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Base class defining a [batch of] linear operator[s].

Inherits From: Module

Subclasses of LinearOperator provide access to common methods on a (batch) matrix, without the need to materialize the matrix. This allows:

  • Matrix free computations
  • Operators that take advantage of special structure, while providing a consistent API to users.


To enable a public method, subclasses should implement the leading-underscore version of the method. The argument signature should be identical except for the omission of name="...". For example, to enable matmul(x, adjoint=False, name="matmul") a subclass should implement _matmul(x, adjoint=False).

Performance contract

Subclasses should only implement the assert methods (e.g. assert_non_singular) if they can be done in less than O(N^3) time.

Class docstrings should contain an explanation of computational complexity. Since this is a high-performance library, attention should be paid to detail, and explanations can include constants as well as Big-O notation.

Shape compatibility

LinearOperator subclasses should operate on a [batch] matrix with compatible shape. Class docstrings should define what is meant by compatible shape. Some subclasses may not support batching.


x is a batch matrix with compatible shape for matmul if

operator.shape = [B1,...,Bb] + [M, N],  b >= 0,
x.shape =   [B1,...,Bb] + [N, R]

rhs is a batch matrix with compatible shape for solve if

operator.shape = [B1,...,Bb] + [M, N],  b >= 0,
rhs.shape =   [B1,...,Bb] + [M, R]

Example docstring for subclasses.

This operator acts like a (batch) matrix A with shape [B1,...,Bb, M, N] for some b >= 0. The first b indices index a batch member. For every batch index (i1,...,ib), A[i1,...,ib, : :] is an m x n matrix. Again, this matrix A may not be materialized, but for purposes of identifying and working with compatible arguments the shape is relevant.


some_tensor = ... shape = ????
operator = MyLinOp(some_tensor)

==> [2, 4, 4]

==> Shape [2] Tensor

x = ... Shape [2, 4, 5] Tensor

==> Shape [2, 4, 5] Tensor

Shape compatibility

This operator acts on batch matrices with compatible shape. FILL IN WHAT IS MEANT BY COMPATIBLE SHAPE



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.

dtype The type of the this LinearOperator. Arguments to matmul and solve will have to be this type.
graph_parents (Deprecated) Python list of graph prerequisites of this LinearOperator Typically tensors that are passed during initialization
is_non_singular Expect that this operator is non-singular.
is_self_adjoint Expect that this operator is equal to its hermitian transpose. If dtype is real, this is equivalent to being symmetric.
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 any member of graph_parents is None or not a Tensor.
ValueError If hints are set incorrectly.

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.

dtype The DType of Tensors handled by this LinearOperator.
graph_parents List of graph dependencies of this LinearOperator. (deprecated)




is_square Return