def _log_abs_determinant(self):
logging.warn(
"Using (possibly slow) default implementation of determinant."
" Requires conversion to a dense matrix and O(N^3) operations.")
if self._can_use_cholesky():
diag = _linalg.diag_part(linalg_ops.cholesky(self.to_dense()))
return 2 * math_ops.reduce_sum(math_ops.log(diag), axis=[-1])
_, log_abs_det = linalg.slogdet(self.to_dense())
return log_abs_det
def _dense_solve(self, rhs, adjoint=False, adjoint_arg=False):
"""Solve by conversion to a dense matrix."""
if self.is_square is False: # pylint: disable=g-bool-id-comparison
raise NotImplementedError(
"Solve is not yet implemented for non-square operators.")
rhs = linalg.adjoint(rhs) if adjoint_arg else rhs
if self._can_use_cholesky():
return linalg_ops.cholesky_solve(
linalg_ops.cholesky(self.to_dense()), rhs)
return linear_operator_util.matrix_solve_with_broadcast(
self.to_dense(), rhs, adjoint=adjoint)
def _solve(self, rhs, adjoint=False, adjoint_arg=False):
"""Default implementation of _solve."""
if self.is_square is False:
raise NotImplementedError(
"Solve is not yet implemented for non-square operators.")
logging.warn(
"Using (possibly slow) default implementation of solve."
" Requires conversion to a dense matrix and O(N^3) operations.")
rhs = linalg.adjoint(rhs) if adjoint_arg else rhs
if self._can_use_cholesky():
return linear_operator_util.cholesky_solve_with_broadcast(
linalg_ops.cholesky(self.to_dense()), rhs)
return linear_operator_util.matrix_solve_with_broadcast(
self.to_dense(), rhs, adjoint=adjoint)
def _assert_positive_definite(self):
"""Default implementation of _assert_positive_definite."""
logging.warn(
"Using (possibly slow) default implementation of "
"assert_positive_definite."
" Requires conversion to a dense matrix and O(N^3) operations.")
# If the operator is self-adjoint, then checking that
# Cholesky decomposition succeeds + results in positive diag is necessary
# and sufficient.
if self.is_self_adjoint:
return check_ops.assert_positive(
_linalg.diag_part(linalg_ops.cholesky(self.to_dense())),
message="Matrix was not positive definite.")
# We have no generic check for positive definite.
raise NotImplementedError("assert_positive_definite is not implemented.")
def _solve(self, rhs, adjoint=False, adjoint_arg=False):
if self.base_operator.is_non_singular is False:
raise ValueError(
"Solve not implemented unless this is a perturbation of a "
"non-singular LinearOperator.")
# The Woodbury formula gives:
# https://en.wikipedia.org/wiki/Woodbury_matrix_identity
# (L + UDV^H)^{-1}
# = L^{-1} - L^{-1} U (D^{-1} + V^H L^{-1} U)^{-1} V^H L^{-1}
# = L^{-1} - L^{-1} U C^{-1} V^H L^{-1}
# where C is the capacitance matrix, C := D^{-1} + V^H L^{-1} U
# Note also that, with ^{-H} being the inverse of the adjoint,
# (L + UDV^H)^{-H}
# = L^{-H} - L^{-H} V C^{-H} U^H L^{-H}
l = self.base_operator
if adjoint:
# If adjoint, U and V have flipped roles in the operator.
v, u = self._get_uv_as_tensors()
