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Combines LinearOperators
into a blockwise lower-triangular matrix.
Inherits From: LinearOperator
, Module
tf.linalg.LinearOperatorBlockLowerTriangular(
operators,
is_non_singular=None,
is_self_adjoint=None,
is_positive_definite=None,
is_square=None,
name='LinearOperatorBlockLowerTriangular'
)
This operator is initialized with a nested list of linear operators, which
are combined into a new LinearOperator
whose underlying matrix
representation is square and has each operator on or below the main diagonal,
and zero's elsewhere. Each element of the outer list is a list of
LinearOperators
corresponding to a row-partition of the blockwise structure.
The number of LinearOperator
s in row-partion i
must be equal to i
.
For example, a blockwise 3 x 3
LinearOperatorBlockLowerTriangular
is
initialized with the list [[op_00], [op_10, op_11], [op_20, op_21, op_22]]
,
where the op_ij
, i < 3, j <= i
, are LinearOperator
instances. The
LinearOperatorBlockLowerTriangular
behaves as the following blockwise
matrix, where 0
represents appropriately-sized [batch] matrices of zeros:
[[op_00, 0, 0],
[op_10, op_11, 0],
[op_20, op_21, op_22]]
Each op_jj
on the diagonal is required to represent a square matrix, and
hence will have shape batch_shape_j + [M_j, M_j]
. LinearOperator
s in row
j
of the blockwise structure must have range_dimension
equal to that of
op_jj
, and LinearOperators
in column j
must have domain_dimension
equal to that of op_jj
.
If each op_jj
on the diagonal has shape batch_shape_j + [M_j, M_j]
, then
the combined operator has shape broadcast_batch_shape + [sum M_j, sum M_j]
,
where broadcast_batch_shape
is the mutual broadcast of batch_shape_j
,
j = 0, 1, ..., J
, assuming the intermediate batch shapes broadcast.
Even if the combined shape is well defined, the combined operator's
methods may fail due to lack of broadcasting ability in the defining
operators' methods.
For example, to create a 4 x 4 linear operator combined of three 2 x 2 operators:
>>> operator_0 = tf.linalg.LinearOperatorFullMatrix([[1., 2.], [3., 4.]])
>>> operator_1 = tf.linalg.LinearOperatorFullMatrix([[1., 0.], [0., 1.]])
>>> operator_2 = tf.linalg.LinearOperatorLowerTriangular([[5., 6.], [7., 8]])
>>> operator = LinearOperatorBlockLowerTriangular(
... [[operator_0], [operator_1, operator_2]])
operator.to_dense()
<tf.Tensor: shape=(4, 4), dtype=float32, numpy=
array([[1., 2., 0., 0.],
[3., 4., 0., 0.],
[1., 0., 5., 0.],
[0., 1., 7., 8.]], dtype=float32)>
operator.shape
TensorShape([4, 4])
operator.log_abs_determinant()
<tf.Tensor: shape=(), dtype=float32, numpy=4.3820267>
x0 = [[1., 6.], [-3., 4.]]
x1 = [[0., 2.], [4., 0.]]
x = tf.concat([x0, x1], 0) # Shape [2, 4] Tensor
operator.matmul(x)
<tf.Tensor: shape=(4, 2), dtype=float32, numpy=
array([[-5., 14.],
[-9., 34.],
[ 1., 16.],
[29., 18.]], dtype=float32)>
The above matmul
is equivalent to:
>>> tf.concat([operator_0.matmul(x0),
... operator_1.matmul(x0) + operator_2.matmul(x1)], axis=0)
<tf.Tensor: shape=(4, 2), dtype=float32, numpy=
array([[-5., 14.],
[-9., 34.],
[ 1., 16.],
[29., 18.]], dtype=float32)>
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.
For example:
Create a [2, 3] batch of 4 x 4 linear operators:
>>> matrix_44 = tf.random.normal(shape=[2, 3, 4, 4])
>>> operator_44 = tf.linalg.LinearOperatorFullMatrix(matrix_44)
Create a [1, 3] batch of 5 x 4 linear operators:
>>> matrix_54 = tf.random.normal(shape=[1, 3, 5, 4])
>>> operator_54 = tf.linalg.LinearOperatorFullMatrix(matrix_54)
Create a [1, 3] batch of 5 x 5 linear operators:
>>> matrix_55 = tf.random.normal(shape=[1, 3, 5, 5])
>>> operator_55 = tf.linalg.LinearOperatorFullMatrix(matrix_55)
Combine to create a [2, 3] batch of 9 x 9 operators:
>>> operator_99 = LinearOperatorBlockLowerTriangular(
... [[operator_44], [operator_54, operator_55]])
>>> operator_99.shape
TensorShape([2, 3, 9, 9])
Create a shape [2, 1, 9] batch of vectors and apply the operator to it.
