Class LinearOperatorFullMatrix
Inherits From: LinearOperator
Defined in tensorflow/python/ops/linalg/linear_operator_full_matrix.py
.
LinearOperator
that wraps a [batch] matrix.
This operator wraps a [batch] matrix A
(which is a Tensor
) 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.
# Create a 2 x 2 linear operator.
matrix = [[1., 2.], [3., 4.]]
operator = LinearOperatorFullMatrix(matrix)
operator.to_dense()
==> [[1., 2.]
[3., 4.]]
operator.shape
==> [2, 2]
operator.log_abs_determinant()
==> scalar Tensor
x = ... Shape [2, 4] Tensor
operator.matmul(x)
==> Shape [2, 4] Tensor
# Create a [2, 3] batch of 4 x 4 linear operators.
matrix = tf.random_normal(shape=[2, 3, 4, 4])
operator = LinearOperatorFullMatrix(matrix)
Shape compatibility
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.
Performance
LinearOperatorFullMatrix
has exactly the same performance as would be
achieved by using standard TensorFlow
matrix ops. Intelligent choices are
made based on the following initialization hints.
- If
dtype
is real, andis_self_adjoint
andis_positive_definite
, a Cholesky factorization is used for the determinant and solve.
In all cases, suppose operator
is a LinearOperatorFullMatrix
of shape
[M, N]
, and x.shape = [N, R]
. Then
operator.matmul(x)
isO(M * N * R)
.- If
M=N
,operator.solve(x)
isO(N^3 * R)
. - If
M=N
,operator.determinant()
isO(N^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
.
Matrix property hints
This LinearOperator
is initialized with boolean flags of the form is_X
,
for X = non_singular, self_adjoint, positive_definite, square
.
These have the following meaning:
- If
is_X == True
, callers should expect the operator to have the propertyX
. 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. - If
is_X == False
, callers should expect the operator to not haveX
. - If
is_X == None
(the default), callers should have no expectation either way.
__init__
__init__(
matrix,
is_non_singular=None,
is_self_adjoint=None,
is_positive_definite=None,
is_square=None,
name='LinearOperatorFullMatrix'
)
Initialize a LinearOperatorFullMatrix
.
Args:
matrix
: Shape[B1,...,Bb, M, N]
withb >= 0
,M, N >= 0
. Allowed dtypes:float16
,float32
,float64
,complex64
,complex128
.is_non_singular
: Expect that this operator is non-singular.is_self_adjoint
: Expect that this operator is equal to its hermitian transpose.is_positive_definite
: Expect that this operator is positive definite, meaning the quadratic formx^H A x
has positive real part for all nonzerox
. Note that we do not require the operator to be self-adjoint to be positive-definite. See: https://en.wikipedia.org/wiki/Positive-definite_matrix#Extension_for_non-symmetric_matricesis_square
: Expect that this operator acts like square [batch] matrices.name
: A name for thisLinearOperator
.
Raises:
TypeError
: Ifdiag.dtype
is not an allowed type.
Properties
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]
Returns:
TensorShape
, statically determined, may be undefined.
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
.
Returns:
Dimension
object.
dtype
The DType
of Tensor
s handled by this LinearOperator
.
graph_parents
List of graph dependencies of this LinearOperator
.
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
.
Returns:
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()
.
Returns:
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
.
Args:
name
: A name for thisOp
.
Returns:
Python integer, or None if the tensor rank is undefined.
Methods
tf.linalg.LinearOperatorFullMatrix.add_to_tensor
add_to_tensor(
x,
name='add_to_tensor'
)
Add matrix represented by this operator to x
. Equivalent to A + x
.
Args:
x
:Tensor
with samedtype
and shape broadcastable toself.shape
.name
: A name to give thisOp
.
Returns:
A Tensor
with broadcast shape and same dtype
as self
.
tf.linalg.LinearOperatorFullMatrix.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
Args:
name
: A string name to prepend to created ops.
Returns:
An Assert
Op
, that, when run, will raise an InvalidArgumentError
if
the operator is singular.
tf.linalg.LinearOperatorFullMatrix.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.
Args:
name
: A name to give thisOp
.
Returns:
An Assert
Op
, that, when run, will raise an InvalidArgumentError
if
the operator is not positive definite.
tf.linalg.LinearOperatorFullMatrix.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.
Args:
name
: A string name to prepend to created ops.
