# tf.linalg.LinearOperatorCirculant

## Class `LinearOperatorCirculant`

`LinearOperator` acting like a circulant matrix.

### Aliases:

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`.

See http://ee.stanford.edu/~gray/toeplitz.pdf

#### 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 that`A 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.

## `__init__`

View source

``````__init__(
spectrum,
input_output_dtype=tf.dtypes.complex64,
is_non_singular=None,
is_positive_definite=None,
is_square=True,
name='LinearOperatorCirculant'
)
``````

Initialize an `LinearOperatorCirculant`.

This `LinearOperator` is initialized to have shape `[B1,...,Bb, N, N]` by providing `spectrum`, a `[B1,...,Bb, N]` `Tensor`.

If `input_output_dtype = DTYPE`:

• Arguments to methods such as `matmul` or `solve` must be `DTYPE`.
• Values returned by all methods, such as `matmul` or `determinant` will be cast to `DTYPE`.

Note that if the spectrum is not Hermitian, then this operator corresponds to a complex matrix with non-zero imaginary part. In this case, setting `input_output_dtype` to a real type will forcibly cast the output to be real, resulting in incorrect results!

If on the other hand the spectrum is Hermitian, then this operator corresponds to a real-valued matrix, and setting `input_output_dtype` to a real type is fine.

#### 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.

## Properties

### `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.

#### Args:

• `name`: A name for this `Op`.

#### Returns:

`LinearOperator` which represents the adjoint of this `LinearOperator`.

### `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]`

#### Returns:

`TensorShape`, statically determined, may be undefined.

### `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 symemtric 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.

#### Returns:

Python `integer`.

### `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`.

#### Returns:

`Dimension` object.

### `dtype`

The `DType` of `Tensor`s handled by this `LinearOperator`.

### `graph_parents`

List of graph dependencies of this `LinearOperator`.

### `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 `A` with `A.shape = [B1,...,Bb, M, N]`, then this returns `M`.

#### Returns:

`Dimension` object.

### `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`.

#### Returns:

`TensorShape`, statically determined, may be undefined.

### `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`.

#### Args:

• `name`: A name for this `Op`.

#### Returns:

Python integer, or None if the tensor rank is undefined.

## Methods

### `add_to_tensor`

View source

``````add_to_tensor(
x,
)
``````

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.

### `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