def _to_dense(self):
num_cols = 0
rows = []
broadcasted_blocks = [
operator.to_dense() for operator in self.operators
]
broadcasted_blocks = linear_operator_util.broadcast_matrix_batch_dims(
broadcasted_blocks)
for block in broadcasted_blocks:
batch_row_shape = array_ops.shape(block)[:-1]
zeros_to_pad_before_shape = array_ops.concat(
[batch_row_shape, [num_cols]], axis=-1)
zeros_to_pad_before = array_ops.zeros(
shape=zeros_to_pad_before_shape, dtype=block.dtype)
num_cols += array_ops.shape(block)[-1]
zeros_to_pad_after_shape = array_ops.concat(
[batch_row_shape, [self.domain_dimension_tensor() - num_cols]],
axis=-1)
zeros_to_pad_after = array_ops.zeros(
shape=zeros_to_pad_after_shape, dtype=block.dtype)
rows.append(
array_ops.concat(
[zeros_to_pad_before, block, zeros_to_pad_after], axis=-1))
mat = array_ops.concat(rows, axis=-2)
mat.set_shape(_ops.TensorShape(self.shape))
return mat
def _matmul(self, x, adjoint=False, adjoint_arg=False):
if self._assert_proper_shapes:
x = linalg.adjoint(x) if adjoint_arg else x
aps = linear_operator_util.assert_compatible_matrix_dimensions(
self, x)
x = control_flow_ops.with_dependencies([aps], x)
if self.is_square:
# Note that adjoint has no effect since this matrix is self-adjoint.
if adjoint_arg:
output_shape = array_ops.concat([
array_ops.shape(x)[:-2],
[array_ops.shape(x)[-1],
array_ops.shape(x)[-2]]
],
axis=0)
else:
output_shape = array_ops.shape(x)
return self._possibly_broadcast_batch_shape(
array_ops.zeros(shape=output_shape, dtype=x.dtype))
x_shape = array_ops.shape(x)
n = self._num_columns if adjoint else self._num_rows
m = x_shape[-2] if adjoint_arg else x_shape[-1]
output_shape = array_ops.concat([x_shape[:-2], [n, m]], axis=0)
zeros = array_ops.zeros(shape=output_shape, dtype=x.dtype)
return self._possibly_broadcast_batch_shape(zeros)
def _vectorize_then_blockify(self, matrix):
"""Shape batch matrix to batch vector, then blockify trailing dimensions."""
# Suppose
# _ops.TensorShape(matrix.shape) = [m0, m1, m2, m3],
# and matrix is a matrix because the final two dimensions are matrix dims.
# self.block_depth = 2,
# self.block_shape = [b0, b1] (note b0 * b1 = m2).
# We will reshape matrix to
# [m3, m0, m1, b0, b1].
# Vectorize: Reshape to batch vector.
# [m0, m1, m2, m3] --> [m3, m0, m1, m2]
# This is called "vectorize" because we have taken the final two matrix dims
# and turned this into a size m3 batch of vectors.
vec = distribution_util.rotate_transpose(matrix, shift=1)
# Blockify: Blockfy trailing dimensions.
# [m3, m0, m1, m2] --> [m3, m0, m1, b0, b1]
if (_ops.TensorShape(vec.shape).is_fully_defined()
and self.block_shape.is_fully_defined()):
# vec_leading_shape = [m3, m0, m1],
# the parts of vec that will not be blockified.
vec_leading_shape = _ops.TensorShape(vec.shape)[:-1]
final_shape = vec_leading_shape.concatenate(self.block_shape)
else:
vec_leading_shape = array_ops.shape(vec)[:-1]
final_shape = array_ops.concat(
(vec_leading_shape, self.block_shape_tensor()), 0)
return array_ops.reshape(vec, final_shape)
def assert_compatible_matrix_dimensions(operator, x):
"""Assert that an argument to solve/matmul has proper domain dimension.
If `_ops.TensorShape(operator.shape)[-2:] = [M, N]`, and `_ops.TensorShape(x.shape)[-2:] = [Q, R]`, then
`operator.matmul(x)` is defined only if `N = Q`. This `Op` returns an
`Assert` that "fires" if this is not the case. Static checks are already
done by the base class `LinearOperator`.
Args:
operator: `LinearOperator`.
x: `Tensor`.
Returns:
`Assert` `Op`.
