def _assert_equal(x, y, data=None, summarize=None, message=None, name=None):
del summarize
del name
x = convert_to_tensor(x)
y = convert_to_tensor(y)
if not np.all(np.equal(x, y)):
raise ValueError('Expected x == y but got {} vs {} {} {}'.format(
x, y, message or '', data or ''))
def __init__(self,
col,
row,
is_non_singular=None,
is_self_adjoint=None,
is_positive_definite=None,
is_square=None,
name="LinearOperatorToeplitz"):
r"""Initialize a `LinearOperatorToeplitz`.
Args:
col: Shape `[B1,...,Bb, N]` `Tensor` with `b >= 0` `N >= 0`.
The first column of the operator. Allowed dtypes: `float16`, `float32`,
`float64`, `complex64`, `complex128`. Note that the first entry of
`col` is assumed to be the same as the first entry of `row`.
row: Shape `[B1,...,Bb, N]` `Tensor` with `b >= 0` `N >= 0`.
The first row of the operator. Allowed dtypes: `float16`, `float32`,
`float64`, `complex64`, `complex128`. Note that the first entry of
`row` is assumed to be the same as the first entry of `col`.
is_non_singular: Expect that this operator is non-singular.
is_self_adjoint: Expect that this operator is equal to its hermitian
transpose. If `diag.dtype` is real, this is auto-set to `True`.
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`.
"""
with ops.name_scope(name, values=[row, col]):
self._row = ops.convert_to_tensor(row, name="row")
self._col = ops.convert_to_tensor(col, name="col")
self._check_row_col(self._row, self._col)
circulant_col = array_ops.concat(
[self._col,
array_ops.zeros_like(self._col[..., 0:1]),
array_ops.reverse(self._row[..., 1:], axis=[-1])], axis=-1)
# To be used for matmul.
self._circulant = linear_operator_circulant.LinearOperatorCirculant(
fft_ops.fft(_to_complex(circulant_col)),
input_output_dtype=self._row.dtype)
if is_square is False: # pylint:disable=g-bool-id-comparison
raise ValueError("Only square Toeplitz operators currently supported.")
is_square = True
super(LinearOperatorToeplitz, self).__init__(
dtype=self._row.dtype,
graph_parents=[self._row, self._col],
is_non_singular=is_non_singular,
is_self_adjoint=is_self_adjoint,
is_positive_definite=is_positive_definite,
is_square=is_square,
name=name)
def matrix_triangular_solve_with_broadcast(matrix,
rhs,
lower=True,
adjoint=False,
name=None):
"""Solves triangular systems of linear equations with by backsubstitution.
Works identically to `tf.linalg.triangular_solve`, but broadcasts batch dims
of `matrix` and `rhs` (by replicating) if they are determined statically to be
different, or if static shapes are not fully defined. Thus, this may result
in an inefficient replication of data.
Args:
matrix: A Tensor. Must be one of the following types:
`float64`, `float32`, `complex64`, `complex128`. Shape is `[..., M, M]`.
rhs: A `Tensor`. Must have the same `dtype` as `matrix`.
Shape is `[..., M, K]`.
lower: An optional `bool`. Defaults to `True`. Indicates whether the
innermost matrices in `matrix` are lower or upper triangular.
adjoint: An optional `bool`. Defaults to `False`. Indicates whether to solve
with matrix or its (block-wise) adjoint.
name: A name for the operation (optional).
Returns:
`Tensor` with same `dtype` as `matrix` and shape `[..., M, K]`.
