def _matmul(self, x, adjoint=False, adjoint_arg=False):
arg_dim = -1 if adjoint_arg else -2
block_dimensions = (self._block_range_dimensions()
if adjoint else self._block_domain_dimensions())
blockwise_arg = linear_operator_util.arg_is_blockwise(
block_dimensions, x, arg_dim)
if blockwise_arg:
split_x = x
else:
split_dim = -1 if adjoint_arg else -2
# Split input by columns if adjoint_arg is True, else rows
split_x = linear_operator_util.split_arg_into_blocks(
self._block_domain_dimensions(),
self._block_domain_dimension_tensors,
x,
axis=split_dim)
result_list = []
# Iterate over row-partitions (i.e. column-partitions of the adjoint).
if adjoint:
for index in range(len(self.operators)):
# Begin with the operator on the diagonal and apply it to the
# respective `rhs` block.
result = self.operators[index][index].matmul(
split_x[index], adjoint=adjoint, adjoint_arg=adjoint_arg)
# Iterate top to bottom over the operators in the remainder of the
# column-partition (i.e. left to right over the row-partition of the
# adjoint), apply the operator to the respective `rhs` block and
# accumulate the sum. For example, given the
# `LinearOperatorBlockLowerTriangular`:
#
# op = [[A, 0, 0],
# [B, C, 0],
# [D, E, F]]
#
# if `index = 1`, the following loop calculates:
# `y_1 = (C.matmul(x_1, adjoint=adjoint) +
# E.matmul(x_2, adjoint=adjoint)`,
# where `x_1` and `x_2` are splits of `x`.
for j in range(index + 1, len(self.operators)):
result += self.operators[j][index].matmul(
split_x[j], adjoint=adjoint, adjoint_arg=adjoint_arg)
result_list.append(result)
else:
for row in self.operators:
# Begin with the left-most operator in the row-partition and apply it
# to the first `rhs` block.
result = row[0].matmul(split_x[0],
adjoint=adjoint,
adjoint_arg=adjoint_arg)
# Iterate left to right over the operators in the remainder of the row
# partition, apply the operator to the respective `rhs` block, and
# accumulate the sum.
for j, operator in enumerate(row[1:]):
result += operator.matmul(split_x[j + 1],
adjoint=adjoint,
adjoint_arg=adjoint_arg)
result_list.append(result)
if blockwise_arg:
return result_list
result_list = linear_operator_util.broadcast_matrix_batch_dims(
result_list)
return array_ops.concat(result_list, axis=-2)
def matmul(self, x, adjoint=False, adjoint_arg=False, name="matmul"):
"""Transform [batch] matrix `x` with left multiplication: `x --> Ax`.
```python
# Make an operator acting like batch matrix A. Assume tensor_shape.TensorShape(A.shape) = [..., M, N]
operator = LinearOperator(...)
tensor_shape.TensorShape(operator.shape) = [..., M, N]
X = ... # shape [..., N, R], batch matrix, R > 0.
Y = operator.matmul(X)
tensor_shape.TensorShape(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]`.
"""
if isinstance(x, linear_operator.LinearOperator):
left_operator = self.adjoint() if adjoint else self
right_operator = x.adjoint() if adjoint_arg else x
if (right_operator.range_dimension is not None
and left_operator.domain_dimension is not None
and right_operator.range_dimension !=
left_operator.domain_dimension):
raise ValueError(
"Operators are incompatible. Expected `x` to have dimension"
" {} but got {}.".format(left_operator.domain_dimension,
right_operator.range_dimension))
with self._name_scope(name):
return linear_operator_algebra.matmul(left_operator,
right_operator)
with self._name_scope(name):
arg_dim = -1 if adjoint_arg else -2
block_dimensions = (self._block_range_dimensions() if adjoint else
self._block_domain_dimensions())
if linear_operator_util.arg_is_blockwise(block_dimensions, x,
arg_dim):
for i, block in enumerate(x):
if not isinstance(block, linear_operator.LinearOperator):
block = ops.convert_to_tensor(block)
# self._check_input_dtype(block)
block_dimensions[i].assert_is_compatible_with(
tensor_shape.TensorShape(block.shape)[arg_dim])
x[i] = block
else:
x = ops.convert_to_tensor(x, name="x")
# self._check_input_dtype(x)
op_dimension = (self.range_dimension
if adjoint else self.domain_dimension)
op_dimension.assert_is_compatible_with(
tensor_shape.TensorShape(x.shape)[arg_dim])
return self._matmul(x, adjoint=adjoint, adjoint_arg=adjoint_arg)
def solve(self, 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:
```python
# Make an operator acting like batch matrix A. Assume tensor_shape.TensorShape(A.shape) = [..., M, N]
operator = LinearOperator(...)
tensor_shape.TensorShape(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 (for blockwise operators). `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.
