def _matmul(self, x, adjoint=False, adjoint_arg=False):
# Given a Toeplitz matrix, we can embed it in a Circulant matrix to perform
# efficient matrix multiplications. Given a Toeplitz matrix with first row
# [t_0, t_1, ... t_{n-1}] and first column [t0, t_{-1}, ..., t_{-(n-1)},
# let C by the circulant matrix with first column [t0, t_{-1}, ...,
# t_{-(n-1)}, 0, t_{n-1}, ..., t_1]. Also adjoin to our input vector `x`
# `n` zeros, to make it a vector of length `2n` (call it y). It can be shown
# that if we take the first n entries of `Cy`, this is equal to the Toeplitz
# multiplication. See:
# http://math.mit.edu/icg/resources/teaching/18.085-spring2015/toeplitz.pdf
# for more details.
x = linalg.adjoint(x) if adjoint_arg else x
expanded_x = array_ops.concat([x, array_ops.zeros_like(x)], axis=-2)
col = ops.convert_to_tensor(self.col)
row = ops.convert_to_tensor(self.row)
circulant_col = array_ops.concat([
col,
array_ops.zeros_like(col[..., 0:1]),
array_ops.reverse(row[..., 1:], axis=[-1])
],
axis=-1)
circulant = linear_operator_circulant.LinearOperatorCirculant(
fft_ops.fft(_to_complex(circulant_col)),
input_output_dtype=row.dtype)
result = circulant.matmul(expanded_x,
adjoint=adjoint,
adjoint_arg=False)
shape = self._shape_tensor(row=row, col=col)
return math_ops.cast(
result[..., :self._domain_dimension_tensor(shape=shape), :],
self.dtype)
def _inverse_circulant(circulant_operator):
# Inverting the spectrum is sufficient to get the inverse.
return linear_operator_circulant.LinearOperatorCirculant(
spectrum=1. / circulant_operator.spectrum,
is_non_singular=circulant_operator.is_non_singular,
is_self_adjoint=circulant_operator.is_self_adjoint,
is_positive_definite=circulant_operator.is_positive_definite,
is_square=True)
def _matmul_linear_operator_circulant_circulant(linop_a, linop_b):
return linear_operator_circulant.LinearOperatorCirculant(
spectrum=linop_a.spectrum * linop_b.spectrum,
is_non_singular=_combined_non_singular_hint(linop_a, linop_b),
is_self_adjoint=_combined_self_adjoint_hint(linop_a, linop_b),
is_positive_definite=_combined_positive_definite_hint(
linop_a, linop_b),
is_square=True)
def __init__(self,
col,
row,
is_non_singular=None,
is_self_adjoint=None,
is_positive_definite=None,
is_square=None,
name="LinearOperatorToeplitz"):
r"""Initialize a `LinearOperatorToeplitz`.
Args:
col: Shape `[B1,...,Bb, N]` `Tensor` with `b >= 0` `N >= 0`.
The first column of the operator. Allowed dtypes: `float16`, `float32`,
`float64`, `complex64`, `complex128`. Note that the first entry of
`col` is assumed to be the same as the first entry of `row`.
row: Shape `[B1,...,Bb, N]` `Tensor` with `b >= 0` `N >= 0`.
The first row of the operator. Allowed dtypes: `float16`, `float32`,
`float64`, `complex64`, `complex128`. Note that the first entry of
`row` is assumed to be the same as the first entry of `col`.
is_non_singular: Expect that this operator is non-singular.
is_self_adjoint: Expect that this operator is equal to its hermitian
transpose. If `diag.dtype` is real, this is auto-set to `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 for this `LinearOperator`.
"""
with ops.name_scope(name, values=[row, col]):
self._row = ops.convert_to_tensor(row, name="row")
self._col = ops.convert_to_tensor(col, name="col")
self._check_row_col(self._row, self._col)
circulant_col = array_ops.concat(
[self._col,
array_ops.zeros_like(self._col[..., 0:1]),
array_ops.reverse(self._row[..., 1:], axis=[-1])], axis=-1)
# To be used for matmul.
self._circulant = linear_operator_circulant.LinearOperatorCirculant(
fft_ops.fft(_to_complex(circulant_col)),
input_output_dtype=self._row.dtype)
if is_square is False: # pylint:disable=g-bool-id-comparison
raise ValueError("Only square Toeplitz operators currently supported.")
is_square = True
super(LinearOperatorToeplitz, self).__init__(
dtype=self._row.dtype,
graph_parents=[self._row, self._col],
is_non_singular=is_non_singular,
is_self_adjoint=is_self_adjoint,
is_positive_definite=is_positive_definite,
is_square=is_square,
name=name)
def _solve_linear_operator_circulant_circulant(linop_a, linop_b):
return linear_operator_circulant.LinearOperatorCirculant(
spectrum=linop_b.spectrum / linop_a.spectrum,
is_non_singular=registrations_util.combined_non_singular_hint(
linop_a, linop_b),
is_self_adjoint=registrations_util.
combined_commuting_self_adjoint_hint(linop_a, linop_b),
is_positive_definite=(
registrations_util.combined_commuting_positive_definite_hint(
linop_a, linop_b)),
is_square=True)
def _adjoint_circulant(circulant_operator):
spectrum = circulant_operator.spectrum
if spectrum.dtype.is_complex:
spectrum = math_ops.conj(spectrum)
# Conjugating the spectrum is sufficient to get the adjoint.
return linear_operator_circulant.LinearOperatorCirculant(
spectrum=spectrum,
is_non_singular=circulant_operator.is_non_singular,
is_self_adjoint=circulant_operator.is_self_adjoint,
is_positive_definite=circulant_operator.is_positive_definite,
is_square=True)
def _operator_from_kernel(kernel, d, **kwargs):
spectrum = linear_operator_circulant._FFT_OP[d](math_ops.cast(
kernel, dtypes.complex64))
if d == 1:
return linear_operator_circulant.LinearOperatorCirculant(
spectrum, **kwargs)
elif d == 2:
return linear_operator_circulant.LinearOperatorCirculant2D(
spectrum, **kwargs)
elif d == 3:
return linear_operator_circulant.LinearOperatorCirculant3D(
spectrum, **kwargs)