"""
with ops.name_scope(name, "MatrixTriangularSolve", [matrix, rhs]):
matrix = ops.convert_to_tensor(matrix, name="matrix")
rhs = ops.convert_to_tensor(rhs, name="rhs", dtype=matrix.dtype)
# If either matrix/rhs has extra dims, we can reshape to get rid of them.
matrix, rhs, reshape_inv, still_need_to_transpose = _reshape_for_efficiency(
matrix, rhs, adjoint_a=adjoint)
# lower indicates whether the matrix is lower triangular. If we have
# manually taken adjoint inside _reshape_for_efficiency, it is now upper tri
if not still_need_to_transpose and adjoint:
lower = not lower
# This will broadcast by brute force if we still need to.
matrix, rhs = broadcast_matrix_batch_dims([matrix, rhs])
solution = linalg_ops.triangular_solve(matrix,
rhs,
lower=lower,
adjoint=adjoint
and still_need_to_transpose)
return reshape_inv(solution)
def matvec(self, x, adjoint=False, name="matvec"):
"""Transform [batch] vector `x` with left multiplication: `x --> Ax`.
```python
# Make an operator acting like batch matric A. Assume _ops.TensorShape(A.shape) = [..., M, N]
operator = LinearOperator(...)
X = ... # shape [..., N], batch vector
Y = operator.matvec(X)
_ops.TensorShape(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`.
"""
with self._name_scope(name):
x = ops.convert_to_tensor(x, name="x")
# self._check_input_dtype(x)
self_dim = -2 if adjoint else -1
tensor_shape.dimension_at_index(_ops.TensorShape(
self.shape), self_dim).assert_is_compatible_with(
_ops.TensorShape(x.shape)[-1])
return self._matvec(x, adjoint=adjoint)
def _assert_positive(x, data=None, summarize=None, message=None, name=None):
del data
del summarize
del name
x = convert_to_tensor(x)
if np.any(x <= 0):
raise ValueError('Condition x > 0 did not hold; got {} {}'.format(
x, message or ''))
def shape_tensor(shape, name=None):
"""Convert Tensor using default type, unless empty list or tuple."""
# Works just like random_ops._ShapeTensor.
if isinstance(shape, (tuple, list)) and not shape:
dtype = dtypes.int32
else:
dtype = None
return ops.convert_to_tensor(shape, dtype=dtype, name=name)
def assert_no_entries_with_modulus_zero(
x, message=None, name="assert_no_entries_with_modulus_zero"):
"""Returns `Op` that asserts Tensor `x` has no entries with modulus zero.
Args:
x: Numeric `Tensor`, real, integer, or complex.
message: A string message to prepend to failure message.
name: A name to give this `Op`.
Returns:
An `Op` that asserts `x` has no entries with modulus zero.
"""
with ops.name_scope(name, values=[x]):
x = ops.convert_to_tensor(x, name="x")
dtype = x.dtype
should_be_nonzero = math_ops.abs(x)
zero = ops.convert_to_tensor(0, dtype=dtypes.real_dtype(dtype))
return check_ops.assert_less(zero, should_be_nonzero, message=message)
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 _ops.TensorShape(A.shape) = [..., M, N]
operator = LinearOperator(...)
_ops.TensorShape(operator.shape) = [..., M, N]
X = ... # shape [..., N, R], batch matrix, R > 0.
Y = operator.matmul(X)
_ops.TensorShape(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`.
"""
if isinstance(x, 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):
x = ops.convert_to_tensor(x, name="x")
# self._check_input_dtype(x)
self_dim = -2 if adjoint else -1
arg_dim = -1 if adjoint_arg else -2
tensor_shape.dimension_at_index(_ops.TensorShape(
self.shape), self_dim).assert_is_compatible_with(
_ops.TensorShape(x.shape)[arg_dim])
return self._matmul(x, adjoint=adjoint, adjoint_arg=adjoint_arg)
def matrix_solve_with_broadcast(matrix, rhs, adjoint=False, name=None):
"""Solve systems of linear equations."""
