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LinearOperator
acting like a circulant matrix.
tf.linalg.LinearOperatorCirculant( spectrum, input_output_dtype=tf.dtypes.complex64, is_non_singular=None, is_self_adjoint=None, is_positive_definite=None, is_square=True, name='LinearOperatorCirculant' )
This operator acts like a circulant matrix A
with shape [B1,...,Bb, N, N]
for some b >= 0
. The first b
indices index a batch member. For every batch index (i1,...,ib)
, A[i1,...,ib, : :]
is an N x N
matrix. This matrix A
is not materialized, but for purposes of broadcasting this shape will be relevant.
Circulant means the entries of A
are generated by a single vector, the convolution kernel h
: A_{mn} := h_{mn mod N}
. With h = [w, x, y, z]
,
A = w z y x x w z y y x w z z y x w
This means that the result of matrix multiplication v = Au
has Lth
column given circular convolution between h
with the Lth
column of u
.
See http://ee.stanford.edu/~gray/toeplitz.pdf
There is an equivalent description in terms of the [batch] spectrum H
and Fourier transforms. Here we consider A.shape = [N, N]
and ignore batch dimensions. Define the discrete Fourier transform (DFT) and its inverse by
DFT[ h[n] ] = H[k] := sum_{n = 0}^{N  1} h_n e^{i 2pi k n / N} IDFT[ H[k] ] = h[n] = N^{1} sum_{k = 0}^{N  1} H_k e^{i 2pi k n / N}
From these definitions, we see that
H[0] = sum_{n = 0}^{N  1} h_n H[1] = "the first positive frequency" H[N  1] = "the first negative frequency"
Loosely speaking, with *
elementwise multiplication, matrix multiplication is equal to the action of a Fourier multiplier: A u = IDFT[ H * DFT[u] ]
. Precisely speaking, given [N, R]
matrix u
, let DFT[u]
be the [N, R]
matrix with rth
column equal to the DFT of the rth
column of u
. Define the IDFT
similarly. Matrix multiplication may be expressed columnwise:
Letting U
be the kth
Euclidean basis vector, and U = IDFT[u]
. The above formulas show thatA U = H_k * U
. We conclude that the elements of H
are the eigenvalues of this operator. Therefore
Real{H} > 0
.A general property of Fourier transforms is the correspondence between Hermitian functions and real valued transforms.
Suppose H.shape = [B1,...,Bb, N]
. We say that H
is a Hermitian spectrum if, with %
meaning modulus division,
H
is Hermitian.H
is real.See e.g. "DiscreteTime Signal Processing", Oppenheim and Schafer.
# spectrum is real ==> operator is selfadjoint # spectrum is positive ==> operator is positive definite spectrum = [6., 4, 2] operator = LinearOperatorCirculant(spectrum) # IFFT[spectrum] operator.convolution_kernel() ==> [4 + 0j, 1 + 0.58j, 1  0.58j] operator.to_dense() ==> [[4 + 0.0j, 1  0.6j, 1 + 0.6j], [1 + 0.6j, 4 + 0.0j, 1  0.6j], [1  0.6j, 1 + 0.6j, 4 + 0.0j]]
# convolution_kernel is real ==> spectrum is Hermitian. convolution_kernel = [1., 2., 1.]] spectrum = tf.signal.fft(tf.cast(convolution_kernel, tf.complex64)) # spectrum is Hermitian ==> operator is real. # spectrum is shape [3] ==> operator is shape [3, 3] # We force the input/output type to be real, which allows this to operate # like a real matrix. operator = LinearOperatorCirculant(spectrum, input_output_dtype=tf.float32) operator.to_dense() ==> [[ 1, 1, 2], [ 2, 1, 1], [ 1, 2, 1]]
# spectrum is shape [3] ==> operator is shape [3, 3] # spectrum is Hermitian ==> operator is real. spectrum = [1, 1j, 1j] operator = LinearOperatorCirculant(spectrum) operator.to_dense() ==> [[ 0.33 + 0j, 0.91 + 0j, 0.24 + 0j], [0.24 + 0j, 0.33 + 0j, 0.91 + 0j], [ 0.91 + 0j, 0.24 + 0j, 0.33 + 0j]
dtype
when spectrum is Hermitian# spectrum is shape [4] ==> operator is shape [4, 4] # spectrum is real ==> operator is selfadjoint # spectrum is Hermitian ==> operator is real # spectrum has positive real part ==> operator is positivedefinite. spectrum = [6., 4, 2, 4] # Force the input dtype to be float32. # Cast the output to float32. This is fine because the operator will be # real due to Hermitian spectrum. operator = LinearOperatorCirculant(spectrum, input_output_dtype=tf.float32) operator.shape ==> [4, 4] operator.to_dense() ==> [[4, 1, 0, 1], [1, 4, 1, 0], [0, 1, 4, 1], [1, 0, 1, 4]] # convolution_kernel = tf.signal.ifft(spectrum) operator.convolution_kernel() ==> [4, 1, 0, 1]
Suppose operator
is a LinearOperatorCirculant
of shape [N, N]
, and x.shape = [N, R]
. Then
operator.matmul(x)
is O(R*N*Log[N])
operator.solve(x)
is O(R*N*Log[N])
operator.determinant()
involves a size N
reduce_prod
.If instead operator
and x
have shape [B1,...,Bb, N, N]
and [B1,...,Bb, N, R]
, every operation increases in complexity by B1*...*Bb
.
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:
is_X == True
, callers should expect the operator to have the property X
. 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.is_X == False
, callers should expect the operator to not have X
.is_X == None
(the default), callers should have no expectation either way.Args  

