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tf.linalg.LinearOperatorCirculant2D

TensorFlow 1 version View source on GitHub

LinearOperator acting like a block circulant matrix.

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

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 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 Tensors handled by this LinearOperator.
graph_parents List of graph dependencies of this LinearOperator. (deprecated)
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_to_tensor(
    x, name='add_to_tensor'
)

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.

cond

View source

cond(
    name='cond'
)

Returns the condition number of this linear operator.

Args
name A name for this Op.

Returns
Shape [B1,...,Bb] Tensor of same dtype as self.

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

Args
name A name for this Op.

Returns
int32 Tensor

eigvals

View source

eigvals(
    name='eigvals'
)

Returns the eigenvalues of this linear operator.

If the operator is marked as self-adjoint (via is_self_adjoint) this computation can be more efficient.

Args
name A name for this Op.

Returns
Shape [B1,...,Bb, N] Tensor of same dtype as self.

inverse

View source

inverse(
    name='inverse'
)

Returns the Inverse of this LinearOperator.

Given A representing this LinearOperator, return a LinearOperator representing A^-1.

Args
name A name scope to use for ops added by this method.

Returns
LinearOperator representing inverse of this matrix.

Raises
ValueError When the LinearOperator is not hinted to be non_singular.

log_abs_determinant

View source

log_abs_determinant(
    name='log_abs_det'
)

Log absolute value of 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.

matmul

View source

matmul(
    x, adjoint=False, adjoint_arg=False, name='matmul'
)

Transform [batch] matrix x with left multiplication: x --> Ax.

# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

X = ... # shape [..., N, R], batch matrix, R > 0.

Y = operator.matmul(X)
Y.shape
==> [..., M, R]

Y[..., :, r] = sum_j A[..., :, j] X[j, r]

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.

Returns
A LinearOperator or Tensor with shape [..., M, R] and same dtype as self.

matvec

View source

matvec(
    x, adjoint=False, name='matvec'
)

Transform [batch] vector x with left multiplication: x --> Ax.

# Make an operator acting like batch matric A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)

X = ... # shape [..., N], batch vector

Y = operator.matvec(X)
Y.shape
==> [..., M]

Y[..., :] = sum_j A[..., :, j] X[..., j]

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.

Returns
A Tensor with shape [..., M] and same dtype as self.

range_dimension_tensor

View source

range_dimension_tensor(
    name='range_dimension_tensor'
)

Dimension (in the sense of vector spaces) of the range 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 M.

Args
name A name for this Op.

Returns
int32 Tensor

shape_tensor

View source

shape_tensor(
    name='shape_tensor'
)

Shape of this LinearOperator, 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, M, N], equivalent to tf.shape(A).

Args
name A name for this Op.

Returns
int32 Tensor

solve

View source

solve(
    rhs, adjoint=False, adjoint_arg=False, name='solve'
)

Solve (exact or approx) R (batch) systems of equations: A X = rhs.

The returned Tensor will be close to an exact solution if A is well conditioned. Otherwise closeness will vary. See class docstring for details.

Examples:

# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

# Solve R > 0 linear systems for every member of the batch.
RHS = ... # shape [..., M, R]

X = operator.solve(RHS)
# X[..., :, r] is the solution to the r'th linear system
# sum_j A[..., :, j] X[..., j, r] = RHS[..., :, r]

operator.matmul(X)
==> RHS

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.

Returns
Tensor with shape [...,N, R] and same dtype as rhs.

Raises
NotImplementedError If self.is_non_singular or is_square is False.

solvevec

View source

solvevec(
    rhs, adjoint=False, name='solve'
)

Solve single equation with best effort: A X = rhs.

The returned Tensor will be close to an exact solution if A is well conditioned. Otherwise closeness will vary. See class docstring for details.

Examples:

# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

# Solve one linear system for every member of the batch.
RHS = ... # shape [..., M]

X = operator.solvevec(RHS)
# X is the solution to the linear system
# sum_j A[..., :, j] X[..., j] = RHS[..., :]

operator.matvec(X)
==> RHS

Args
rhs Tensor with same dtype as this operator. rhs is treated like a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility regarding batch dimensions.
adjoint Python bool. If True, solve the system involving the adjoint of this LinearOperator: A^H X = rhs.
name A name scope to use for ops added by this method.

Returns
Tensor with shape [...,N] and same dtype as rhs.

Raises
NotImplementedError If self.is_non_singular or is_square is False.

tensor_rank_tensor

View source

tensor_rank_tensor(
    name='tensor_rank_tensor'
)

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