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# tf.linalg.LinearOperatorCirculant2D

`LinearOperator` acting like a block circulant matrix.

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)` 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.

`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.

`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 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.

`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 `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 `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 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

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

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`.