"""
if self.is_non_singular is False:
raise NotImplementedError(
"Exact solve not implemented for an operator that is expected to "
"be singular.")
if self.is_square is False:
raise NotImplementedError(
"Exact solve not implemented for an operator that is expected to "
"not be square.")
if isinstance(rhs, linear_operator.LinearOperator):
left_operator = self.adjoint() if adjoint else self
right_operator = rhs.adjoint() if adjoint_arg else rhs
if (right_operator.range_dimension is not None and
left_operator.domain_dimension is not None and
right_operator.range_dimension != left_operator.domain_dimension):
raise ValueError(
"Operators are incompatible. Expected `rhs` to have dimension"
" {} but got {}.".format(
left_operator.domain_dimension, right_operator.range_dimension))
with self._name_scope(name):
return linear_operator_algebra.solve(left_operator, right_operator)
with self._name_scope(name):
block_dimensions = (self._block_domain_dimensions() if adjoint
else self._block_range_dimensions())
arg_dim = -1 if adjoint_arg else -2
blockwise_arg = linear_operator_util.arg_is_blockwise(
block_dimensions, rhs, arg_dim)
if blockwise_arg:
split_rhs = rhs
for i, block in enumerate(split_rhs):
if not isinstance(block, linear_operator.LinearOperator):
block = ops.convert_to_tensor(block)
# self._check_input_dtype(block)
block_dimensions[i].assert_is_compatible_with(tensor_shape.TensorShape(block.shape)[arg_dim])
split_rhs[i] = block
else:
rhs = ops.convert_to_tensor(rhs, name="rhs")
# self._check_input_dtype(rhs)
op_dimension = (self.domain_dimension if adjoint
else self.range_dimension)
op_dimension.assert_is_compatible_with(tensor_shape.TensorShape(rhs.shape)[arg_dim])
split_dim = -1 if adjoint_arg else -2
# Split input by rows normally, and otherwise columns.
split_rhs = linear_operator_util.split_arg_into_blocks(
self._block_domain_dimensions(),
self._block_domain_dimension_tensors,
rhs, axis=split_dim)
solution_list = []
for index, operator in enumerate(self.operators):
solution_list = solution_list + [operator.solve(
split_rhs[index], adjoint=adjoint, adjoint_arg=adjoint_arg)]
if blockwise_arg:
return solution_list
solution_list = linear_operator_util.broadcast_matrix_batch_dims(
solution_list)
return array_ops.concat(solution_list, axis=-2)
def solvevec(self, 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:
```python
# Make an operator acting like batch matrix A. Assume tensor_shape.TensorShape(A.shape) = [..., M, N]
operator = LinearOperator(...)
tensor_shape.TensorShape(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.
"""
with self._name_scope(name):
block_dimensions = (self._block_domain_dimensions() if adjoint
else self._block_range_dimensions())
if linear_operator_util.arg_is_blockwise(block_dimensions, rhs, -1):
for i, block in enumerate(rhs):
if not isinstance(block, linear_operator.LinearOperator):
block = ops.convert_to_tensor(block)
# self._check_input_dtype(block)
block_dimensions[i].assert_is_compatible_with(tensor_shape.TensorShape(block.shape)[-1])
rhs[i] = block
rhs_mat = [array_ops.expand_dims(block, axis=-1) for block in rhs]
solution_mat = self.solve(rhs_mat, adjoint=adjoint)
return [array_ops.squeeze(x, axis=-1) for x in solution_mat]
rhs = ops.convert_to_tensor(rhs, name="rhs")
# self._check_input_dtype(rhs)
op_dimension = (self.domain_dimension if adjoint
else self.range_dimension)
op_dimension.assert_is_compatible_with(tensor_shape.TensorShape(rhs.shape)[-1])
rhs_mat = array_ops.expand_dims(rhs, axis=-1)
solution_mat = self.solve(rhs_mat, adjoint=adjoint)
return array_ops.squeeze(solution_mat, axis=-1)
def solve(self, 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:
```python
# Make an operator acting like batch matrix A. Assume tensor_shape.TensorShape(A.shape) = [..., M, N]
operator = LinearOperator(...)
tensor_shape.TensorShape(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.