"""
# Static checks are done in the base class. Only tensor asserts here.
assert_same_dd = check_ops.assert_equal(
array_ops.shape(x)[-2],
operator.domain_dimension_tensor(),
# This error message made to look similar to error raised by static check
# in the base class.
message=("Dimensions are not compatible. "
"shape[-2] of argument to be the same as this operator"))
return assert_same_dd
def _unvec_by(y, num_col):
"""Unstack vector to form a matrix, with a specified amount of columns."""
return _linalg.matrix_transpose(
array_ops.reshape(
y,
array_ops.concat(
[array_ops.shape(y)[:-1], [num_col, -1]], axis=0)))
def _shape_tensor(self):
# See _ops.TensorShape(self.shape) for explanation of steps
s_shape = array_ops.shape(self._spectrum)
batch_shape = s_shape[:-self.block_depth]
trailing_dims = s_shape[-self.block_depth:]
n = math_ops.reduce_prod(trailing_dims)
n_x_n = [n, n]
return array_ops.concat((batch_shape, n_x_n), 0)
def reshape_inv(y):
# Expand the extra dims hanging off the end, "b_extra_sh".
# Note we use y_sh[:-1] + [b_main_sh[-1]] rather than b_main_sh, because y
# Could have different batch dims than a and b, because of broadcasting.
y_extra_shape = array_ops.concat(
(array_ops.shape(y)[:-1], [b_main_sh[-1]], b_extra_sh), 0)
y_extra_on_end = array_ops.reshape(y, y_extra_shape)
inverse_perm = np.argsort(perm)
return array_ops.transpose(y_extra_on_end, perm=inverse_perm)
def _to_dense(self):
product = self.operators[0].to_dense()
for operator in self.operators[1:]:
# Product has shape [B, R1, 1, C1].
product = product[
..., :, array_ops.newaxis, :, array_ops.newaxis]
# Operator has shape [B, 1, R2, 1, C2].
op_to_mul = operator.to_dense()[
..., array_ops.newaxis, :, array_ops.newaxis, :]
# This is now [B, R1, R2, C1, C2].
product *= op_to_mul
# Now merge together dimensions to get [B, R1 * R2, C1 * C2].
product = array_ops.reshape(
product,
shape=array_ops.concat(
[array_ops.shape(product)[:-4],
[array_ops.shape(product)[-4] * array_ops.shape(product)[-3],
array_ops.shape(product)[-2] * array_ops.shape(product)[-1]]
], axis=0))
product.set_shape(_ops.TensorShape(self.shape))
return product
def _diag_part(self):
diag_part = self.operators[0].diag_part()
for operator in self.operators[1:]:
diag_part = diag_part[..., :, array_ops.newaxis]
op_diag_part = operator.diag_part()[..., array_ops.newaxis, :]
diag_part *= op_diag_part
diag_part = array_ops.reshape(
diag_part,
shape=array_ops.concat(
[array_ops.shape(diag_part)[:-2], [-1]], axis=0))
if self.range_dimension > self.domain_dimension:
diag_dimension = self.domain_dimension
else:
diag_dimension = self.range_dimension
diag_part.set_shape(
self.batch_shape.concatenate(diag_dimension))
return diag_part
def _broadcast_batch_dims(self, x, spectrum):
"""Broadcast batch dims of batch matrix `x` and spectrum."""
# _ops.TensorShape(spectrum.shape) = batch_shape + block_shape
# First make spectrum a batch matrix with
# _ops.TensorShape(spectrum.shape) = batch_shape + [prod(block_shape), 1]
spec_mat = array_ops.reshape(
spectrum,
array_ops.concat((self.batch_shape_tensor(), [-1, 1]), axis=0))
# Second, broadcast, possibly requiring an addition of array of zeros.
x, spec_mat = linear_operator_util.broadcast_matrix_batch_dims(
(x, spec_mat))
# Third, put the block shape back into spectrum.
batch_shape = array_ops.shape(x)[:-2]
spectrum = array_ops.reshape(
spec_mat,
array_ops.concat((batch_shape, self.block_shape_tensor()), axis=0))
return x, spectrum
def _set_diag_operators(self, diag_update, is_diag_update_positive):
"""Set attributes self._diag_update and self._diag_operator."""
if diag_update is not None:
self._diag_operator = linear_operator_diag.LinearOperatorDiag(
self._diag_update,
is_positive_definite=is_diag_update_positive)
self._diag_inv_operator = linear_operator_diag.LinearOperatorDiag(
1. / self._diag_update,
is_positive_definite=is_diag_update_positive)
else:
if tensor_shape.dimension_value(
_ops.TensorShape(self.u.shape)[-1]) is not None:
r = tensor_shape.dimension_value(
_ops.TensorShape(self.u.shape)[-1])
else:
r = array_ops.shape(self.u)[-1]
self._diag_operator = linear_operator_identity.LinearOperatorIdentity(
num_rows=r, dtype=self.dtype)
self._diag_inv_operator = self._diag_operator
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 _unblockify_then_matricize(self, vec):
"""Flatten the block dimensions then reshape to a batch matrix."""
