tf.linalg.LinearOperatorLowerTriangular

Class LinearOperatorLowerTriangular

Inherits From: LinearOperator

Defined in tensorflow/python/ops/linalg/linear_operator_lower_triangular.py.

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

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.
operator.to_dense()
==> [[1., 0.]
     [3., 4.]]

operator.shape
==> [2, 2]

operator.log_abs_determinant()
==> scalar Tensor

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

Performance

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.

__init__

__init__(
    tril,
    is_non_singular=None,
    is_self_adjoint=None,
    is_positive_definite=None,
    is_square=None,
    name='LinearOperatorLowerTriangular'
)

Initialize a LinearOperatorLowerTriangular.

Args:

  • 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: https://en.wikipedia.org/wiki/Positive-definite_matrix#Extension_for_non-symmetric_matrices
  • is_square: Expect that this operator acts like square [batch] matrices.
  • name: A name for this LinearOperator.

Raises:

  • ValueError: If is_square is False.

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

is_square

Return True/False depending on if this operator is square.

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

tf.linalg.LinearOperatorLowerTriangular.add_to_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.

tf.linalg.LinearOperatorLowerTriangular.assert_non_singular

assert_non_singular(name='assert_non_singular')

Returns an Op that asserts this operator is non singular.

This operator is considered non-singular if

ConditionNumber < max{100, range_dimension, domain_dimension} * eps,
eps := np.finfo(self.dtype.as_numpy_dtype).eps

Args:

  • name: A string name to prepend to created ops.

Returns:

An Assert Op, that, when run, will raise an InvalidArgumentError if the operator is singular.

tf.linalg.LinearOperatorLowerTriangular.assert_positive_definite

assert_positive_definite(name='assert_positive_definite')

Returns an Op that asserts this operator is positive definite.

Here, positive definite means that 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.

Args:

  • name: A name to give this Op.

Returns:

An Assert Op, that, when run, will raise an InvalidArgumentError if the operator is not positive definite.

tf.linalg.LinearOperatorLowerTriangular.assert_self_adjoint

assert_self_adjoint(name='assert_self_adjoint')

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

Here we check that this operator is exactly equal to its hermitian transpose.

Args:

  • name: A string name to prepend to created ops.

Returns:

An Assert Op, that, when run, will raise an InvalidArgumentError if the operator is not self-adjoint.

tf.linalg.LinearOperatorLowerTriangular.batch_shape_tensor

batch_shape_tensor(name='batch_shape_tensor')

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

tf.linalg.LinearOperatorLowerTriangular.cholesky

cholesky(name='cholesky')

Returns a Cholesky factor as a LinearOperator.

Given A representing this LinearOperator, if A is positive definite self-adjoint, return L, where A = L L^T, i.e. the cholesky decomposition.

Args:

  • name: A name for this Op.

Returns:

LinearOperator which represents the lower triangular matrix in the Cholesky decomposition.

Raises:

  • ValueError: When the LinearOperator is not hinted to be positive definite and self adjoint.

tf.linalg.LinearOperatorLowerTriangular.determinant

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.

Raises:

  • NotImplementedError: If self.is_square is False.

tf.linalg.LinearOperatorLowerTriangular.diag_part

diag_part(name='diag_part')

Efficiently get the [batch] diagonal part of this operator.

If this operator has shape [B1,...,Bb, M, N], this returns a Tensor diagonal, of shape [B1,...,Bb, min(M, N)], where diagonal[b1,...,bb, i] = self.to_dense()[b1,...,bb, i, i].

my_operator = LinearOperatorDiag([1., 2.])

# Efficiently get the diagonal
my_operator.diag_part()
==> [1., 2.]

# Equivalent, but inefficient method
tf.matrix_diag_part(my_operator.to_dense())
==> [1., 2.]

Args:

  • name: A name for this Op.

Returns:

  • diag_part: A Tensor of same dtype as self.

tf.linalg.LinearOperatorLowerTriangular.domain_dimension_tensor

domain_dimension_tensor(name='domain_dimension_tensor')

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

tf.linalg.LinearOperatorLowerTriangular.log_abs_determinant

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.