with ops.name_scope(name, "MatrixSolveWithBroadcast", [matrix, rhs]):
matrix = ops.convert_to_tensor(matrix, name="matrix")
rhs = ops.convert_to_tensor(rhs, name="rhs", dtype=matrix.dtype)
# If either matrix/rhs has extra dims, we can reshape to get rid of them.
matrix, rhs, reshape_inv, still_need_to_transpose = _reshape_for_efficiency(
matrix, rhs, adjoint_a=adjoint)
# This will broadcast by brute force if we still need to.
matrix, rhs = broadcast_matrix_batch_dims([matrix, rhs])
solution = linalg_ops.matrix_solve(matrix,
rhs,
adjoint=adjoint
and still_need_to_transpose)
return reshape_inv(solution)
def _check_spectrum_and_return_tensor(self, spectrum):
"""Static check of spectrum. Then return `Tensor` version."""
spectrum = ops.convert_to_tensor(spectrum, name="spectrum")
if _ops.TensorShape(spectrum.shape).ndims is not None:
if _ops.TensorShape(spectrum.shape).ndims < self.block_depth:
raise ValueError(
"Argument spectrum must have at least %d dimensions. Found: %s"
% (self.block_depth, spectrum))
return spectrum
def assert_zero_imag_part(x, message=None, name="assert_zero_imag_part"):
"""Returns `Op` that asserts Tensor `x` has no non-zero imaginary parts.
Args:
x: Numeric `Tensor`, real, integer, or complex.
message: A string message to prepend to failure message.
name: A name to give this `Op`.
Returns:
An `Op` that asserts `x` has no entries with modulus zero.
"""
with ops.name_scope(name, values=[x]):
x = ops.convert_to_tensor(x, name="x")
dtype = x.dtype
if dtype.is_floating:
return control_flow_ops.no_op()
zero = ops.convert_to_tensor(0, dtype=dtypes.real_dtype(dtype))
return check_ops.assert_equal(zero, math_ops.imag(x), message=message)
def add_to_tensor(self, 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 `_ops.TensorShape(self.shape)`.
name: A name to give this `Op`.
Returns:
A `Tensor` with broadcast shape and same `dtype` as `self`.
"""
with self._name_scope(name):
x = ops.convert_to_tensor(x, name="x")
# self._check_input_dtype(x)
return self._add_to_tensor(x)
def add_to_tensor(self, mat, name="add_to_tensor"):
"""Add matrix represented by this operator to `mat`. Equiv to `I + mat`.
Args:
mat: `Tensor` with same `dtype` and shape broadcastable to `self`.
name: A name to give this `Op`.
Returns:
A `Tensor` with broadcast shape and same `dtype` as `self`.
"""
with self._name_scope(name):
mat = ops.convert_to_tensor(mat, name="mat")
mat_diag = _linalg.diag_part(mat)
new_diag = 1 + mat_diag
return _linalg.set_diag(mat, new_diag)
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 _ops.TensorShape(A.shape) = [..., M, N]
operator = LinearOperator(...)
_ops.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.
`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.
"""
with self._name_scope(name):
rhs = ops.convert_to_tensor(rhs, name="rhs")
# self._check_input_dtype(rhs)
self_dim = -1 if adjoint else -2
tensor_shape.dimension_at_index(_ops.TensorShape(
self.shape), self_dim).assert_is_compatible_with(
_ops.TensorShape(rhs.shape)[-1])
return self._solvevec(rhs, adjoint=adjoint)
def tensor_rank_tensor(self, 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
`_ops.TensorShape(A.shape) = [B1,...,Bb, M, N]`, then this returns `b + 2`.
Args:
name: A name for this `Op`.
Returns:
`int32` `Tensor`, determined at runtime.
"""
# Derived classes get this "for free" once .shape() is implemented.
with self._name_scope(name):
# Prefer to use statically defined shape if available.
if self.tensor_rank is not None:
return ops.convert_to_tensor(self.tensor_rank)
else:
return array_ops.size(self.shape_tensor())
def _matmul(self, x, adjoint=False, adjoint_arg=False):
