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)
tensorshape_util.set_shape(mat, tensor_shape.TensorShape(self.shape))
return mat
def _possibly_broadcast_batch_shape(self, x):
"""Return 'x', possibly after broadcasting the leading dimensions."""
# If we have no batch shape, our batch shape broadcasts with everything!
if self._batch_shape_arg is None:
return x
# Static attempt:
# If we determine that no broadcast is necessary, pass x through
# If we need a broadcast, add to an array of zeros.
#
# special_shape is the shape that, when broadcast with x's shape, will give
# the correct broadcast_shape. Note that
# We have already verified the second to last dimension of tensor_shape.TensorShape(self.shape)
# matches x's shape in assert_compatible_matrix_dimensions.
# Also, the final dimension of 'x' can have any shape.
# Therefore, the final two dimensions of special_shape are 1's.
special_shape = self.batch_shape.concatenate([1, 1])
bshape = _ops.broadcast_static_shape(tensor_shape.TensorShape(x.shape),
special_shape)
if special_shape.is_fully_defined():
# bshape.is_fully_defined iff special_shape.is_fully_defined.
if bshape == tensor_shape.TensorShape(x.shape):
return x
# Use the built in broadcasting of addition.
zeros = array_ops.zeros(shape=special_shape, dtype=self.dtype)
return x + zeros
# Dynamic broadcast:
# Always add to an array of zeros, rather than using a "cond", since a
# cond would require copying data from GPU --> CPU.
special_shape = array_ops.concat((self.batch_shape_tensor(), [1, 1]),
0)
zeros = array_ops.zeros(shape=special_shape, dtype=self.dtype)
return x + zeros
def _trace(self):
# Get Tensor of all zeros of same shape as self.batch_shape.
if self.batch_shape.is_fully_defined():
return array_ops.zeros(shape=self.batch_shape, dtype=self.dtype)
else:
return array_ops.zeros(shape=self.batch_shape_tensor(),
dtype=self.dtype)
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 = distribution_util.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 = prefer_static.concat([
prefer_static.shape(x)[:-2],
[prefer_static.shape(x)[-1],
prefer_static.shape(x)[-2]]
],
axis=0)
else:
output_shape = prefer_static.shape(x)
return self._possibly_broadcast_batch_shape(
array_ops.zeros(shape=output_shape, dtype=x.dtype))
x_shape = prefer_static.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 = prefer_static.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 _shape_tensor(self):
# Avoid messy broadcasting if possible.
if tensor_shape.TensorShape(self.shape).is_fully_defined():
return ops.convert_to_tensor(
tensor_shape.TensorShape(self.shape).as_list(), dtype=dtypes.int32, name="shape")
domain_dimension = sum(self._block_domain_dimension_tensors())
range_dimension = sum(self._block_range_dimension_tensors())
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 = 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 _zeros_diag(self):
"""Returns the diagonal of this operator as all zeros."""
if tensor_shape.TensorShape(self.shape).is_fully_defined():
d_shape = self.batch_shape.concatenate([self._min_matrix_dim()])
else:
d_shape = prefer_static.concat(
[self.batch_shape_tensor(), [self._min_matrix_dim_tensor()]],
axis=0)
return array_ops.zeros(shape=d_shape, dtype=self.dtype)
def _shape_tensor(self):
# Avoid messy broadcasting if possible.
if tensor_shape.TensorShape(self.shape).is_fully_defined():
return ops.convert_to_tensor(
tensor_shape.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 _to_dense(self):
num_cols = 0
dense_rows = []
flat_broadcast_operators = linear_operator_util.broadcast_matrix_batch_dims(
[op.to_dense() for row in self.operators for op in row]) # pylint: disable=g-complex-comprehension
broadcast_operators = [
flat_broadcast_operators[i * (i + 1) // 2:(i + 1) * (i + 2) // 2]
for i in range(len(self.operators))]
for row_blocks in broadcast_operators:
batch_row_shape = prefer_static.shape(row_blocks[0])[:-1]
num_cols = num_cols + prefer_static.shape(row_blocks[-1])[-1]
zeros_to_pad_after_shape = prefer_static.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=self.dtype)
row_blocks.append(zeros_to_pad_after)
dense_rows.append(prefer_static.concat(row_blocks, axis=-1))
mat = prefer_static.concat(dense_rows, axis=-2)
tensorshape_util.set_shape(mat, tensor_shape.TensorShape(self.shape))
return mat
def _solve(self, rhs, adjoint=False, adjoint_arg=False):
