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 _assert_positive_definite(self):
# This operator has the action Ax = F^H D F x,
# where D is the diagonal matrix with self.spectrum on the diag. Therefore,
# <x, Ax> = <Fx, DFx>,
# Since F is bijective, the condition for positive definite is the same as
# for a diagonal matrix, i.e. real part of spectrum is positive.
message = (
"Not positive definite: Real part of spectrum was not all positive.")
return check_ops.assert_positive(
math_ops.real(self.spectrum), message=message)
def _assert_positive_definite(self):
if np.issubdtype(self.dtype, np.complexfloating):
message = (
"Diagonal operator had diagonal entries with non-positive real part, "
"thus was not positive definite.")
else:
message = (
"Real diagonal operator had non-positive diagonal entries, "
"thus was not positive definite.")
return check_ops.assert_positive(math_ops.real(self._diag),
message=message)
def _assert_positive_definite(self):
return check_ops.assert_positive(
math_ops.real(self.multiplier),
message="LinearOperator was not positive definite.")