# Given a vector `v`, we would like to reflect `x` about the hyperplane
# orthogonal to `v` going through the origin. We first project `x` to `v`
# to get v * dot(v, x) / dot(v, v). After we project, we can reflect the
# projection about the hyperplane by flipping sign to get
# -v * dot(v, x) / dot(v, v). Finally, we can add back the component
# that is orthogonal to v. This is invariant under reflection, since the
# whole hyperplane is invariant. This component is equal to x - v * dot(v,
# x) / dot(v, v), giving the formula x - 2 * v * dot(v, x) / dot(v, v)
# for the reflection.
# Note that because this is a reflection, it lies in O(n) (for real vector
# spaces) or U(n) (for complex vector spaces), and thus is its own adjoint.
reflection_axis = ops.convert_to_tensor(self.reflection_axis)
x = linalg.adjoint(x) if adjoint_arg else x
normalized_axis = reflection_axis / linalg.norm(
reflection_axis, axis=-1, keepdims=True)
mat = normalized_axis[..., array_ops.newaxis]
x_dot_normalized_v = _linalg.matmul(mat, x, adjoint_a=True)
return x - 2 * mat * x_dot_normalized_v
def range_dimension_tensor(self, 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
`_ops.TensorShape(A.shape) = [B1,...,Bb, M, N]`, then this returns `M`.
Args:
name: A name for this `Op`.
Returns:
`int32` `Tensor`
"""
# Derived classes get this "for free" once .shape() is implemented.
with self._name_scope(name):
# Prefer to use statically defined shape if available.
dim_value = tensor_shape.dimension_value(self.range_dimension)
if dim_value is not None:
return ops.convert_to_tensor(dim_value)
else:
return self.shape_tensor()[-2]
def _shape_tensor(self):
# Avoid messy broadcasting if possible.
if _ops.TensorShape(self.shape).is_fully_defined():
return ops.convert_to_tensor(_ops.TensorShape(
self.shape).as_list(),
dtype=dtypes.int32,
name="shape")
# Don't check the matrix dimensions. That would add unnecessary Asserts to
# the graph. Things will fail at runtime naturally if shapes are
# incompatible.
matrix_shape = array_ops.stack([
self.operators[0].range_dimension_tensor(),
self.operators[-1].domain_dimension_tensor()
])
# Dummy Tensor of zeros. Will never be materialized.
zeros = array_ops.zeros(shape=self.operators[0].batch_shape_tensor())
for operator in self.operators[1:]:
zeros += array_ops.zeros(shape=operator.batch_shape_tensor())
batch_shape = array_ops.shape(zeros)
return array_ops.concat((batch_shape, matrix_shape), 0)
def _shape_tensor(self):
# Avoid messy broadcasting if possible.
if _ops.TensorShape(self.shape).is_fully_defined():
return ops.convert_to_tensor(_ops.TensorShape(
self.shape).as_list(),
dtype=dtypes.int32,
name="shape")
domain_dimension = self.operators[0].domain_dimension_tensor()
range_dimension = self.operators[0].range_dimension_tensor()
for operator in self.operators[1:]:
domain_dimension += operator.domain_dimension_tensor()
range_dimension += operator.range_dimension_tensor()
matrix_shape = array_ops.stack([domain_dimension, range_dimension])
# Dummy Tensor of zeros. Will never be materialized.
zeros = array_ops.zeros(shape=self.operators[0].batch_shape_tensor())
for operator in self.operators[1:]:
zeros += array_ops.zeros(shape=operator.batch_shape_tensor())
batch_shape = array_ops.shape(zeros)
return array_ops.concat((batch_shape, matrix_shape), 0)
def _check_matrix(self, matrix):
"""Static check of the `matrix` argument."""
allowed_dtypes = [
dtypes.float16,
dtypes.float32,
dtypes.float64,
dtypes.complex64,
dtypes.complex128,
]
matrix = ops.convert_to_tensor(matrix, name="matrix")
dtype = matrix.dtype
if dtype not in allowed_dtypes:
raise TypeError(
"Argument matrix must have dtype in %s. Found: %s" %
(allowed_dtypes, dtype))
if _ops.TensorShape(
matrix.shape).ndims is not None and _ops.TensorShape(
matrix.shape).ndims < 2:
raise ValueError(
"Argument matrix must have at least 2 dimensions. Found: %s" %
matrix)
def add_to_tensor(self, mat, name="add_to_tensor"):
"""Add matrix represented by this operator to `mat`. Equiv to `I + mat`.