# Here we follow the same use of Roth's column lemma as in `matmul`, with
# the key difference that we replace all `matmul` instances with `solve`.
# This follows from the property that inv(A x B) = inv(A) x inv(B).
# Below we document the shape manipulation for adjoint=False,
# adjoint_arg=False, but the general case of different adjoints is still
# handled.
if adjoint_arg:
rhs = linalg.adjoint(rhs)
# Always add a batch dimension to enable broadcasting to work.
batch_shape = self._compute_ones_matrix_shape()
rhs = rhs + array_ops.zeros(batch_shape, dtype=rhs.dtype)
# rhs has shape [B, R, C], where B represent some number of batch
# dimensions,
# R represents the number of rows, and C represents the number of columns.
# In order to apply Roth's column lemma, we need to operate on a batch of
# column vectors, so we reshape into a batch of column vectors. We put it
# at the front to ensure that broadcasting between operators to the batch
# dimensions B still works.
output = _rotate_last_dim(rhs, rotate_right=True)
# Also expand the shape to be [A, C, B, R]. The first dimension will be
# used to accumulate dimensions from each operator matmul.
output = output[_ops.newaxis, ...]
# In this loop, A is going to refer to the value of the accumulated
# dimension. A = 1 at the start, and will end up being self.range_dimension.
# V will refer to the last dimension. V = R at the start, and will end up
# being 1 in the end.
for operator in self.operators[:-1]:
# Reshape output from [A, C, B, V] to be
# [A, C, B, V / op.domain_dimension, op.domain_dimension]
if adjoint:
operator_dimension = operator.range_dimension_tensor()
else:
operator_dimension = operator.domain_dimension_tensor()
output = _unvec_by(output, operator_dimension)
# We are computing (XA^-1^T) = (A^-1 X^T)^T.
# output has [A, C, B, V / op.domain_dimension, op.domain_dimension],
# which is being converted to:
# [A, C, B, V / op.domain_dimension, op.range_dimension]
output = _linalg.matrix_transpose(output)
output = operator.solve(output, adjoint=adjoint, adjoint_arg=False)
output = _linalg.matrix_transpose(output)
# Rearrange it to [A * op.range_dimension, C, B, V / op.domain_dimension]
output = _rotate_last_dim(output, rotate_right=False)
output = _vec(output)
output = _rotate_last_dim(output, rotate_right=True)
# After the loop, we will have
# A = self.range_dimension / op[-1].range_dimension
# V = op[-1].domain_dimension
# We convert that using matvec to get:
# [A, C, B, op[-1].range_dimension]
output = self.operators[-1].solvevec(output, adjoint=adjoint)
# Rearrange shape to be [B1, ... Bn, self.range_dimension, C]
output = _rotate_last_dim(output, rotate_right=False)
output = _vec(output)
output = _rotate_last_dim(output, rotate_right=False)
if tensor_shape.TensorShape(rhs.shape).is_fully_defined():
column_dim = tensor_shape.TensorShape(rhs.shape)[-1]
broadcast_batch_shape = common_shapes.broadcast_shape(
tensor_shape.TensorShape(rhs.shape)[:-2], self.batch_shape)
if adjoint:
matrix_dimensions = [self.domain_dimension, column_dim]
else:
matrix_dimensions = [self.range_dimension, column_dim]
tensorshape_util.set_shape(
output, broadcast_batch_shape.concatenate(matrix_dimensions))
return output
def _matmul(self, x, adjoint=False, adjoint_arg=False):