# Suppose
# _ops.TensorShape(vec.shape) = [v0, v1, v2, v3],
# self.block_depth = 2.
# Then
# leading shape = [v0, v1]
# block shape = [v2, v3].
# We will reshape vec to
# [v1, v2*v3, v0].
# Un-blockify: Flatten block dimensions. Reshape
# [v0, v1, v2, v3] --> [v0, v1, v2*v3].
if _ops.TensorShape(vec.shape).is_fully_defined():
# vec_shape = [v0, v1, v2, v3]
vec_shape = _ops.TensorShape(vec.shape).as_list()
# vec_leading_shape = [v0, v1]
vec_leading_shape = vec_shape[:-self.block_depth]
# vec_block_shape = [v2, v3]
vec_block_shape = vec_shape[-self.block_depth:]
# flat_shape = [v0, v1, v2*v3]
flat_shape = vec_leading_shape + [np.prod(vec_block_shape)]
else:
vec_shape = array_ops.shape(vec)
vec_leading_shape = vec_shape[:-self.block_depth]
vec_block_shape = vec_shape[-self.block_depth:]
flat_shape = array_ops.concat(
(vec_leading_shape, [math_ops.reduce_prod(vec_block_shape)]),
0)
vec_flat = array_ops.reshape(vec, flat_shape)
# Matricize: Reshape to batch matrix.
# [v0, v1, v2*v3] --> [v1, v2*v3, v0],
# representing a shape [v1] batch of [v2*v3, v0] matrices.
matrix = distribution_util.rotate_transpose(vec_flat, shift=-1)
return matrix
def _shape_tensor(self):
d_shape = array_ops.shape(self._diag)
k = d_shape[-1]
return array_ops.concat((d_shape, [k]), 0)
def _shape_tensor(self):
v_shape = array_ops.broadcast_dynamic_shape(
array_ops.shape(self.row),
array_ops.shape(self.col))
k = v_shape[-1]
return array_ops.concat((v_shape, [k]), 0)
def __init__(self,
spectrum,
block_depth,
input_output_dtype=dtypes.complex64,
is_non_singular=None,
is_self_adjoint=None,
is_positive_definite=None,
is_square=True,
name="LinearOperatorCirculant"):
r"""Initialize an `_BaseLinearOperatorCirculant`.
Args:
spectrum: Shape `[B1,...,Bb, N]` `Tensor`. Allowed dtypes: `float16`,
`float32`, `float64`, `complex64`, `complex128`. Type can be different
than `input_output_dtype`
block_depth: Python integer, either 1, 2, or 3. Will be 1 for circulant,
2 for block circulant, and 3 for nested block circulant.
input_output_dtype: `dtype` for input/output.
is_non_singular: Expect that this operator is non-singular.
is_self_adjoint: Expect that this operator is equal to its hermitian
transpose. If `spectrum` is real, this will always be 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 to prepend to all ops created by this class.
Raises:
ValueError: If `block_depth` is not an allowed value.
TypeError: If `spectrum` is not an allowed type.
"""
allowed_block_depths = [1, 2, 3]
self._name = name
if block_depth not in allowed_block_depths:
raise ValueError("Expected block_depth to be in %s. Found: %s." %
(allowed_block_depths, block_depth))
self._block_depth = block_depth
with ops.name_scope(name, values=[spectrum]):
self._spectrum = self._check_spectrum_and_return_tensor(spectrum)
# Check and auto-set hints.
if not np.issubdtype(self.spectrum.dtype, np.complexfloating):
if is_self_adjoint is False:
raise ValueError(
"A real spectrum always corresponds to a self-adjoint operator."
)
is_self_adjoint = True
if is_square is False:
raise ValueError(
"A [[nested] block] circulant operator is always square.")
is_square = True
# If _ops.TensorShape(spectrum.shape) = [s0, s1, s2], and block_depth = 2,
# block_shape = [s1, s2]
s_shape = array_ops.shape(self.spectrum)
self._block_shape_tensor = s_shape[-self.block_depth:]
# Add common variants of spectrum to the graph.
self._spectrum_complex = _to_complex(self.spectrum)
self._abs_spectrum = math_ops.abs(self.spectrum)
self._conj_spectrum = math_ops.conj(self._spectrum_complex)
super(_BaseLinearOperatorCirculant,
self).__init__(dtype=dtypes.as_dtype(input_output_dtype),
graph_parents=[self.spectrum],
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 _vec(x):
"""Stacks column of matrix to form a single column."""
return array_ops.reshape(
_linalg.matrix_transpose(x),
array_ops.concat(
[array_ops.shape(x)[:-2], [-1]], axis=0))
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
def _reshape_for_efficiency(a,
b,
transpose_a=False,
transpose_b=False,
adjoint_a=False,
adjoint_b=False):
"""Maybe reshape a, b, and return an inverse map. For matmul/solve."""
