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# tf.linalg.LinearOperatorCirculant

`LinearOperator` acting like a circulant matrix.

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:

```
```

#### 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,

```
```
• 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.