# Here we heavily rely on Roth's column Lemma [1]:
# (A x B) * vec X = vec BXA^T,
# where vec stacks all the columns of the matrix under each other. In our
# case, x represents a batch of vec X (i.e. we think of x as a batch of
# column vectors, rather than a matrix). Each member of the batch can be
# reshaped to a matrix (hence we get a batch of matrices).
# We can iteratively apply this lemma by noting that if B is a Kronecker
# product, then we can apply the lemma again.
# [1] W. E. Roth, "On direct product matrices,"
# Bulletin of the American Mathematical Society, vol. 40, pp. 461-468,
# 1934
# Efficiency
# Naively doing the Kronecker product, by calculating the dense matrix and
# applying it will can take cubic time in the size of domain_dimension
# (assuming a square matrix). The other issue is that calculating the dense
# matrix can be prohibitively expensive, in that it can take a large amount
# of memory.
#
# This implementation avoids this memory blow up by only computing matmuls
# with the factors. In this way, we don't have to realize the dense matrix.
# In terms of complexity, if we have Kronecker Factors of size:
# (n1, n1), (n2, n2), (n3, n3), ... (nJ, nJ), with N = \prod n_i, and we
# have as input a [N, M] matrix, the naive approach would take O(N^2 M).
# With this approach (ignoring reshaping of tensors and transposes for now),
# the time complexity can be O(M * (\sum n_i) * N). There is also the
# benefit of batched multiplication (In this example, the batch size is
# roughly M * N) so this can be much faster. However, not factored in are
# the costs of the several transposing of tensors, which can affect cache
# behavior.
# Below we document the shape manipulation for adjoint=False,
# adjoint_arg=False, but the general case of different adjoints is still
# handled.
if adjoint_arg:
x = linalg.adjoint(x)
# Always add a batch dimension to enable broadcasting to work.
batch_shape = self._compute_ones_matrix_shape()
x = x + array_ops.zeros(batch_shape, dtype=x.dtype)
# x has shape [B, R, C], where B represent some number of batch dimensions,
# R represents the number of rows, and C represents the number of columns.
# In order to apply Roth's column lemma, we need to operate on a batch of
# column vectors, so we reshape into a batch of column vectors. We put it
# at the front to ensure that broadcasting between operators to the batch
# dimensions B still works.
output = _rotate_last_dim(x, rotate_right=True)
# Also expand the shape to be [A, C, B, R]. The first dimension will be
# used to accumulate dimensions from each operator matmul.
output = output[_ops.newaxis, ...]
# In this loop, A is going to refer to the value of the accumulated
# dimension. A = 1 at the start, and will end up being self.range_dimension.
# V will refer to the last dimension. V = R at the start, and will end up
# being 1 in the end.
for operator in self.operators[:-1]:
# Reshape output from [A, C, B, V] to be
# [A, C, B, V / op.domain_dimension, op.domain_dimension]
if adjoint:
operator_dimension = operator.range_dimension_tensor()
else:
operator_dimension = operator.domain_dimension_tensor()
output = _unvec_by(output, operator_dimension)
# We are computing (XA^T) = (AX^T)^T.
# output has [A, C, B, V / op.domain_dimension, op.domain_dimension],
# which is being converted to:
# [A, C, B, V / op.domain_dimension, op.range_dimension]
output = _linalg.matrix_transpose(output)
output = operator.matmul(output,
adjoint=adjoint,
adjoint_arg=False)
output = _linalg.matrix_transpose(output)
# Rearrange it to [A * op.range_dimension, C, B, V / op.domain_dimension]
output = _rotate_last_dim(output, rotate_right=False)
output = _vec(output)
output = _rotate_last_dim(output, rotate_right=True)
# After the loop, we will have
# A = self.range_dimension / op[-1].range_dimension
# V = op[-1].domain_dimension
# We convert that using matvec to get:
# [A, C, B, op[-1].range_dimension]
output = self.operators[-1].matvec(output, adjoint=adjoint)
# Rearrange shape to be [B1, ... Bn, self.range_dimension, C]
output = _rotate_last_dim(output, rotate_right=False)
output = _vec(output)
output = _rotate_last_dim(output, rotate_right=False)
if tensor_shape.TensorShape(x.shape).is_fully_defined():
column_dim = tensor_shape.TensorShape(x.shape)[-1]
broadcast_batch_shape = common_shapes.broadcast_shape(
tensor_shape.TensorShape(x.shape)[:-2], self.batch_shape)
if adjoint:
matrix_dimensions = [self.domain_dimension, column_dim]
else:
matrix_dimensions = [self.range_dimension, column_dim]
tensorshape_util.set_shape(
output, broadcast_batch_shape.concatenate(matrix_dimensions))
return output
def _log_abs_determinant(self): # Orthogonal matrix -> log|Q| = 0. return array_ops.zeros(shape=self.batch_shape_tensor(), dtype=self.dtype)
def _log_abs_determinant(self):
return array_ops.zeros(shape=self.batch_shape_tensor(),
dtype=self.dtype)
def _determinant(self):
if self.batch_shape.is_fully_defined():
return array_ops.zeros(shape=self.batch_shape, dtype=self.dtype)
else:
return array_ops.zeros(shape=self.batch_shape_tensor(),
dtype=self.dtype)
