is None or isinstance(M, MatrixLinearOperator)):
method = "exacteig"
else:
# TODO: implement robust LOBPCG and put it here
method = "exacteig"
if neig is None:
neig = A.shape[-1]
# perform expensive check if debug mode is enabled
if is_debug_enabled():
A.check()
if M is not None:
M.check()
if method == "exacteig":
return exacteig(A, neig, mode, M)
else:
fwd_options["method"] = method
# get the unique parameters of A & M
params = A.getlinopparams()
mparams = M.getlinopparams() if M is not None else []
na = len(params)
return symeig_torchfcn.apply(A, neig, mode, M, fwd_options,
bck_options, na, *params, *mparams)
def symeig(A: LinearOperator,
neig: Optional[int] = None,
mode: str = "lowest",
M: Optional[LinearOperator] = None,
bck_options: Mapping[str, Any] = {},
method: Union[str, Callable, None] = None,
**fwd_options) -> Tuple[torch.Tensor, torch.Tensor]:
r"""
Obtain ``neig`` lowest eigenvalues and eigenvectors of a linear operator,
.. math::
\mathbf{AX = MXE}
where :math:`\mathbf{A}, \mathbf{M}` are linear operators,
:math:`\mathbf{E}` is a diagonal matrix containing the eigenvalues, and
:math:`\mathbf{X}` is a matrix containing the eigenvectors.
This function can handle derivatives for degenerate cases by setting non-zero
``degen_atol`` and ``degen_rtol`` in the backward option using the expressions
in [1]_.
Arguments
---------
A: xitorch.LinearOperator
The linear operator object on which the eigenpairs are constructed.
It must be a Hermitian linear operator with shape ``(*BA, q, q)``
neig: int or None
The number of eigenpairs to be retrieved. If ``None``, all eigenpairs are
retrieved
mode: str
``"lowest"`` or ``"uppermost"``/``"uppest"``. If ``"lowest"``,
it will take the lowest ``neig`` eigenpairs.
If ``"uppest"``, it will take the uppermost ``neig``.
M: xitorch.LinearOperator
The transformation on the right hand side. If ``None``, then ``M=I``.
If specified, it must be a Hermitian with shape ``(*BM, q, q)``.
bck_options: dict
Method-specific options for :func:`solve` which used in backpropagation
calculation with some additional arguments for computing the backward
derivatives:
* ``degen_atol`` (``float`` or None): Minimum absolute difference between
two eigenvalues to be treated as degenerate. If None, it is
``torch.finfo(dtype).eps**0.6``. If 0.0, no special treatment on
degeneracy is applied. (default: None)
* ``degen_rtol`` (``float`` or None): Minimum relative difference between
two eigenvalues to be treated as degenerate. If None, it is
``torch.finfo(dtype).eps**0.4``. If 0.0, no special treatment on
degeneracy is applied. (default: None)
Note: the default values of ``degen_atol`` and ``degen_rtol`` are going
to change in the future. So, for future compatibility, please specify
the specific values.
method: str or callable or None
Method for the eigendecomposition. If ``None``, it will choose
``"exacteig"``.
**fwd_options
Method-specific options (see method section below).
Returns
-------
tuple of tensors (eigenvalues, eigenvectors)
It will return eigenvalues and eigenvectors with shapes respectively
``(*BAM, neig)`` and ``(*BAM, na, neig)``, where ``*BAM`` is the
broadcasted shape of ``*BA`` and ``*BM``.
References
----------
.. [1] Muhammad F. Kasim,
"Derivatives of partial eigendecomposition of a real symmetric matrix for degenerate cases".
arXiv:2011.04366 (2020)
`https://arxiv.org/abs/2011.04366 <https://arxiv.org/abs/2011.04366>`_
"""
assert_runtime(A.is_hermitian, "The linear operator A must be Hermitian")
assert_runtime(
not torch.is_grad_enabled() or A.is_getparamnames_implemented,
"The _getparamnames(self, prefix) of linear operator A must be "
"implemented if using symeig with grad enabled")
if M is not None:
assert_runtime(M.is_hermitian,
"The linear operator M must be Hermitian")
assert_runtime(
M.shape[-1] == A.shape[-1],
"The shape of A & M must match (A: %s, M: %s)" %
(A.shape, M.shape))
assert_runtime(
not torch.is_grad_enabled() or M.is_getparamnames_implemented,
"The _getparamnames(self, prefix) of linear operator M must be "
"implemented if using symeig with grad enabled")
mode = mode.lower()
if mode == "uppermost":
mode = "uppest"
if method is None:
if isinstance(A, MatrixLinearOperator) and \
(M is None or isinstance(M, MatrixLinearOperator)):
method = "exacteig"
else:
# TODO: implement robust LOBPCG and put it here
method = "exacteig"
if neig is None:
neig = A.shape[-1]
# perform expensive check if debug mode is enabled
if is_debug_enabled():
A.check()
if M is not None:
M.check()
if method == "exacteig":
return exacteig(A, neig, mode, M)
else:
fwd_options["method"] = method
# get the unique parameters of A & M
params = A.getlinopparams()
mparams = M.getlinopparams() if M is not None else []
na = len(params)
return symeig_torchfcn.apply(A, neig, mode, M, fwd_options,
bck_options, na, *params, *mparams)
def custom_exacteig(A, neig, mode, M=None, **options):
return exacteig(A, neig, mode, M)
def custom_exacteig(A, params, neig, mode, M=None, mparams=[], **options):
with A.uselinopparams(*params), M.uselinopparams(
*mparams) if M is not None else dummy_context_manager():
return exacteig(A, neig, mode, M)