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LinearOperator
that wraps a [batch] matrix.
Inherits From: LinearOperator
tf.linalg.LinearOperatorFullMatrix( matrix, is_non_singular=None, is_self_adjoint=None, is_positive_definite=None, is_square=None, name='LinearOperatorFullMatrix' )
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)
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.
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.
dtype
is real, and is_self_adjoint
and is_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)
is O(M * N * R)
.M=N
, operator.solve(x)
is O(N^3 * R)
.M=N
, operator.determinant()
is O(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
.
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:
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.Args | |
---|---|
matrix | Shape [B1,...,Bb, M, N] with b >= 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 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. See: https://en.wikipedia.org/wiki/Positive-definite_matrix#Extension_for_non-symmetric_matrices |
is_square | Expect that this operator acts like square [batch] matrices. |
name | A name for this LinearOperator . |
Raises | |
---|---|
TypeError | If diag.dtype is not an allowed type. |
Attributes | |
---|---|
H | Returns the adjoint of the current LinearOperator . Given |
batch_shape | TensorShape of batch dimensions of this LinearOperator . If this operator acts like the batch matrix |
domain_dimension | Dimension (in the sense of vector spaces) of the domain of this operator. If this operator acts like the batch matrix |
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. |
range_dimension | Dimension (in the sense of vector spaces) of the range of this operator. If this operator acts like the batch matrix |
shape | TensorShape of this LinearOperator . If this operator acts like the batch matrix |
tensor_rank | Rank (in the sense of tensors) of matrix corresponding to this operator. If this operator acts like the batch matrix |
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 same dtype and shape broadcastable to self.shape . |
name | A name to give this Op . |
Returns | |
---|---|
A Tensor with broadcast shape and same dtype as self . |
adjoint
adjoint( name='adjoint' )
Returns the adjoint of the current LinearOperator
.
Given A
representing this LinearOperator
, return A*
. Note that calling self.adjoint()
and self.H
are equivalent.
Args | |
---|---|
name | A name for this Op . |
Returns | |
---|---|
LinearOperator which represents the adjoint of this LinearOperator . |
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. |
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 this Op . |
Returns | |
---|---|
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.
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. |
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 this Op . |
Returns | |
---|---|
int32 Tensor |
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 this Op . |
Returns | |
---|---|
LinearOperator which represents the lower triangular matrix in the Cholesky decomposition. |
Raises | |
---|---|
ValueError | When the LinearOperator is not hinted to be positive definite and self adjoint. |
determinant
determinant( name='det' )
Determinant for every batch member.
Args | |
---|---|
name | A name for this Op . |
Returns | |
---|---|
Tensor with shape self.batch_shape and same dtype as self . |
Raises | |
---|---|
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.linalg.diag_part(my_operator.to_dense()) ==> [1., 2.]
Args | |
---|---|
name | A name for this Op . |
Returns | |
---|---|
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
.
Args | |
---|---|
name | A name for this Op . |
Returns | |
---|---|
int32 Tensor |
inverse
inverse( name='inverse' )
Returns the Inverse of this LinearOperator
.
Given A
representing this LinearOperator
, return a LinearOperator
representing A^-1
.
Args | |
---|---|
name | A name scope to use for ops added by this method. |
Returns | |
---|---|
LinearOperator representing inverse of this matrix. |
Raises | |
---|---|
ValueError | When the LinearOperator is not hinted to be non_singular . |
log_abs_determinant
log_abs_determinant( name='log_abs_det' )
Log absolute value of determinant for every batch member.
Args | |
---|---|
name | A name for this Op . |
Returns | |
---|---|
Tensor with shape self.batch_shape and same dtype as self . |
Raises | |
---|---|
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]
Args | |
---|---|
x | LinearOperator or 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 . |
Returns | |
---|---|
A LinearOperator or 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]
Args | |
---|---|
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 . |
Returns | |
---|---|
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
.
Args | |
---|---|
name | A name for this Op . |
Returns | |
---|---|
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)
.
Args | |
---|---|
name | A name for this Op . |
Returns | |
---|---|
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.
# 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 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. |
Returns | |
---|---|
Tensor with shape [...,N, R] and same dtype as rhs . |
Raises | |
---|---|
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.
# 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 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. |
Returns | |
---|---|
Tensor with shape [...,N] and same dtype as rhs . |
Raises | |
---|---|
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
.
Args | |
---|---|
name | A name for this Op . |
Returns | |
---|---|
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.
Args | |
---|---|
name | A name for this Op . |
Returns | |
---|---|
Shape [B1,...,Bb] Tensor of same dtype as self . |
© 2020 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/versions/r1.15/api_docs/python/tf/linalg/LinearOperatorFullMatrix