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 _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,
input_output_dtype=circulant_operator.dtype)
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=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 np.issubdtype(spectrum.dtype, np.complexfloating):
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)