Returns:
An Assert
Op
, that, when run, will raise an InvalidArgumentError
if
the operator is not self-adjoint.
tf.linalg.LinearOperatorFullMatrix.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]
.
Args:
name
: A name for thisOp
.
Returns:
int32
Tensor
tf.linalg.LinearOperatorFullMatrix.cholesky
cholesky(name='cholesky')
Returns a Cholesky factor as a LinearOperator
.
Given A
representing this LinearOperator
, if A
is positive definite
self-adjoint, return L
, where A = L L^T
, i.e. the cholesky
decomposition.
Args:
name
: A name for thisOp
.
Returns:
LinearOperator
which represents the lower triangular matrix
in the Cholesky decomposition.
Raises:
ValueError
: When theLinearOperator
is not hinted to be positive definite and self adjoint.
tf.linalg.LinearOperatorFullMatrix.determinant
determinant(name='det')
Determinant for every batch member.
Args:
name
: A name for thisOp
.
Returns:
Tensor
with shape self.batch_shape
and same dtype
as self
.
Raises:
NotImplementedError
: Ifself.is_square
isFalse
.
tf.linalg.LinearOperatorFullMatrix.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.]
Args:
name
: A name for thisOp
.
Returns:
diag_part
: ATensor
of samedtype
as self.
tf.linalg.LinearOperatorFullMatrix.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
.
Args:
name
: A name for thisOp
.
Returns:
int32
Tensor
tf.linalg.LinearOperatorFullMatrix.log_abs_determinant
log_abs_determinant(name='log_abs_det')
Log absolute value of determinant for every batch member.
Args:
name
: A name for thisOp
.
Returns:
Tensor
with shape self.batch_shape
and same dtype
as self
.
Raises:
NotImplementedError
: Ifself.is_square
isFalse
.
tf.linalg.LinearOperatorFullMatrix.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]
Args:
x
:LinearOperator
orTensor
with compatible shape and samedtype
asself
. See class docstring for definition of compatibility.adjoint
: Pythonbool
. IfTrue
, left multiply by the adjoint:A^H x
.adjoint_arg
: Pythonbool
. IfTrue
, computeA x^H
wherex^H
is the hermitian transpose (transposition and complex conjugation).name
: A name for thisOp
.
Returns:
A LinearOperator
or Tensor
with shape [..., M, R]
and same dtype
as self
.
tf.linalg.LinearOperatorFullMatrix.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]
Args:
x
:Tensor
with compatible shape and samedtype
asself
.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
: Pythonbool
. IfTrue
, left multiply by the adjoint:A^H x
.name
: A name for thisOp
.
Returns:
A Tensor
with shape [..., M]
and same dtype
as self
.
tf.linalg.LinearOperatorFullMatrix.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
.
Args:
name
: A name for thisOp
.
Returns:
int32
Tensor
tf.linalg.LinearOperatorFullMatrix.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)
.
Args:
name
: A name for thisOp
.
Returns:
int32
Tensor
tf.linalg.LinearOperatorFullMatrix.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
Args:
rhs
:Tensor
with samedtype
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
: Pythonbool
. IfTrue
, solve the system involving the adjoint of thisLinearOperator
:A^H X = rhs
.adjoint_arg
: Pythonbool
. IfTrue
, solveA X = rhs^H
whererhs^H
is the hermitian transpose (transposition and complex conjugation).name
: A name scope to use for ops added by this method.
Returns:
Tensor
with shape [...,N, R]
and same dtype
as rhs
.
Raises:
NotImplementedError
: Ifself.is_non_singular
oris_square
is False.
tf.linalg.LinearOperatorFullMatrix.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
Args:
rhs
:Tensor
with samedtype
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
: Pythonbool
. IfTrue
, solve the system involving the adjoint of thisLinearOperator
:A^H X = rhs
.name
: A name scope to use for ops added by this method.
Returns:
Tensor
with shape [...,N]
and same dtype
as rhs
.
Raises:
NotImplementedError
: Ifself.is_non_singular
oris_square
is False.
tf.linalg.LinearOperatorFullMatrix.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
.
Args:
name
: A name for thisOp
.
Returns:
int32
Tensor
, determined at runtime.
tf.linalg.LinearOperatorFullMatrix.to_dense
to_dense(name='to_dense')
Return a dense (batch) matrix representing this operator.
tf.linalg.LinearOperatorFullMatrix.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.
Args:
name
: A name for thisOp
.
Returns:
Shape [B1,...,Bb]
Tensor
of same dtype
as self
.