# Capacitance should still be computed with u=self.u and v=self.v, which
# after the "flip" on the line above means u=v, v=u. I.e. no need to
# "flip" in the capacitance call, since the call to
# matrix_solve_with_broadcast below is done with the `adjoint` argument,
# and this takes care of things.
capacitance = self._make_capacitance(u=v, v=u)
else:
u, v = self._get_uv_as_tensors()
capacitance = self._make_capacitance(u=u, v=v)
# L^{-1} rhs
linv_rhs = l.solve(rhs, adjoint=adjoint, adjoint_arg=adjoint_arg)
# V^H L^{-1} rhs
vh_linv_rhs = _linalg.matmul(v, linv_rhs, adjoint_a=True)
# C^{-1} V^H L^{-1} rhs
if self._use_cholesky:
capinv_vh_linv_rhs = linalg_ops.cholesky_solve(
linalg_ops.cholesky(capacitance), vh_linv_rhs)
else:
capinv_vh_linv_rhs = linear_operator_util.matrix_solve_with_broadcast(
capacitance, vh_linv_rhs, adjoint=adjoint)
# U C^{-1} V^H M^{-1} rhs
u_capinv_vh_linv_rhs = _linalg.matmul(u, capinv_vh_linv_rhs)
# L^{-1} U C^{-1} V^H L^{-1} rhs
linv_u_capinv_vh_linv_rhs = l.solve(u_capinv_vh_linv_rhs,
adjoint=adjoint)
# L^{-1} - L^{-1} U C^{-1} V^H L^{-1}
return linv_rhs - linv_u_capinv_vh_linv_rhs
def _log_abs_determinant(self):
# Recall
# det(L + UDV^H) = det(D^{-1} + V^H L^{-1} U) det(D) det(L)
# = det(C) det(D) det(L)
log_abs_det_d = self.diag_operator.log_abs_determinant()
log_abs_det_l = self.base_operator.log_abs_determinant()
if self._use_cholesky:
chol_cap_diag = _linalg.diag_part(
linalg_ops.cholesky(self._make_capacitance()))
log_abs_det_c = 2 * math_ops.reduce_sum(
math_ops.log(chol_cap_diag), axis=[-1])
else:
det_c = _linalg.det(self._make_capacitance())
log_abs_det_c = math_ops.log(math_ops.abs(det_c))
if np.issubdtype(self.dtype, np.complexfloating):
log_abs_det_c = _ops.cast(log_abs_det_c, dtype=self.dtype)
return log_abs_det_c + log_abs_det_d + log_abs_det_l
def _solve(self, rhs, adjoint=False, adjoint_arg=False):
if self.base_operator.is_non_singular is False:
raise ValueError(
"Solve not implemented unless this is a perturbation of a "
"non-singular LinearOperator.")
# The Woodbury formula gives:
# https://en.wikipedia.org/wiki/Woodbury_matrix_identity
# (L + UDV^H)^{-1}
# = L^{-1} - L^{-1} U (D^{-1} + V^H L^{-1} U)^{-1} V^H L^{-1}
# = L^{-1} - L^{-1} U C^{-1} V^H L^{-1}
# where C is the capacitance matrix, C := D^{-1} + V^H L^{-1} U
# Note also that, with ^{-H} being the inverse of the adjoint,
# (L + UDV^H)^{-H}
# = L^{-H} - L^{-H} V C^{-H} U^H L^{-H}
l = self.base_operator
if adjoint:
v = self.u
u = self.v
else:
v = self.v
u = self.u
# L^{-1} rhs
linv_rhs = l.solve(rhs, adjoint=adjoint, adjoint_arg=adjoint_arg)
# V^H L^{-1} rhs
vh_linv_rhs = _linalg.matmul(v, linv_rhs, adjoint_a=True)
# C^{-1} V^H L^{-1} rhs
if self._use_cholesky:
capinv_vh_linv_rhs = linalg_ops.cholesky_solve(
linalg_ops.cholesky(self._make_capacitance()), vh_linv_rhs)
else:
capinv_vh_linv_rhs = linear_operator_util.matrix_solve_with_broadcast(
self._make_capacitance(), vh_linv_rhs, adjoint=adjoint)
# U C^{-1} V^H M^{-1} rhs
u_capinv_vh_linv_rhs = _linalg.matmul(u, capinv_vh_linv_rhs)
# L^{-1} U C^{-1} V^H L^{-1} rhs
linv_u_capinv_vh_linv_rhs = l.solve(u_capinv_vh_linv_rhs,
adjoint=adjoint)
# L^{-1} - L^{-1} U C^{-1} V^H L^{-1}
return linv_rhs - linv_u_capinv_vh_linv_rhs
def _cholesky_linear_operator(linop):
return linear_operator_lower_triangular.LinearOperatorLowerTriangular(
linalg_ops.cholesky(linop.to_dense()),
is_non_singular=True,
is_self_adjoint=False,
is_square=True)
def __init__(self,
base_operator,
u,
diag_update=None,
v=None,
is_diag_update_positive=None,
is_non_singular=None,
is_self_adjoint=None,
is_positive_definite=None,
is_square=None,
name="LinearOperatorLowRankUpdate"):
"""Initialize a `LinearOperatorLowRankUpdate`.
This creates a `LinearOperator` of the form `A = L + U D V^H`, with
`L` a `LinearOperator`, `U, V` both [batch] matrices, and `D` a [batch]
diagonal matrix.
If `L` is non-singular, solves and determinants are available.
Solves/determinants both involve a solve/determinant of a `K x K` system.
In the event that L and D are self-adjoint positive-definite, and U = V,
this can be done using a Cholesky factorization. The user should set the
`is_X` matrix property hints, which will trigger the appropriate code path.