Args:
mat: `Tensor` with same `dtype` and shape broadcastable to `self`.
name: A name to give this `Op`.
Returns:
A `Tensor` with broadcast shape and same `dtype` as `self`.
"""
with self._name_scope(name):
# Shape [B1,...,Bb, 1]
multiplier_vector = array_ops.expand_dims(self.multiplier, -1)
# Shape [C1,...,Cc, M, M]
mat = ops.convert_to_tensor(mat, name="mat")
# Shape [C1,...,Cc, M]
mat_diag = _linalg.diag_part(mat)
# multiplier_vector broadcasts here.
new_diag = multiplier_vector + mat_diag
return _linalg.set_diag(mat, new_diag)
def __init__(self,
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.
Args:
base_operator: Shape `[B1,...,Bb, M, N]`.
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`.
Raises:
ValueError: If `is_X` flags are set in an inconsistent way.
"""
dtype = base_operator.dtype
if diag_update is not None:
if is_diag_update_positive and np.issubdtype(
dtype, np.complexfloating):
logging.warn(
"Note: setting is_diag_update_positive with a complex "
"dtype means that diagonal is real and positive.")
if diag_update is None:
if is_diag_update_positive is False:
raise ValueError(
"Default diagonal is the identity, which is positive. However, "
"user set 'is_diag_update_positive' to False.")
is_diag_update_positive = True
# In this case, we can use a Cholesky decomposition to help us solve/det.
self._use_cholesky = (base_operator.is_positive_definite
and base_operator.is_self_adjoint
and is_diag_update_positive and v is None)
# Possibly auto-set some characteristic flags from None to True.
# If the Flags were set (by the user) incorrectly to False, then raise.
if base_operator.is_self_adjoint and v is None and not np.issubdtype(
dtype, np.complexfloating):
if is_self_adjoint is False:
raise ValueError(
"A = L + UDU^H, with L self-adjoint and D real diagonal. Since"
" UDU^H is self-adjoint, this must be a self-adjoint operator."
)
is_self_adjoint = True
# The condition for using a cholesky is sufficient for SPD, and
# we no weaker choice of these hints leads to SPD. Therefore,
# the following line reads "if hints indicate SPD..."
if self._use_cholesky:
if (is_positive_definite is False or is_self_adjoint is False
or is_non_singular is False):
raise ValueError(
"Arguments imply this is self-adjoint positive-definite operator."
)
is_positive_definite = True
is_self_adjoint = True
values = base_operator.graph_parents + [u, diag_update, v]
with ops.name_scope(name, values=values):
# Create U and V.
self._u = ops.convert_to_tensor(u, name="u")
if v is None:
self._v = self._u
else:
self._v = ops.convert_to_tensor(v, name="v")
if diag_update is None:
self._diag_update = None
else:
self._diag_update = ops.convert_to_tensor(diag_update,
name="diag_update")
# Create base_operator L.
self._base_operator = base_operator
graph_parents = base_operator.graph_parents + [
self.u, self._diag_update, self.v
]
graph_parents = [p for p in graph_parents if p is not None]
super(LinearOperatorLowRankUpdate,
self).__init__(dtype=self._base_operator.dtype,
graph_parents=graph_parents,
is_non_singular=is_non_singular,
is_self_adjoint=is_self_adjoint,
is_positive_definite=is_positive_definite,
is_square=is_square,
name=name)
