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LinearOperator acting like a [batch] square lower triangular matrix.

Inherits From: LinearOperator, Module

Used in the notebooks

Used in the tutorials

This operator acts like a [batch] lower triangular matrix A 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.

LinearOperatorLowerTriangular is initialized with a Tensor having dimensions [B1,...,Bb, N, N]. The upper triangle of the last two dimensions is ignored.

# Create a 2 x 2 lower-triangular linear operator.
tril = [[1., 2.], [3., 4.]]
operator = LinearOperatorLowerTriangular(tril)

# The upper triangle is ignored.
==> [[1., 0.]
     [3., 4.]]

==> [2, 2]

==> scalar Tensor

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

# Create a [2, 3] batch of 4 x 4 linear operators.
tril = tf.random.normal(shape=[2, 3, 4, 4])
operator = LinearOperatorLowerTriangular(tril)

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 =        [B1,...,Bb] + [N, R],  with R >= 0.


Suppose operator is a LinearOperatorLowerTriangular of shape [N, N], and x.shape = [N, R]. Then

  • operator.matmul(x) involves N^2 * R multiplications.
  • operator.solve(x) involves N * R size N back-substitutions.
  • operator.determinant() involves a size N reduce_prod.

If instead operator and x have shape [B1,...,Bb, N, N] and [B1,...,Bb, N, R], every operation increases in complexity by B1*...*Bb.

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.

tril Shape [B1,...,Bb, N, N] with b >= 0, N >= 0. The lower triangular part of tril defines this operator. The strictly upper triangle is ignored.
is_non_singular Expect that this operator is non-singular. This operator is non-singular if and only if its diagonal elements are all non-zero.
is_self_adjoint Expect that this operator is equal to its hermitian transpose. This operator is self-adjoint only if it is diagonal with real-valued diagonal entries. In this case it is advised to use LinearOperatorDiag.
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:
is_square Expect that this operator acts like square [batch] matrices.
name A name for this LinearOperator.

ValueError If is_square is False.