tf.linalg.LinearOperatorFullMatrix

View source on GitHub

LinearOperator that wraps a [batch] matrix.

Inherits From: LinearOperator

tf.linalg.LinearOperatorFullMatrix(
    matrix, is_non_singular=None, is_self_adjoint=None, is_positive_definite=None,
    is_square=None, name='LinearOperatorFullMatrix'
)

This operator wraps a [batch] matrix A (which is a Tensor) 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.

# Create a 2 x 2 linear operator.
matrix = [[1., 2.], [3., 4.]]
operator = LinearOperatorFullMatrix(matrix)

operator.to_dense()
==> [[1., 2.]
     [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.
matrix = tf.random.normal(shape=[2, 3, 4, 4])
operator = LinearOperatorFullMatrix(matrix)

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

Performance

LinearOperatorFullMatrix has exactly the same performance as would be achieved by using standard TensorFlow matrix ops. Intelligent choices are made based on the following initialization hints.

• If dtype is real, and is_self_adjoint and is_positive_definite, a Cholesky factorization is used for the determinant and solve.

In all cases, suppose operator is a LinearOperatorFullMatrix of shape [M, N], and x.shape = [N, R]. Then

• operator.matmul(x) is O(M * N * R).
• If M=N, operator.solve(x) is O(N^3 * R).
• If M=N, operator.determinant() is O(N^3).

If instead operator and x have shape [B1,...,Bb, M, 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.

Args:

• matrix: Shape [B1,...,Bb, M, N] with b >= 0, M, N >= 0. 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.
• 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.

Attributes:

• 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)

  Warning: THIS FUNCTION IS DEPRECATED. It will be removed in a future version. Instructions for updating: Do not call graph_parents.

• is_non_singular

• is_positive_definite

• is_self_adjoint

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

• 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.

• 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.shape.

• 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.

Raises:

• TypeError: If diag.dtype is not an allowed type.

Methods

add_to_tensor

View source

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.

adjoint

View source

adjoint(
    name='adjoint'
)

Returns the adjoint of the current LinearOperator.

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

Args:

• name: A name for this Op.

Returns:

LinearOperator which represents the adjoint of this LinearOperator.

assert_non_singular

View source

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.

assert_positive_definite

View source

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.

assert_self_adjoint

View source

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.

batch_shape_tensor

View source

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

cholesky

View source

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.

determinant

View source

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.

diag_part

View source

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.linalg.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.

domain_dimension_tensor

View source

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

eigvals

View source

eigvals(
    name='eigvals'
)

Returns the eigenvalues of this linear operator.

If the operator is marked as self-adjoint (via is_self_adjoint) this computation can be more efficient.

Note: This currently only supports self-adjoint operators.

Args:

• name: A name for this Op.

Returns:

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

inverse

View source

inverse(
    name='inverse'
)

Returns the Inverse of this LinearOperator.

Given A representing this LinearOperator, return a LinearOperator representing A^-1.

Args:

• name: A name scope to use for ops added by this method.

Returns:

LinearOperator representing inverse of this matrix.

Raises:

• ValueError: When the LinearOperator is not hinted to be non_singular.

log_abs_determinant

View source

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.

matmul

View source

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.

matvec

View source

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.

range_dimension_tensor

View source

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

shape_tensor

View source

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

solve

View source

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.

solvevec

View source

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.

tensor_rank_tensor

View source

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.

to_dense

View source

to_dense(
    name='to_dense'
)

Return a dense (batch) matrix representing this operator.

trace

View source

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.