spectrum  Shape [B1,...,Bb, N] Tensor . Allowed dtypes: float16 , float32 , float64 , complex64 , complex128 . Type can be different than input_output_dtype 
input_output_dtype  dtype for input/output. 
is_non_singular  Expect that this operator is nonsingular. 
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 selfadjoint to be positivedefinite. See: https://en.wikipedia.org/wiki/Positivedefinite_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. 
Attributes  

H  Returns the adjoint of the current LinearOperator . Given 
batch_shape  TensorShape of batch dimensions of this LinearOperator . If this operator acts like the batch matrix 
block_depth  Depth of recursively defined circulant blocks defining this Operator . With
A = w z y x x w z y y x w z z y x w
A = W Z Y X X W Z Y Y X W Z Z Y X W

block_shape  
domain_dimension  Dimension (in the sense of vector spaces) of the domain of this operator. If this operator acts like the batch matrix 
dtype  The DType of Tensor s handled by this LinearOperator . 
graph_parents  List of graph dependencies of this LinearOperator . 
is_non_singular  
is_positive_definite  
is_self_adjoint  
is_square  Return True/False depending on if this operator is square. 
range_dimension  Dimension (in the sense of vector spaces) of the range of this operator. If this operator acts like the batch matrix 
shape  TensorShape of this LinearOperator . If this operator acts like the batch matrix 
spectrum  
tensor_rank  Rank (in the sense of tensors) of matrix corresponding to this operator. If this operator acts like the batch matrix 
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_hermitian_spectrum
assert_hermitian_spectrum( name='assert_hermitian_spectrum' )
Returns an Op
that asserts this operator has Hermitian spectrum.
This operator corresponds to a realvalued matrix if and only if its spectrum is Hermitian.
Args  

name  A name to give this Op . 
Returns  

An Op that asserts this operator has Hermitian spectrum. 
assert_non_singular
assert_non_singular( name='assert_non_singular' )
Returns an Op
that asserts this operator is non singular.
This operator is considered nonsingular 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 selfadjoint 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 selfadjoint.
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 selfadjoint. 
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 
block_shape_tensor
block_shape_tensor()
Shape of the block dimensions of self.spectrum
.
cholesky
cholesky( name='cholesky' )
Returns a Cholesky factor as a LinearOperator
.
Given A
representing this LinearOperator
, if A
is positive definite selfadjoint, 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. 
convolution_kernel
convolution_kernel( name='convolution_kernel' )
Convolution kernel corresponding to self.spectrum
.
The D
dimensional DFT of this kernel is the frequency domain spectrum of this operator.
Args  

name  A name to give this Op . 
Returns  

Tensor with dtype self.dtype . 
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 
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 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 . 
matvec
matvec( x, adjoint=False, name='matvec' )
Transform [batch] vector x
with left multiplication: x > Ax
.
# Make an operator acting like batch matric 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 . 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 . 
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.
# 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. 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. 
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.
# 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. 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. 
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 . 
© 2020 The TensorFlow Authors. All rights reserved.
Licensed under the Creative Commons Attribution License 3.0.
Code samples licensed under the Apache 2.0 License.
https://www.tensorflow.org/versions/r1.15/api_docs/python/tf/linalg/LinearOperatorCirculant