Args:
base_operator: Shape `[B1,...,Bb, M, N]`.
u: Shape `[B1,...,Bb, M, K]` `Tensor` of same `dtype` as `base_operator`.
This is `U` above.
diag_update: Optional shape `[B1,...,Bb, K]` `Tensor` with same `dtype`
as `base_operator`. This is the diagonal of `D` above.
Defaults to `D` being the identity operator.
v: Optional `Tensor` of same `dtype` as `u` and shape `[B1,...,Bb, N, K]`
Defaults to `v = u`, in which case the perturbation is symmetric.
If `M != N`, then `v` must be set since the perturbation is not square.
is_diag_update_positive: Python `bool`.
If `True`, expect `diag_update > 0`.
is_non_singular: Expect that this operator is non-singular.
Default is `None`, unless `is_positive_definite` is auto-set to be
`True` (see below).
is_self_adjoint: Expect that this operator is equal to its hermitian
transpose. Default is `None`, unless `base_operator` is self-adjoint
and `v = None` (meaning `u=v`), in which case this defaults to `True`.
is_positive_definite: Expect that this operator is positive definite.
Default is `None`, unless `base_operator` is positive-definite
`v = None` (meaning `u=v`), and `is_diag_update_positive`, in which case
this defaults to `True`.
Note that we say an operator is positive definite when the quadratic
form `x^H A x` has positive real part for all nonzero `x`.
is_square: Expect that this operator acts like square [batch] matrices.
name: A name for this `LinearOperator`.
Raises:
ValueError: If `is_X` flags are set in an inconsistent way.
"""
dtype = base_operator.dtype
if diag_update is not None:
if is_diag_update_positive and np.issubdtype(
dtype, np.complexfloating):
logging.warn(
"Note: setting is_diag_update_positive with a complex "
"dtype means that diagonal is real and positive.")
if diag_update is None:
if is_diag_update_positive is False:
raise ValueError(
"Default diagonal is the identity, which is positive. However, "
"user set 'is_diag_update_positive' to False.")
is_diag_update_positive = True
# In this case, we can use a Cholesky decomposition to help us solve/det.
self._use_cholesky = (base_operator.is_positive_definite
and base_operator.is_self_adjoint
and is_diag_update_positive and v is None)
# Possibly auto-set some characteristic flags from None to True.
# If the Flags were set (by the user) incorrectly to False, then raise.
if base_operator.is_self_adjoint and v is None and not np.issubdtype(
dtype, np.complexfloating):
if is_self_adjoint is False:
raise ValueError(
"A = L + UDU^H, with L self-adjoint and D real diagonal. Since"
" UDU^H is self-adjoint, this must be a self-adjoint operator."
)
is_self_adjoint = True
# The condition for using a cholesky is sufficient for SPD, and
# we no weaker choice of these hints leads to SPD. Therefore,
# the following line reads "if hints indicate SPD..."
if self._use_cholesky:
if (is_positive_definite is False or is_self_adjoint is False
or is_non_singular is False):
raise ValueError(
"Arguments imply this is self-adjoint positive-definite operator."
)
is_positive_definite = True
is_self_adjoint = True
values = base_operator.graph_parents + [u, diag_update, v]
with ops.name_scope(name, values=values):
# Create U and V.
self._u = ops.convert_to_tensor(u, name="u")
if v is None:
self._v = self._u
else:
self._v = ops.convert_to_tensor(v, name="v")
if diag_update is None:
self._diag_update = None
else:
self._diag_update = ops.convert_to_tensor(diag_update,
name="diag_update")
# Create base_operator L.
self._base_operator = base_operator
graph_parents = base_operator.graph_parents + [
self.u, self._diag_update, self.v
]
graph_parents = [p for p in graph_parents if p is not None]
super(LinearOperatorLowRankUpdate,
self).__init__(dtype=self._base_operator.dtype,
graph_parents=graph_parents,
is_non_singular=is_non_singular,
is_self_adjoint=is_self_adjoint,
is_positive_definite=is_positive_definite,
is_square=is_square,
name=name)
# Create the diagonal operator D.
self._set_diag_operators(diag_update, is_diag_update_positive)
self._is_diag_update_positive = is_diag_update_positive
self._check_shapes()
# Pre-compute the so-called "capacitance" matrix
# C := D^{-1} + V^H L^{-1} U
self._capacitance = self._make_capacitance()
if self._use_cholesky:
self._chol_capacitance = linalg_ops.cholesky(self._capacitance)