tf.linalg.LinearOperatorCirculant     View source on GitHub    LinearOperator acting like a circulant matrix. Inherits From: LinearOperator, Module  View aliases  Compat aliases for migration 
See Migration guide for more details. tf.compat.v1.linalg.LinearOperatorCirculant  
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. Description in terms of circulant matrices Circulant means the entries of A are generated by a single vector, the convolution kernel h: A_{mn} := h_{m-n 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. Description in terms of the frequency spectrum 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 * element-wise 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: (A u)_r = IDFT[ H * (DFT[u])_r ] Operator properties deduced from the spectrum. 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  This operator is positive definite if and only if 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[..., n % N] = ComplexConjugate[ H[..., (-n) % N] ]  This operator corresponds to a real matrix if and only if H is Hermitian. This operator is self-adjoint if and only if H is real.  See e.g. "Discrete-Time Signal Processing", Oppenheim and Schafer. Example of a self-adjoint positive definite operator # spectrum is real ==> operator is self-adjoint
# 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]]
 Example of defining in terms of a real convolution kernel # 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]]
 Example of Hermitian spectrum # 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]
 Example of forcing real dtype when spectrum is Hermitian # spectrum is shape [4] ==> operator is shape [4, 4]
# spectrum is real ==> operator is self-adjoint
# spectrum is Hermitian ==> operator is real
# spectrum has positive real part ==> operator is positive-definite.
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]
 Performance 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. Matrix property hints 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:  If 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. If is_X == False, callers should expect the operator to not have X. If is_X == None (the default), callers should have no expectation either way.  References: Toeplitz and Circulant Matrices - A Review: Gray, 2006 (pdf)
 


 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 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.   
 


 Attributes
  H   Returns the adjoint of the current LinearOperator. Given A representing this LinearOperator, return A*. Note that calling self.adjoint() and self.H are equivalent. 
 
  batch_shape   TensorShape of batch dimensions of this LinearOperator. If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns TensorShape([B1,...,Bb]), equivalent to A.shape[:-2] 
 
  block_depth   Depth of recursively defined circulant blocks defining this Operator. With A the dense representation of this Operator, block_depth = 1 means A is symmetric circulant. For example, A = |w z y x|
|x w z y|
|y x w z|
|z y x w|
 block_depth = 2 means A is block symmetric circulant with symmetric circulant blocks. For example, with W, X, Y, Z symmetric circulant, A = |W Z Y X|
|X W Z Y|
|Y X W Z|
|Z Y X W|
 block_depth = 3 means A is block symmetric circulant with block symmetric circulant blocks. 
 
  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 A with A.shape = [B1,...,Bb, M, N], then this returns N. 
 
  dtype   The DType of Tensors handled by this LinearOperator.  
  graph_parents   List of graph dependencies of this LinearOperator. (deprecated) Warning: THIS FUNCTION IS DEPRECATED. It will be removed in a future version. Instructions for updating: Do not call graph_parents. 
 
  is_non_singular  
 
  is_positive_definite  
 
  is_self_adjoint  
 
  is_square   Return True/False depending on if this operator is square.  
  parameters   Dictionary of parameters used to instantiate this LinearOperator.  
  range_dimension   Dimension (in the sense of vector spaces) of the range of this operator. If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns M. 
 
  shape   TensorShape of this LinearOperator. If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns TensorShape([B1,...,Bb, M, N]), equivalent to A.shape. 
 
  spectrum  
 
  tensor_rank   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. 
   Methods add_to_tensor View source 
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 View source 
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 View source 
assert_hermitian_spectrum(
    name='assert_hermitian_spectrum'
)
 Returns an Op that asserts this operator has Hermitian spectrum. This operator corresponds to a real-valued 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 View source 
assert_non_singular(
    name='assert_non_singular'
)
 Returns an Op that asserts this operator is non singular. This operator is considered non-singular 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 View source 
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 self-adjoint 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 View source 
assert_self_adjoint(
    name='assert_self_adjoint'
)
 Returns an Op that asserts this operator is self-adjoint. 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 self-adjoint.  
 batch_shape_tensor View source 
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 View source 
block_shape_tensor()
 Shape of the block dimensions of self.spectrum. cholesky View source 
cholesky(
    name='cholesky'
)
 Returns a Cholesky factor as a LinearOperator. Given A representing this LinearOperator, if A is positive definite self-adjoint, 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.    cond View source 
cond(
    name='cond'
)
 Returns the condition number of this linear operator.
 


 Args
  name   A name for this Op.   
 


 Returns   Shape [B1,...,Bb] Tensor of same dtype as self.  
 convolution_kernel View source 
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 View source 
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 View source 
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 View source 
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  
 eigvals View source 
eigvals(
    name='eigvals'
)
 Returns the eigenvalues of this linear operator. If the operator is marked as self-adjoint (via is_self_adjoint) this computation can be more efficient. 
Note: This currently only supports self-adjoint operators.

 


 Args
  name   A name for this Op.   
 


 Returns   Shape [B1,...,Bb, N] Tensor of same dtype as self.  
 inverse View source 
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 View source 
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 View source 
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 View source 
matvec(
    x, adjoint=False, name='matvec'
)
 Transform [batch] vector x with left multiplication: x --> Ax. # Make an operator acting like batch matrix 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 View source 
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 View source 
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 View source 
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. Examples: # 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 View source 
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. Examples: # 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 View source 
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 View source 
to_dense(
    name='to_dense'
)
 Return a dense (batch) matrix representing this operator. trace View source 
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.  
 __matmul__ View source 
__matmul__(
    other
)