Raises:

  • NotImplementedError: If self.is_square is False.

tf.linalg.LinearOperatorLowerTriangular.matmul

matmul(
    x,
    adjoint=False,
    adjoint_arg=False,
    name='matmul'
)

Transform [batch] matrix x with left multiplication: x --> Ax.

# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

X = ... # shape [..., N, R], batch matrix, R > 0.

Y = operator.matmul(X)
Y.shape
==> [..., M, R]

Y[..., :, r] = sum_j A[..., :, j] X[j, r]

Args:

  • x: LinearOperator or 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: A^H x.
  • adjoint_arg: Python bool. If True, compute A x^H where x^H is the hermitian transpose (transposition and complex conjugation).
  • name: A name for this Op.

Returns:

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

tf.linalg.LinearOperatorLowerTriangular.matvec

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

Transform [batch] vector x with left multiplication: x --> Ax.

# Make an operator acting like batch matric A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)

X = ... # shape [..., N], batch vector

Y = operator.matvec(X)
Y.shape
==> [..., M]

Y[..., :] = sum_j A[..., :, j] X[..., j]

Args:

  • x: Tensor with compatible shape and same dtype as self. x is treated as a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility.
  • adjoint: Python bool. If True, left multiply by the adjoint: A^H x.
  • name: A name for this Op.

Returns:

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

tf.linalg.LinearOperatorLowerTriangular.range_dimension_tensor

range_dimension_tensor(name='range_dimension_tensor')

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

tf.linalg.LinearOperatorLowerTriangular.shape_tensor

shape_tensor(name='shape_tensor')

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

tf.linalg.LinearOperatorLowerTriangular.solve

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

Solve (exact or approx) R (batch) systems of equations: A X = rhs.

The returned Tensor will be close to an exact solution if A is well conditioned. Otherwise closeness will vary. See class docstring for details.

Examples:

# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

# Solve R > 0 linear systems for every member of the batch.
RHS = ... # shape [..., M, R]

X = operator.solve(RHS)
# X[..., :, r] is the solution to the r'th linear system
# sum_j A[..., :, j] X[..., j, r] = RHS[..., :, r]

operator.matmul(X)
==> RHS

Args:

  • rhs: Tensor with same dtype as this operator and compatible shape. rhs is treated like a [batch] matrix meaning for every set of leading dimensions, the last two dimensions defines a matrix. See class docstring for definition of compatibility.
  • adjoint: Python bool. If True, solve the system involving the adjoint of this LinearOperator: A^H X = rhs.
  • adjoint_arg: Python bool. If True, solve A X = rhs^H where rhs^H is the hermitian transpose (transposition and complex conjugation).
  • name: A name scope to use for ops added by this method.

Returns:

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

Raises:

  • NotImplementedError: If self.is_non_singular or is_square is False.

tf.linalg.LinearOperatorLowerTriangular.solvevec

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

Solve single equation with best effort: A X = rhs.

The returned Tensor will be close to an exact solution if A is well conditioned. Otherwise closeness will vary. See class docstring for details.

Examples:

# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

# Solve one linear system for every member of the batch.
RHS = ... # shape [..., M]

X = operator.solvevec(RHS)
# X is the solution to the linear system
# sum_j A[..., :, j] X[..., j] = RHS[..., :]

operator.matvec(X)
==> RHS

Args:

  • rhs: Tensor with same dtype as this operator. rhs is treated like a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility regarding batch dimensions.
  • adjoint: Python bool. If True, solve the system involving the adjoint of this LinearOperator: A^H X = rhs.
  • name: A name scope to use for ops added by this method.

Returns:

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

Raises:

  • NotImplementedError: If self.is_non_singular or is_square is False.

tf.linalg.LinearOperatorLowerTriangular.tensor_rank_tensor

tensor_rank_tensor(name='tensor_rank_tensor')

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.

tf.linalg.LinearOperatorLowerTriangular.to_dense

to_dense(name='to_dense')

Return a dense (batch) matrix representing this operator.

tf.linalg.LinearOperatorLowerTriangular.trace

trace(name='trace')

Trace of the linear operator, equal to sum of self.diag_part().

If the operator is square, this is also the sum of the eigenvalues.

Args:

  • name: A name for this Op.

Returns:

Shape [B1,...,Bb] Tensor of same dtype as self.