# Create the diagonal operator D.
self._set_diag_operators(diag_update, is_diag_update_positive)
self._is_diag_update_positive = is_diag_update_positive
self._check_shapes()
# Pre-compute the so-called "capacitance" matrix
# C := D^{-1} + V^H L^{-1} U
self._capacitance = self._make_capacitance()
if self._use_cholesky:
self._chol_capacitance = linalg_ops.cholesky(self._capacitance)
return np.conjugate(x) if conjugate else x
def _zeros_like(input, dtype=None, name=None): # pylint: disable=redefined-builtin
s = _shape(input)
if isinstance(s, (np.ndarray, onp.generic)):
return np.zeros(s, utils.numpy_dtype(dtype or input.dtype))
return tf.zeros(s, dtype or s.dtype, name)
# --- Begin Public Functions --------------------------------------------------
concat = utils.copy_docstring(
tf.concat,
lambda values, axis, name='concat': ( # pylint: disable=g-long-lambda
np.concatenate([ops.convert_to_tensor(v) for v in values], axis)))
expand_dims = utils.copy_docstring(
tf.expand_dims, lambda input, axis, name=None: np.expand_dims(input, axis))
fill = utils.copy_docstring(
tf.fill,
lambda dims, value, name=None: value * np.ones(dims,
np.array(value).dtype))
gather = utils.copy_docstring(tf.gather, _gather)
gather_nd = utils.copy_docstring(tf.gather_nd, _gather_nd)
reverse = utils.copy_docstring(tf.reverse, _reverse)
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 _ops.TensorShape(A.shape) = [..., M, N]
operator = LinearOperator(...)
_ops.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.
`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.
"""
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, 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):
rhs = ops.convert_to_tensor(rhs, name="rhs")
# self._check_input_dtype(rhs)
self_dim = -1 if adjoint else -2
arg_dim = -1 if adjoint_arg else -2
tensor_shape.dimension_at_index(_ops.TensorShape(
self.shape), self_dim).assert_is_compatible_with(
_ops.TensorShape(rhs.shape)[arg_dim])
return self._solve(rhs, adjoint=adjoint, adjoint_arg=adjoint_arg)
def convert_nonref_to_tensor(value, dtype=None, dtype_hint=None, name=None):
"""Converts the given `value` to a `Tensor` if input is nonreference type.
This function converts Python objects of various types to `Tensor` objects
except if the input has nonreference semantics. Reference semantics are
characterized by `is_ref` and is any object which is a
`tf.Variable` or instance of `tf.Module`. This function accepts any input
which `tf.convert_to_tensor` would also.
Note: This function diverges from default Numpy behavior for `float` and
`string` types when `None` is present in a Python list or scalar. Rather
than silently converting `None` values, an error will be thrown.
Args:
value: An object whose type has a registered `Tensor` conversion function.
dtype: Optional element type for the returned tensor. If missing, the
type is inferred from the type of `value`.
dtype_hint: Optional element type for the returned tensor,
used when dtype is None. In some cases, a caller may not have a
dtype in mind when converting to a tensor, so dtype_hint
can be used as a soft preference. If the conversion to
`dtype_hint` is not possible, this argument has no effect.
name: Optional name to use if a new `Tensor` is created.
Returns:
tensor: A `Tensor` based on `value`.
Raises:
TypeError: If no conversion function is registered for `value` to `dtype`.
RuntimeError: If a registered conversion function returns an invalid value.
ValueError: If the `value` is a tensor not of given `dtype` in graph mode.
#### Examples:
```python
x = tf.Variable(0.)
y = convert_nonref_to_tensor(x)
x is y
# ==> True
x = tf.constant(0.)
y = convert_nonref_to_tensor(x)
x is y
# ==> True
x = np.array(0.)
y = convert_nonref_to_tensor(x)
x is y
# ==> False
tf.is_tensor(y)
# ==> True
x = tfp.util.DeferredTensor(lambda x: x, 13.37)
y = convert_nonref_to_tensor(x)
x is y
# ==> True
tf.is_tensor
# ==> False
tf.equal(y, 13.37)
# ==> True
```
"""
# We explicitly do not use a tf.name_scope to avoid graph clutter.
if value is None:
return None
if is_ref(value):
if dtype is None:
return value
dtype_base = base_dtype(dtype)
value_dtype_base = base_dtype(value.dtype)
if dtype_base != value_dtype_base:
raise TypeError(
'Mutable type must be of dtype "{}" but is "{}".'.format(
dtype_name(dtype_base), dtype_name(value_dtype_base)))
return value
return ops.convert_to_tensor(value,
dtype=dtype,
dtype_hint=dtype_hint,
name=name)
def broadcast_matrix_batch_dims(batch_matrices, name=None):
"""Broadcast leading dimensions of zero or more [batch] matrices.
