LinearOperator acting like a [batch] of Householder transformations.

Inherits From: LinearOperator

This operator acts like a [batch] of householder reflections with shape [B1,...,Bb, N, N] for some b >= 0. The first b indices index a batch member. For every batch index (i1,...,ib), A[i1,...,ib, : :] is an N x N matrix. This matrix A is not materialized, but for purposes of broadcasting this shape will be relevant.

LinearOperatorHouseholder is initialized with a (batch) vector.

A Householder reflection, defined via a vector v, which reflects points in R^n about the hyperplane orthogonal to v and through the origin.

# Create a 2 x 2 householder transform.
vec = [1 / np.sqrt(2), 1. / np.sqrt(2)]
operator = LinearOperatorHouseholder(vec)

==> [[0.,  -1.]
     [-1., -0.]]

==> [2, 2]

==> scalar Tensor

x = ... Shape [2, 4] Tensor
==> Shape [2, 4] Tensor

Shape compatibility

This operator acts on [batch] matrix with compatible shape. x is a batch matrix with compatible shape for matmul and solve if

operator.shape = [B1,...,Bb] + [N, N],  with b >= 0
x.shape =   [C1,...,Cc] + [N, R],
and [C1,...,Cc] broadcasts with [B1,...,Bb] to [D1,...,Dd]

Matrix property hints

This LinearOperator is initialized with boolean flags of the form is_X, for X = non_singular, self_adjoint, positive_definite, square. These have the following meaning:

  • If is_X == True, callers should expect the operator to have the property X. This is a promise that should be fulfilled, but is not a runtime assert. For example, finite floating point precision may result in these promises being violated.
  • If is_X == False, callers should expect the operator to not have X.
  • If is_X == None (the default), callers should have no expectation either way.

reflection_axis Shape [B1,...,Bb, N] Tensor with b >= 0 N >= 0. The vector defining the hyperplane to reflect about. Allowed dtypes: float16, float32, float64, complex64, complex128.
is_non_singular Expect that this operator is non-singular.
is_self_adjoint Expect that this operator is equal to its hermitian transpose. This is autoset 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: This is autoset to false.
is_square Expect that this operator acts like square [batch] matrices. This is autoset to true.
name A name for this LinearOperator.

ValueError is_self_adjoint is not True, is_positive_definite is not False or is_square is not True.

H Returns the adjoint of the current LinearOperator.

Given A representing this LinearOperator, return A*. Note that calling self.adjoint() and self.H are equivalent.

batch_shape TensorShape of batch dimensions of this LinearOperator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns TensorShape([B1,...,Bb]), equivalent to A.shape[:-2]

domain_dimension Dimension (in the sense of vector spaces) of the domain of this operator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns N.