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LinearOperator
acting like a block circulant matrix.
tf.linalg.LinearOperatorCirculant2D(
spectrum, input_output_dtype=tf.dtypes.complex64, is_non_singular=None,
is_self_adjoint=None, is_positive_definite=None, is_square=True,
name='LinearOperatorCirculant2D'
)
This operator acts like a block 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 block circulant matrices
If A
is block circulant, with block sizes N0, N1
(N0 * N1 = N
):
A
has a block circulant structure, composed of N0 x N0
blocks, with each
block an N1 x N1
circulant matrix.
For example, with W
, X
, Y
, Z
each circulant,
A = |W Z Y X|
|X W Z Y|
|Y X W Z|
|Z Y X W|
Note that A
itself will not in general be circulant.
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.
If H.shape = [N0, N1]
, (N0 * N1 = N
):
Loosely speaking, matrix multiplication is equal to the action of a
Fourier multiplier: A u = IDFT2[ H DFT2[u] ]
.
Precisely speaking, given [N, R]
matrix u
, let DFT2[u]
be the
[N0, N1, R]
Tensor
defined by re-shaping u
to [N0, N1, R]
and taking
a two dimensional DFT across the first two dimensions. Let IDFT2
be the
inverse of DFT2
. Matrix multiplication may be expressed columnwise:
Operator properties deduced from the spectrum.
- 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, N0, N1]
, we say that H
is a Hermitian
spectrum if, with %
indicating modulus division,
H[..., n0 % N0, n1 % N1] = ComplexConjugate[ H[..., (-n0) % N0, (-n1) % N1 ].
- 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 = [[1., 2., 3.],
[4., 5., 6.],
[7., 8., 9.]]
operator = LinearOperatorCirculant2D(spectrum)
# IFFT[spectrum]
operator.convolution_kernel()
==> [[5.0+0.0j, -0.5-.3j, -0.5+.3j],
[-1.5-.9j, 0, 0],
[-1.5+.9j, 0, 0]]
operator.to_dense()
==> Complex self adjoint 9 x 9 matrix.
Example of defining in terms of a real convolution kernel,
# convolution_kernel is real ==> spectrum is Hermitian.
convolution_kernel = [[1., 2., 1.], [5., -1., 1.]]
spectrum = tf.signal.fft2d(tf.cast(convolution_kernel, tf.complex64))
# spectrum is shape [2, 3] ==> operator is shape [6, 6]
# spectrum is Hermitian ==> operator is real.
operator = LinearOperatorCirculant2D(spectrum, input_output_dtype=tf.float32)
Performance
Suppose operator
is a LinearOperatorCirculant
of shape [N, N]
,
and x.shape = [N, R]
. Then
operator.matmul(x)
isO(R*N*Log[N])
operator.solve(x)
isO(R*N*Log[N])
operator.determinant()
involves a sizeN
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 propertyX
. 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 haveX
. - If
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 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 |
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
|
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 . (deprecated)
|
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 |
Methods
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 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
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
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 ope |