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 _trace(self):
# The diagonal of the [[nested] block] circulant operator is the mean of
# the spectrum.
# Proof: For the [0,...,0] element, this follows from the IDFT formula.
# Then the result follows since all diagonal elements are the same.
# Therefore, the trace is the sum of the spectrum.
# Get shape of diag along with the axis over which to reduce the spectrum.
# We will reduce the spectrum over all block indices.
if tensor_shape.TensorShape(self.spectrum.shape).is_fully_defined():
spec_rank = tensor_shape.TensorShape(self.spectrum.shape).ndims
axis = np.arange(spec_rank - self.block_depth, spec_rank, dtype=np.int32)
else:
spec_rank = array_ops.rank(self.spectrum)
axis = math_ops.range(spec_rank - self.block_depth, spec_rank)
# Real diag part "re_d".
# Suppose tensor_shape.TensorShape(spectrum.shape) = [B1,...,Bb, N1, N2]
# tensor_shape.TensorShape(self.shape) = [B1,...,Bb, N, N], with N1 * N2 = N.
# tensor_shape.TensorShape(re_d_value.shape) = [B1,...,Bb]
re_d_value = math_ops.reduce_sum(math_ops.real(self.spectrum), axis=axis)
if not np.issubdtype(self.dtype, np.complexfloating):
return _ops.cast(re_d_value, self.dtype)
# Imaginary part, "im_d".
if self.is_self_adjoint:
im_d_value = array_ops.zeros_like(re_d_value)
else:
im_d_value = math_ops.reduce_sum(math_ops.imag(self.spectrum), axis=axis)
return _ops.cast(math_ops.complex(re_d_value, im_d_value), self.dtype)
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)
result = self._circulant.matmul(expanded_x,
adjoint=adjoint,
adjoint_arg=False)
return _ops.cast(result[..., :self.domain_dimension_tensor(), :],
self.dtype)
def _assert_self_adjoint(self):
imag_multiplier = math_ops.imag(self.multiplier)
return check_ops.assert_equal(
array_ops.zeros_like(imag_multiplier),
imag_multiplier,
message="LinearOperator was not self-adjoint")
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)
