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.")