LinearOperatorLowRankUpdate
Inherits From: LinearOperator
tf.contrib.linalg.LinearOperatorLowRankUpdate
tf.linalg.LinearOperatorLowRankUpdate
Defined in tensorflow/python/ops/linalg/linear_operator_low_rank_update.py
.
See the guide: Linear Algebra (contrib) > LinearOperator
Perturb a LinearOperator
with a rank K
update.
This operator acts like a [batch] matrix A
with shape [B1,...,Bb, M, N]
for some b >= 0
. The first b
indices index a batch member. For every batch index (i1,...,ib)
, A[i1,...,ib, : :]
is an M x N
matrix.
LinearOperatorLowRankUpdate
represents A = L + U D V^H
, where
L, is a LinearOperator representing [batch] M x N matrices U, is a [batch] M x K matrix. Typically K << M. D, is a [batch] K x K matrix. V, is a [batch] N x K matrix. Typically K << N. V^H is the Hermitian transpose (adjoint) of V.
If M = N
, determinants and solves are done using the matrix determinant lemma and Woodbury identities, and thus require L and D to be non-singular.
Solves and determinants will be attempted unless the "is_non_singular" property of L and D is False.
In the event that L and D are positive-definite, and U = V, solves and determinants can be done using a Cholesky factorization.
# Create a 3 x 3 diagonal linear operator. diag_operator = LinearOperatorDiag( diag_update=[1., 2., 3.], is_non_singular=True, is_self_adjoint=True, is_positive_definite=True) # Perturb with a rank 2 perturbation operator = LinearOperatorLowRankUpdate( operator=diag_operator, u=[[1., 2.], [-1., 3.], [0., 0.]], diag_update=[11., 12.], v=[[1., 2.], [-1., 3.], [10., 10.]]) operator.shape ==> [3, 3] operator.log_abs_determinant() ==> scalar Tensor x = ... Shape [3, 4] Tensor operator.matmul(x) ==> Shape [3, 4] Tensor
This operator acts on [batch] matrix with compatible shape. x
is a batch matrix with compatible shape for matmul
and solve
if
operator.shape = [B1,...,Bb] + [M, N], with b >= 0 x.shape = [B1,...,Bb] + [N, R], with R >= 0.
Suppose operator
is a LinearOperatorLowRankUpdate
of shape [M, N]
, made from a rank K
update of base_operator
which performs .matmul(x)
on x
having x.shape = [N, R]
with O(L_matmul*N*R)
complexity (and similarly for solve
, determinant
. Then, if x.shape = [N, R]
,
operator.matmul(x)
is O(L_matmul*N*R + K*N*R)
and if M = N
,
operator.solve(x)
is O(L_matmul*N*R + N*K*R + K^2*R + K^3)
operator.determinant()
is O(L_determinant + L_solve*N*K + K^2*N + K^3)
If instead operator
and x
have shape [B1,...,Bb, M, N]
and [B1,...,Bb, N, R]
, every operation increases in complexity by B1*...*Bb
.
This LinearOperator
is initialized with boolean flags of the form is_X
, for X = non_singular
, self_adjoint
, positive_definite
, diag_update_positive
and square
. These have the following meaning:
is_X == True
, callers should expect the operator to have the property X
. This is a promise that should be fulfilled, but is not a runtime assert. For example, finite floating point precision may result in these promises being violated.is_X == False
, callers should expect the operator to not have X
.is_X == None
(the default), callers should have no expectation either way.base_operator
If this operator is A = L + U D V^H
, this is the L
.
batch_shape
TensorShape
of batch dimensions of this LinearOperator
.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns TensorShape([B1,...,Bb])
, equivalent to A.get_shape()[:-2]
TensorShape
, statically determined, may be undefined.
diag_operator
If this operator is A = L + U D V^H
, this is D
.
diag_update
If this operator is A = L + U D V^H
, this is the diagonal of D
.
domain_dimension
Dimension (in the sense of vector spaces) of the domain of this operator.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns N
.
Dimension
object.
dtype
The DType
of Tensor
s handled by this LinearOperator
.
graph_parents
List of graph dependencies of this LinearOperator
.
is_diag_update_positive
If this operator is A = L + U D V^H
, this hints D > 0
elementwise.
is_non_singular
is_positive_definite
is_self_adjoint
is_square
Return True/False
depending on if this operator is square.
name
Name prepended to all ops created by this LinearOperator
.
range_dimension
Dimension (in the sense of vector spaces) of the range of this operator.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns M
.
Dimension
object.
shape
TensorShape
of this LinearOperator
.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns TensorShape([B1,...,Bb, M, N])
, equivalent to A.get_shape()
.