def identity(x):
return x
# At this point, we have not taken transpose/adjoint of a/b.
still_need_to_transpose = True
if _ops.TensorShape(a.shape).ndims is None or _ops.TensorShape(
b.shape).ndims is None:
return a, b, identity, still_need_to_transpose
# This could be handled in the future, but seems less common.
if _ops.TensorShape(a.shape).ndims >= _ops.TensorShape(b.shape).ndims:
return a, b, identity, still_need_to_transpose
# From now on, we might modify b, but will not modify a.
# Suppose:
# _ops.TensorShape(a.shape) = C + [m, n], _ops.TensorShape(b.shape) =
# _ops.TensorShape(b.shape) = S + C + [n, r]
b_extra_ndims = _ops.TensorShape(b.shape).ndims - _ops.TensorShape(
a.shape).ndims
# b_extra_sh = S, b_main_sh = C + [n, r]
b_extra_sh = array_ops.shape(b)[:b_extra_ndims]
b_main_sh = array_ops.shape(b)[b_extra_ndims:]
# No reason to flip unless the extra dims of b are big enough. Why?
# Assume adjoint/transpose = False. Then...
# By not flipping, we have to replicate a to shape
# b_extra_sh + _ops.TensorShape(a.shape),
# which could use extra memory. But in all cases, the final output has shape
# b_extra_sh + _ops.TensorShape(a.shape)[:-1] + _ops.TensorShape([b.shape)[-1]]
# So we only end up creating a larger object if the end dim of b is smaller
# than the end dim of a. This often happens, e.g. if b was a vector that was
# expanded to a matrix (by appending a singleton).
# Since adjoint/transpose may not be False, we must make adjustments here.
# The dim of b that holds the multiple equations.
a_domain_sz_ = _ops.TensorShape(
a.shape)[-2 if adjoint_a or transpose_a else -1]
b_eq_sz_ = _ops.TensorShape(
b.shape)[-2 if adjoint_b or transpose_b else -1]
b_extra_sz_ = (
np.prod(_ops.TensorShape(b.shape)[:b_extra_ndims].as_list()) if
_ops.TensorShape(b.shape)[:b_extra_ndims].is_fully_defined() else None)
if (a_domain_sz_ is not None and b_eq_sz_ is not None
and b_extra_sz_ is not None):
if b_extra_sz_ < 2 or a_domain_sz_ <= b_eq_sz_:
return a, b, identity, still_need_to_transpose
# At this point, we're flipping for sure!
# Any transposes/adjoints will happen here explicitly, rather than in calling
# code. Why? To avoid having to write separate complex code for each case.
if adjoint_a:
a = linalg.adjoint(a)
elif transpose_a:
a = linalg.transpose(a)
if adjoint_b:
b = linalg.adjoint(b)
elif transpose_b:
b = linalg.transpose(b)
still_need_to_transpose = False
# Recompute shapes, since the transpose/adjoint may have changed them.
b_extra_sh = array_ops.shape(b)[:b_extra_ndims]
b_main_sh = array_ops.shape(b)[b_extra_ndims:]
# Permutation to put the extra dims at the end.
perm = (np.concatenate(
(np.arange(b_extra_ndims,
_ops.TensorShape(b.shape).ndims), np.arange(
0, b_extra_ndims)), 0))
b_extra_on_end = array_ops.transpose(b, perm=perm)
# Now squash this end into one long dim.
b_squashed_end = array_ops.reshape(
b_extra_on_end, array_ops.concat((b_main_sh[:-1], [-1]), 0))
def reshape_inv(y):
# Expand the extra dims hanging off the end, "b_extra_sh".
# Note we use y_sh[:-1] + [b_main_sh[-1]] rather than b_main_sh, because y
# Could have different batch dims than a and b, because of broadcasting.
y_extra_shape = array_ops.concat(
(array_ops.shape(y)[:-1], [b_main_sh[-1]], b_extra_sh), 0)
y_extra_on_end = array_ops.reshape(y, y_extra_shape)
inverse_perm = np.argsort(perm)
return array_ops.transpose(y_extra_on_end, perm=inverse_perm)
return a, b_squashed_end, reshape_inv, still_need_to_transpose
def _shape_tensor(self):
batch_shape = array_ops.broadcast_dynamic_shape(
self.base_operator.batch_shape_tensor(),
array_ops.shape(self.u)[:-2])
return array_ops.concat(
[batch_shape, self.base_operator.shape_tensor()[-2:]], axis=0)
def _shape_tensor(self):
d_shape = array_ops.shape(self._reflection_axis)
k = d_shape[-1]
return array_ops.concat((d_shape, [k]), 0)
def _shape_tensor(self):
return array_ops.shape(self._matrix)
def _shape_tensor(self):
matrix_shape = array_ops.stack((self._num_rows, self._num_rows),
axis=0)
batch_shape = array_ops.shape(self.multiplier)
return array_ops.concat((batch_shape, matrix_shape), 0)