"""
if self.is_non_singular is False:
raise NotImplementedError(
"Exact solve not implemented for an operator that is expected to "
"be singular.")
if self.is_square is False:
raise NotImplementedError(
"Exact solve not implemented for an operator that is expected to "
"not be square.")
if isinstance(rhs, linear_operator.LinearOperator):
left_operator = self.adjoint() if adjoint else self
right_operator = rhs.adjoint() if adjoint_arg else rhs
if (right_operator.range_dimension is not None
and left_operator.domain_dimension is not None
and right_operator.range_dimension !=
left_operator.domain_dimension):
raise ValueError(
"Operators are incompatible. Expected `rhs` to have dimension"
" {} but got {}.".format(left_operator.domain_dimension,
right_operator.range_dimension))
with self._name_scope(name):
return linear_operator_algebra.solve(left_operator,
right_operator)
with self._name_scope(name):
block_dimensions = (self._block_domain_dimensions()
if adjoint else self._block_range_dimensions())
arg_dim = -1 if adjoint_arg else -2
blockwise_arg = linear_operator_util.arg_is_blockwise(
block_dimensions, rhs, arg_dim)
if blockwise_arg:
for i, block in enumerate(rhs):
if not isinstance(block, linear_operator.LinearOperator):
block = ops.convert_to_tensor(block)
# self._check_input_dtype(block)
block_dimensions[i].assert_is_compatible_with(
tensor_shape.TensorShape(block.shape)[arg_dim])
rhs[i] = block
if adjoint_arg:
split_rhs = [linalg.adjoint(y) for y in rhs]
else:
split_rhs = rhs
else:
rhs = ops.convert_to_tensor(rhs, name="rhs")
# self._check_input_dtype(rhs)
op_dimension = (self.domain_dimension
if adjoint else self.range_dimension)
op_dimension.assert_is_compatible_with(
tensor_shape.TensorShape(rhs.shape)[arg_dim])
rhs = linalg.adjoint(rhs) if adjoint_arg else rhs
split_rhs = linear_operator_util.split_arg_into_blocks(
self._block_domain_dimensions(),
self._block_domain_dimension_tensors,
rhs,
axis=-2)
solution_list = []
if adjoint:
# For an adjoint blockwise lower-triangular linear operator, the system
# must be solved bottom to top. Iterate backwards over rows of the
# adjoint (i.e. columns of the non-adjoint operator).
for index in reversed(range(len(self.operators))):
y = split_rhs[index]
# Iterate top to bottom over the operators in the off-diagonal portion
# of the column-partition (i.e. row-partition of the adjoint), apply
# the operator to the respective block of the solution found in
# previous iterations, and subtract the result from the `rhs` block.
# For example,let `A`, `B`, and `D` be the linear operators in the top
# row-partition of the adjoint of
# `LinearOperatorBlockLowerTriangular([[A], [B, C], [D, E, F]])`,
# and `x_1` and `x_2` be blocks of the solution found in previous
# iterations of the outer loop. The following loop (when `index == 0`)
# expresses
# `Ax_0 + Bx_1 + Dx_2 = y_0` as `Ax_0 = y_0*`, where
# `y_0* = y_0 - Bx_1 - Dx_2`.
for j in reversed(range(index + 1, len(self.operators))):
y -= self.operators[j][index].matmul(
solution_list[len(self.operators) - 1 - j],
adjoint=adjoint)
# Continuing the example above, solve `Ax_0 = y_0*` for `x_0`.
solution_list.append(self._diagonal_operators[index].solve(
y, adjoint=adjoint))
solution_list.reverse()
else:
# Iterate top to bottom over the row-partitions.
for row, y in zip(self.operators, split_rhs):
# Iterate left to right over the operators in the off-diagonal portion
# of the row-partition, apply the operator to the block of the
# solution found in previous iterations, and subtract the result from
# the `rhs` block. For example, let `D`, `E`, and `F` be the linear
# operators in the bottom row-partition of
# `LinearOperatorBlockLowerTriangular([[A], [B, C], [D, E, F]])` and
# `x_0` and `x_1` be blocks of the solution found in previous
# iterations of the outer loop. The following loop
# (when `index == 2`), expresses
# `Dx_0 + Ex_1 + Fx_2 = y_2` as `Fx_2 = y_2*`, where
# `y_2* = y_2 - D_x0 - Ex_1`.
for i, operator in enumerate(row[:-1]):
y -= operator.matmul(solution_list[i], adjoint=adjoint)
# Continuing the example above, solve `Fx_2 = y_2*` for `x_2`.
solution_list.append(row[-1].solve(y, adjoint=adjoint))
if blockwise_arg:
return solution_list
solution_list = linear_operator_util.broadcast_matrix_batch_dims(
solution_list)
return array_ops.concat(solution_list, axis=-2)
