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tf.linalg.LinearOperatorCirculant3D

Class LinearOperatorCirculant3D

Defined in tensorflow/python/ops/linalg/linear_operator_circulant.py.

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:

(A u)_r = IDFT3[ H * (DFT3[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, 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.

__init__

__init__(
    spectrum,
    input_output_dtype=tf.complex64,
    is_non_singular=None,
    is_self_adjoint=None,
    is_positive_definite=None,
    is_square=True,
    name='LinearOperatorCirculant3D'
)

Initialize an LinearOperatorCirculant.

This LinearOperator is initialized to have shape [B1,...,Bb, N, N] by providing spectrum, a [B1,...,Bb, N0, N1, N2] Tensor with N0*N1*N2 = N.

If input_output_dtype = DTYPE:

  • Arguments to methods such as matmul or solve must be DTYPE.
  • Values returned by all methods, such as matmul or determinant will be cast to DTYPE.

Note that if the spectrum is not Hermitian, then this operator corresponds to a complex matrix with non-zero imaginary part. In this case, setting input_output_dtype to a real type will forcibly cast the output to be real, resulting in incorrect results!

If on the other hand the spectrum is Hermitian, then this operator corresponds to a real-valued matrix, and setting input_output_dtype to a real type is fine.

Args:

  • spectrum: Shape [B1,...,Bb, N] Tensor. Allowed dtypes are float32, complex64. Type can be different than input_output_dtype
  • input_output_dtype: dtype for input/output. Must be either float32 or complex64.
  • 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.

Properties

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.get_shape()[:-2]

Returns:

TensorShape, statically determined, may be undefined.

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.

Returns:

Python integer.

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.

Returns:

Dimension object.

dtype

The DType of Tensors 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.

name

Name prepended to all ops created by 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.

Returns:

Dimension object.

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.get_shape().

Returns:

TensorShape, statically determined, may be undefined.

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:

  • name: A name for this `Op.

Returns:

Python integer, or None if the tensor rank is undefined.

Methods

add_to_tensor

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.

assert_hermitian_spectrum

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

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

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

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

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

block_shape_tensor()

Shape of the block dimensions of self.spectrum.

convolution_kernel

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

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

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.matrix_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

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

log_abs_determinant

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

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: 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 Tensor with shape [..., M, R] and same dtype as self.

matvec

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

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

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

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

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

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:

int32 Tensor, determined at runtime.

to_dense

to_dense(name='to_dense')

Return a dense (batch) matrix representing this operator.

trace

trace(name='trace')

Trace of the linear operator, equal to sum of self.diag_part().

If the operator is square, this is also the sum of the eigenvalues.

Args:

  • name: A name for this Op.

Returns:

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