>>> x = tf.random.normal(shape=[2, 1, 9])
>>> y = operator_99.matvec(x)
>>> y.shape
TensorShape([2, 3, 9])
Create a blockwise list of vectors and apply the operator to it. A blockwise list is returned.
>>> x4 = tf.random.normal(shape=[2, 1, 4])
>>> x5 = tf.random.normal(shape=[2, 3, 5])
>>> y_blockwise = operator_99.matvec([x4, x5])
>>> y_blockwise[0].shape
TensorShape([2, 3, 4])
>>> y_blockwise[1].shape
TensorShape([2, 3, 5])
Performance
Suppose operator
is a LinearOperatorBlockLowerTriangular
consisting of D
row-partitions and D
column-partitions, such that the total number of
operators is N = D * (D + 1) // 2
.
operator.matmul
has complexity equal to the sum of thematmul
complexities of the individual operators.operator.solve
has complexity equal to the sum of thesolve
complexities of the operators on the diagonal and thematmul
complexities of the operators off the diagonal.operator.determinant
has complexity equal to the sum of thedeterminant
complexities of the operators on the diagonal.
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.
Raises | |
---|---|
TypeError
|
If all operators do not have the same dtype .
|
ValueError
|
If operators is empty, contains an erroneous number of
elements, or contains operators with incompatible shapes.
|
Methods
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.
|
cond
cond(
name='cond'
)
Returns the condition number of this linear operator.
Args | |
---|---|
name
|
A name for this Op .
|
Returns | |
---|---|
Shape [B1,...,Bb] Tensor of same dtype as self .
|
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
|
eigvals
eigvals(
name='eigvals'
)
Returns the eigenvalues of this linear operator.
If the operator is marked as self-adjoint (via is_self_adjoint
)
this computation can be more efficient.
Args | |
---|---|
name
|
A name for this Op .
|
Returns | |
---|---|
Shape [B1,...,Bb, N] Tensor of same dtype as self .
|
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 , Tensor with compatible shape and same dtype as
self , or a blockwise iterable of LinearOperator s or Tensor s. See
class docstring for definition of shape 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 , or if x is blockwise, a list of Tensor s with shapes that
concatenate to [..., M, R] .
|
matvec
matvec(
x, adjoint=False, name='matvec'
)
Transform [batch] vector x
with left multiplication: x --> Ax
.
# Make an operator acting like batch matrix 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 , or an
iterable of Tensor s. Tensor s are treated a [batch] vectors, 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.
Given the blockwise n + 1
-by-n + 1
linear operator:
op = [[A_00 0 ... 0 ... 0], [A_10 A_11 ... 0 ... 0], ... [A_k0 A_k1 ... A_kk ... 0], ... [A_n0 A_n1 ... A_nk ... A_nn]]
we find x = op.solve(y)
by observing that
y_k = A_k0.matmul(x_0) + A_k1.matmul(x_1) + ... + A_kk.matmul(x_k)
and therefore
x_k = A_kk.solve(y_k -
A_k0.matmul(x_0) - ... - A_k(k-1).matmul(x_(k-1)))
where x_k
and y_k
are the k
th blocks obtained by decomposing x
and y
along their appropriate axes.
We first solve x_0 = A_00.solve(y_0)
. Proceeding inductively, we solve
for x_k
, k = 1..n
, given x_0..x_(k-1)
.
The adjoint case is solved similarly, beginning with
x_n = A_nn.solve(y_n, adjoint=True)
and proceeding backwards.
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 same dtype as this operator and compatible shape,
or a list of Tensor s. Tensor s are treated like a [batch] matrices
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.
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 same dtype as this operator, or list of Tensor s
(for blockwise operators). Tensor s are treated as [batch] vectors,
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 .
|
__getitem__
__getitem__(
slices
)
__matmul__
__matmul__(
other
)