TensorShape
, statically determined, may be undefined.
tensor_rank
Rank (in the sense of tensors) of matrix corresponding to this operator.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns b + 2
.
name
: A name for this `Op.Python integer, or None if the tensor rank is undefined.
u
If this operator is A = L + U D V^H
, this is the U
.
v
If this operator is A = L + U D V^H
, this is the V
.
__init__
__init__( 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.
base_operator
: Shape [B1,...,Bb, M, N]
real float16
, float32
or float64
LinearOperator
. This is L
above.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
.ValueError
: If is_X
flags are set in an inconsistent way.add_to_tensor
add_to_tensor( x, name='add_to_tensor' )
Add matrix represented by this operator to x
. Equivalent to A + x
.
x
: Tensor
with same dtype
and shape broadcastable to self.shape
.name
: A name to give this Op
.A Tensor
with broadcast shape and same dtype
as self
.
assert_non_singular
assert_non_singular(name='assert_non_singular')
Returns an Op
that asserts this operator is non singular.
This operator is considered non-singular if
ConditionNumber < max{100, range_dimension, domain_dimension} * eps, eps := np.finfo(self.dtype.as_numpy_dtype).eps
name
: A string name to prepend to created ops.An Assert
Op
, that, when run, will raise an InvalidArgumentError
if the operator is singular.
assert_positive_definite
assert_positive_definite(name='assert_positive_definite')
Returns an Op
that asserts this operator is positive definite.
Here, positive definite means that the quadratic form x^H A x
has positive real part for all nonzero x
. Note that we do not require the operator to be self-adjoint to be positive definite.
name
: A name to give this Op
.An Assert
Op
, that, when run, will raise an InvalidArgumentError
if the operator is not positive definite.
assert_self_adjoint
assert_self_adjoint(name='assert_self_adjoint')
Returns an Op
that asserts this operator is self-adjoint.
Here we check that this operator is exactly equal to its hermitian transpose.
name
: A string name to prepend to created ops.An Assert
Op
, that, when run, will raise an InvalidArgumentError
if the operator is not self-adjoint.
batch_shape_tensor
batch_shape_tensor(name='batch_shape_tensor')
Shape of batch dimensions of this operator, determined at runtime.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns a Tensor
holding [B1,...,Bb]
.
name
: A name for this `Op.int32
Tensor
determinant
determinant(name='det')
Determinant for every batch member.
name
: A name for this `Op.Tensor
with shape self.batch_shape
and same dtype
as self
.
NotImplementedError
: If self.is_square
is False
.diag_part
diag_part(name='diag_part')
Efficiently get the [batch] diagonal part of this operator.
If this operator has shape [B1,...,Bb, M, N]
, this returns a Tensor
diagonal
, of shape [B1,...,Bb, min(M, N)]
, where diagonal[b1,...,bb, i] = self.to_dense()[b1,...,bb, i, i]
.
my_operator = LinearOperatorDiag([1., 2.]) # Efficiently get the diagonal my_operator.diag_part() ==> [1., 2.] # Equivalent, but inefficient method tf.matrix_diag_part(my_operator.to_dense()) ==> [1., 2.]
name
: A name for this Op
.diag_part
: A Tensor
of same dtype
as self.domain_dimension_tensor
domain_dimension_tensor(name='domain_dimension_tensor')
Dimension (in the sense of vector spaces) of the domain of this operator.
Determined at runtime.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns N
.
name
: A name for this Op
.int32
Tensor
log_abs_determinant
log_abs_determinant(name='log_abs_det')
Log absolute value of determinant for every batch member.
name
: A name for this `Op.Tensor
with shape self.batch_shape
and same dtype
as self
.
NotImplementedError
: If self.is_square
is False
.matmul
matmul( x, adjoint=False, adjoint_arg=False, name='matmul' )
Transform [batch] matrix x
with left multiplication: x --> Ax
.
# Make an operator acting like batch matrix A. Assume A.shape = [..., M, N] operator = LinearOperator(...) operator.shape = [..., M, N] X = ... # shape [..., N, R], batch matrix, R > 0. Y = operator.matmul(X) Y.shape ==> [..., M, R] Y[..., :, r] = sum_j A[..., :, j] X[j, r]
x
: Tensor
with compatible shape and same dtype
as self
. See class docstring for definition of compatibility.adjoint
: Python bool
. If True
, left multiply by the adjoint: A^H x
.adjoint_arg
: Python bool
. If True
, compute A x^H
where x^H
is the hermitian transpose (transposition and complex conjugation).name
: A name for this `Op.A Tensor
with shape [..., M, R]
and same dtype
as self
.
matvec
matvec( x, adjoint=False, name='matvec' )
Transform [batch] vector x
with left multiplication: x --> Ax
.
