tf.contrib.linalg.LinearOperator

class tf.contrib.linalg.LinearOperator

See the guide: Linear Algebra (contrib) > LinearOperator

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

Subclasses of LinearOperator provide a 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.

Subclassing

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 apply(x, adjoint=False, name="apply") a subclass should implement _apply(x, adjoint=False).

Performance contract

Subclasses should implement a method only if it can be done with a reasonable performance increase over generic dense operations, either in time, parallel scalability, or memory usage. For example, if the determinant can only be computed using tf.matrix_determinant(self.to_dense()), then determinants should not be implemented.

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

An example is:

x is a batch matrix with compatible shape for apply 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.

Examples:

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

operator.shape()
==> [2, 4, 4]

operator.log_determinant()
==> Shape [2] Tensor

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

operator.apply(x)
==> Shape [2, 4, 5] Tensor

Shape compatibility

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

Performance

FILL THIS IN

Matrix property hints

This LinearOperator is initialized with boolean flags of the form is_X, for X = non_singular, self_adjoint, positive_definite. 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.

Properties

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.get_shape()[:-2]

Returns:

TensorShape, statically determined, may be undefined.

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.

Returns:

Dimension object.

dtype

The DType of Tensors handled by this LinearOperator.

graph_parents

List of graph dependencies of this LinearOperator.

is_non_singular

is_positive_definite

is_self_adjoint

name

Name prepended to all ops created by this LinearOperator.

range_dimension

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

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

Returns:

Dimension object.

shape

TensorShape 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, M, N]), equivalent to A.get_shape().

Returns:

TensorShape, statically determined, may be undefined.

tensor_rank

Rank (in the sense of tensors) of matrix corresponding to this operator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns b + 2.

Args:

• name: A name for this `Op.

Returns:

Python integer, or None if the tensor rank is undefined.

Methods

__init__(dtype, graph_parents=None, is_non_singular=None, is_self_adjoint=None, is_positive_definite=None, name=None)

Initialize the LinearOperator.

This is a private method for subclass use. Subclasses should copy-paste this __init__ documentation.

Args:

• dtype: The type of the this LinearOperator. Arguments to apply and solve will have to be this type.
• graph_parents: 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 real part of all eigenvalues is positive. We do not require the operator to be self-adjoint to be positive-definite. See: https://en.wikipedia.org/wiki/Positive-definite_matrix\ #Extension_for_non_symmetric_matrices
• name: A name for this LinearOperator.

Raises:

• ValueError: if any member of graph_parents is None or not a Tensor.

add_to_tensor(x, name='add_to_tensor')

Add matrix represented by this operator to x. Equivalent to A + x.

Args:

• x: Tensor with same dtype and shape broadcastable to self.shape.
• name: A name to give this Op.

Returns:

A Tensor with broadcast shape and same dtype as self.

apply(x, adjoint=False, name='apply')

Transform x with left multiplication: x --> Ax.

Args:

• x: Tensor with compatible shape and same dtype as self. See class docstring for definition of compatibility.
• adjoint: Python bool. If True, left multiply by the adjoint.
• name: A name for this `Op.

Returns:

A Tensor with shape [..., M, R] and same dtype as self.

assert_non_singular(name='assert_non_singular')

Returns an Op that asserts this operator is non singular.

assert_positive_definite(name='assert_positive_definite')

Returns an Op that asserts this operator is positive definite.

Here, positive definite means the real part of all eigenvalues is positive. We do not require the operator to be self-adjoint.

Args:

• name: A name to give this Op.

Returns:

An Op that asserts this operator is positive definite.

assert_self_adjoint(name='assert_self_adjoint')

Returns an Op that asserts this operator is self-adjoint.

batch_shape_dynamic(name='batch_shape_dynamic')

Shape of batch dimensions of this operator, determined at runtime.

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

Args:

• name: A name for this `Op.

Returns:

int32 Tensor

determinant(name='det')

Determinant for every batch member.

Args:

• name: A name for this `Op.

Returns:

Tensor with shape self.batch_shape and same dtype as self.

domain_dimension_dynamic(name='domain_dimension_dynamic')

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

Determined at runtime.

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

Args:

• name: A name for this Op.

Returns:

int32 Tensor

log_abs_determinant(name='log_abs_det')

Log absolute value of determinant for every batch member.

Args:

• name: A name for this `Op.

Returns:

Tensor with shape self.batch_shape and same dtype as self.

range_dimension_dynamic(name='range_dimension_dynamic')

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

Determined at runtime.

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

Args:

• name: A name for this Op.

Returns:

int32 Tensor

shape_dynamic(name='shape_dynamic')

Shape of this LinearOperator, determined at runtime.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns a Tensor holding [B1,...,Bb, M, N], equivalent to tf.shape(A).

Args:

• name: A name for this `Op.

Returns:

int32 Tensor

solve(rhs, adjoint=False, name='solve')

Solve R (batch) systems of equations exactly: A X = rhs.

Examples:

# Create an operator acting like a 10 x 2 x 2 matrix.
operator = LinearOperator(...)
operator.shape # = 10 x 2 x 2

# Solve one linear system (R = 1) for every member of the length 10 batch.
RHS = ... # shape 10 x 2 x 1
X = operator.solve(RHS)  # shape 10 x 2 x 1

# Solve five linear systems (R = 5) for every member of the length 10 batch.
RHS = ... # shape 10 x 2 x 5
X = operator.solve(RHS)
X[3, :, 2]  # Solution to the linear system A[3, :, :] X = RHS[3, :, 2]

Args:

• rhs: Tensor with same dtype as this operator and compatible shape. See class docstring for definition of compatibility.
• adjoint: Python bool. If True, solve the system involving the adjoint of this LinearOperator.
• name: A name scope to use for ops added by this method.

Returns:

Tensor with shape [...,N, R] and same dtype as rhs.

Raises:

• ValueError: If self.is_non_singular is False.

tensor_rank_dynamic(name='tensor_rank_dynamic')

Rank (in the sense of tensors) of matrix corresponding to this operator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns b + 2.

Args:

• name: A name for this `Op.

Returns:

int32 Tensor, determined at runtime.

to_dense(name='to_dense')

Return a dense (batch) matrix representing this operator.

Defined in tensorflow/contrib/linalg/python/ops/linear_operator.py.