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# tf.linalg.LinearOperatorCirculant3D

`LinearOperator` acting like a nested 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 nested block circulant, with block sizes `N0, N1, N2` (`N0 * N1 * N2 = N`): `A` has a block structure, composed of `N0 x N0` blocks, with each block an `N1 x N1` block circulant matrix.

For example, with `W`, `X`, `Y`, `Z` each block 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, N2]`, (`N0 * N1 * N2 = N`): Loosely speaking, matrix multiplication is equal to the action of a Fourier multiplier: `A u = IDFT3[ H DFT3[u] ]`. Precisely speaking, given `[N, R]` matrix `u`, let `DFT3[u]` be the `[N0, N1, N2, R]` `Tensor` defined by re-shaping `u` to `[N0, N1, N2, R]` and taking a three dimensional DFT across the first three dimensions. Let `IDFT3` be the inverse of `DFT3`. 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, N2]`, we say that `H` is a Hermitian spectrum if, with `%` meaning modulus division,

``````H[..., n0 % N0, n1 % N1, n2 % N2]
= ComplexConjugate[ H[..., (-n0) % N0, (-n1) % N1, (-n2) % N2] ].
``````
• 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.

### Examples

See `LinearOperatorCirculant` and `LinearOperatorCirculant2D` for examples.

#### 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 real part of all eigenvalues is positive. 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`.

Returns
An `Op` that asserts this operator has Hermitian spectrum.

### `assert_non_singular`

View source

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

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

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

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

Shape of the block dimensions of `self.spectrum`.

### `cholesky`

View source

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 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`