# Make an operator acting like batch matric A. Assume A.shape = [..., M, N] operator = LinearOperator(...) X = ... # shape [..., N], batch vector Y = operator.matvec(X) Y.shape ==> [..., M] Y[..., :] = sum_j A[..., :, j] X[..., j]
x
: Tensor
with compatible shape and same dtype
as self
. x
is treated as a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility.adjoint
: Python bool
. If True
, left multiply by the adjoint: A^H x
.name
: A name for this `Op.A Tensor
with shape [..., M]
and same dtype
as self
.
range_dimension_tensor
range_dimension_tensor(name='range_dimension_tensor')
Dimension (in the sense of vector spaces) of the range of this operator.
Determined at runtime.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns M
.
name
: A name for this Op
.int32
Tensor
shape_tensor
shape_tensor(name='shape_tensor')
Shape of this LinearOperator
, determined at runtime.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns a Tensor
holding [B1,...,Bb, M, N]
, equivalent to tf.shape(A)
.
name
: A name for this `Op.int32
Tensor
solve
solve( rhs, adjoint=False, adjoint_arg=False, name='solve' )
Solve (exact or approx) R
(batch) systems of equations: A X = rhs
.
The returned Tensor
will be close to an exact solution if A
is well conditioned. Otherwise closeness will vary. See class docstring for details.
Examples:
# Make an operator acting like batch matrix A. Assume A.shape = [..., M, N] operator = LinearOperator(...) operator.shape = [..., M, N] # Solve R > 0 linear systems for every member of the batch. RHS = ... # shape [..., M, R] X = operator.solve(RHS) # X[..., :, r] is the solution to the r'th linear system # sum_j A[..., :, j] X[..., j, r] = RHS[..., :, r] operator.matmul(X) ==> RHS
rhs
: Tensor
with same dtype
as this operator and compatible shape. rhs
is treated like a [batch] matrix meaning for every set of leading dimensions, the last two dimensions defines a matrix. See class docstring for definition of compatibility.adjoint
: Python bool
. If True
, solve the system involving the adjoint of this LinearOperator
: A^H X = rhs
.adjoint_arg
: Python bool
. If True
, solve A X = rhs^H
where rhs^H
is the hermitian transpose (transposition and complex conjugation).name
: A name scope to use for ops added by this method.Tensor
with shape [...,N, R]
and same dtype
as rhs
.
NotImplementedError
: If self.is_non_singular
or is_square
is False.solvevec
solvevec( rhs, adjoint=False, name='solve' )
Solve single equation with best effort: A X = rhs
.
The returned Tensor
will be close to an exact solution if A
is well conditioned. Otherwise closeness will vary. See class docstring for details.
Examples:
# Make an operator acting like batch matrix A. Assume A.shape = [..., M, N] operator = LinearOperator(...) operator.shape = [..., M, N] # Solve one linear system for every member of the batch. RHS = ... # shape [..., M] X = operator.solvevec(RHS) # X is the solution to the linear system # sum_j A[..., :, j] X[..., j] = RHS[..., :] operator.matvec(X) ==> RHS
rhs
: Tensor
with same dtype
as this operator. rhs
is treated like a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility regarding batch dimensions.adjoint
: Python bool
. If True
, solve the system involving the adjoint of this LinearOperator
: A^H X = rhs
.name
: A name scope to use for ops added by this method.Tensor
with shape [...,N]
and same dtype
as rhs
.
NotImplementedError
: If self.is_non_singular
or is_square
is False.tensor_rank_tensor
tensor_rank_tensor(name='tensor_rank_tensor')
Rank (in the sense of tensors) of matrix corresponding to this operator.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns b + 2
.
name
: A name for this `Op.int32
Tensor
, determined at runtime.
to_dense
to_dense(name='to_dense')
Return a dense (batch) matrix representing this operator.
trace
trace(name='trace')
Trace of the linear operator, equal to sum of self.diag_part()
.
If the operator is square, this is also the sum of the eigenvalues.
name
: A name for this Op
.Shape [B1,...,Bb]
Tensor
of same dtype
as self
.
© 2018 The TensorFlow Authors. All rights reserved.
Licensed under the Creative Commons Attribution License 3.0.
Code samples licensed under the Apache 2.0 License.
https://www.tensorflow.org/api_docs/python/tf/linalg/LinearOperatorLowRankUpdate