tf.linalg.LinearOperatorCirculant2D

LinearOperator acting like a block circulant matrix.

Inherits From: LinearOperator, Module

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Compat aliases for migration

tf.compat.v1.linalg.LinearOperatorCirculant2D

tf.linalg.LinearOperatorCirculant2D( spectrum: tf.Tensor, input_output_dtype=tf.dtypes.complex64, is_non_singular: bool = None, is_self_adjoint: bool = None, is_positive_definite: bool = None, is_square: bool = 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:

(A u)_r = IDFT2[ H * (DFT2[u])_r ]

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.

Args
`spectrum`	Shape `[B1,...,Bb, N0, N1]` `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 `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 symmetric 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`. (deprecated) Deprecated: THIS FUNCTION IS DEPRECATED. It will be removed in a future version. Instructions for updating: Do not call `graph_parents`.
`is_non_singular`
`is_positive_definite`
`is_self_adjoint`
`is_square`	Return `True/False` depending on if this operator is square.
`parameters`	Dictionary of parameters used to instantiate this `LinearOperator`.
`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`.

Args
`x`	`Tensor` with same `dtype` and shape broadcastable to `self.shape`.
`name`	A name to give this `Op`.

Args
`x`	`LinearOperator` or `Tensor` with compatible shape and same `dtype` as `self`. See class docstring for definition of compatibility.
`adjoint`	Python `bool`. If `True`, left multiply by the adjoint: `A^H x`.
`adjoint_arg`	Python `bool`. If `True`, compute `A x^H` where `x^H` is the hermitian transpose (transposition and complex conjugation).
`name`	A name for this `Op`.

Args
`x`	`Tensor` with compatible shape and same `dtype` as `self`. `x` is treated as a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility.
`adjoint`	Python `bool`. If `True`, left multiply by the adjoint: `A^H x`.
`name`	A name for this `Op`.

Args
`rhs`	`Tensor` with same `dtype` as this operator and compatible shape. `rhs` is treated like a [batch] matrix meaning for every set of leading dimensions, the last two dimensions defines a matrix. See class docstring for definition of compatibility.
`adjoint`	Python `bool`. If `True`, solve the system involving the adjoint of this `LinearOperator`: `A^H X = rhs`.
`adjoint_arg`	Python `bool`. If `True`, solve `A X = rhs^H` where `rhs^H` is the hermitian transpose (transposition and complex conjugation).
`name`	A name scope to use for ops added by this method.

tf.linalg.LinearOperatorCirculant2D Stay organized with collections Save and categorize content based on your preferences.

View aliases

Description in terms of block circulant matrices

Description in terms of the frequency spectrum

Operator properties deduced from the spectrum.

Example of a self-adjoint positive definite operator

Example of defining in terms of a real convolution kernel,

Performance

Matrix property hints

Args

Attributes

Methods

add_to_tensor

adjoint

assert_hermitian_spectrum

assert_non_singular

assert_positive_definite

assert_self_adjoint

batch_shape_tensor

block_shape_tensor

cholesky

cond

convolution_kernel

determinant

diag_part

domain_dimension_tensor

eigvals

inverse

log_abs_determinant

matmul

matvec

range_dimension_tensor

shape_tensor

solve

Examples:

solvevec

Examples:

tensor_rank_tensor

to_dense

trace

__getitem__

__matmul__

tf.linalg.LinearOperatorCirculant2D

`add_to_tensor`

`adjoint`

`assert_hermitian_spectrum`

`assert_non_singular`

`assert_positive_definite`

`assert_self_adjoint`

`batch_shape_tensor`

`block_shape_tensor`

`cholesky`

`cond`

`convolution_kernel`

`determinant`

`diag_part`

`domain_dimension_tensor`

`eigvals`

`inverse`

`log_abs_determinant`

`matmul`

`matvec`

`range_dimension_tensor`

`shape_tensor`

`solve`

`solvevec`

`tensor_rank_tensor`

`to_dense`

`trace`

`getitem`

`matmul`