Example broadcasting one batch dim of two simple matrices.
```python
x = [[1, 2],
[3, 4]] # Shape [2, 2], no batch dims
y = [[[1]]] # Shape [1, 1, 1], 1 batch dim of shape [1]
x_bc, y_bc = broadcast_matrix_batch_dims([x, y])
x_bc
==> [[[1, 2],
[3, 4]]] # Shape [1, 2, 2], 1 batch dim of shape [1].
y_bc
==> same as y
```
Example broadcasting many batch dims
```python
x = tf.random.normal(shape=(2, 3, 1, 4, 4))
y = tf.random.normal(shape=(1, 3, 2, 5, 5))
x_bc, y_bc = broadcast_matrix_batch_dims([x, y])
_ops.TensorShape(x_bc.shape)
==> (2, 3, 2, 4, 4)
_ops.TensorShape(y_bc.shape)
==> (2, 3, 2, 5, 5)
```
Args:
batch_matrices: Iterable of `Tensor`s, each having two or more dimensions.
name: A string name to prepend to created ops.
Returns:
bcast_matrices: List of `Tensor`s, with `bcast_matricies[i]` containing
the values from `batch_matrices[i]`, with possibly broadcast batch dims.
Raises:
ValueError: If any input `Tensor` is statically determined to have less
than two dimensions.
"""
with ops.name_scope(name or "broadcast_matrix_batch_dims",
values=batch_matrices):
check_ops.assert_proper_iterable(batch_matrices)
batch_matrices = list(batch_matrices)
for i, mat in enumerate(batch_matrices):
batch_matrices[i] = ops.convert_to_tensor(mat)
assert_is_batch_matrix(batch_matrices[i])
if len(batch_matrices) < 2:
return batch_matrices
# Try static broadcasting.
# bcast_batch_shape is the broadcast batch shape of ALL matrices.
# E.g. if batch_matrices = [x, y], with
# _ops.TensorShape(x.shape) = [2, j, k] (batch shape = [2])
# _ops.TensorShape(y.shape) = [3, 1, l, m] (batch shape = [3, 1])
# ==> bcast_batch_shape = [3, 2]
bcast_batch_shape = _ops.TensorShape(batch_matrices[0].shape)[:-2]
for mat in batch_matrices[1:]:
bcast_batch_shape = _ops.broadcast_static_shape_as_tensorshape(
bcast_batch_shape,
_ops.TensorShape(mat.shape)[:-2])
if bcast_batch_shape.is_fully_defined():
for i, mat in enumerate(batch_matrices):
if _ops.TensorShape(mat.shape)[:-2] != bcast_batch_shape:
bcast_shape = array_ops.concat([
bcast_batch_shape.as_list(),
array_ops.shape(mat)[-2:]
],
axis=0)
batch_matrices[i] = _ops.broadcast_to(mat, bcast_shape)
return batch_matrices
# Since static didn't work, do dynamic, which always copies data.
bcast_batch_shape = array_ops.shape(batch_matrices[0])[:-2]
for mat in batch_matrices[1:]:
bcast_batch_shape = array_ops.broadcast_dynamic_shape(
bcast_batch_shape,
array_ops.shape(mat)[:-2])
for i, mat in enumerate(batch_matrices):
batch_matrices[i] = _ops.broadcast_to(
mat,
array_ops.concat(
[bcast_batch_shape,
array_ops.shape(mat)[-2:]], axis=0))
return batch_matrices
