tf.linalg.LinearOperatorTridiag

View source on GitHub

LinearOperator acting like a [batch] square tridiagonal matrix.

Inherits From: LinearOperator

tf.linalg.LinearOperatorTridiag(
    diagonals, diagonals_format=_COMPACT, is_non_singular=None,
    is_self_adjoint=None, is_positive_definite=None, is_square=None,
    name='LinearOperatorTridiag'
)

This operator acts like a [batch] square tridiagonal 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 M matrix. This matrix A is not materialized, but for purposes of broadcasting this shape will be relevant.

Example usage:

Create a 3 x 3 tridiagonal linear operator.

superdiag = [3., 4., 5.] 
diag = [1., -1., 2.] 
subdiag = [6., 7., 8] 
operator = tf.linalg.LinearOperatorTridiag( 
   [superdiag, diag, subdiag], 
   diagonals_format='sequence') 
operator.to_dense() 
<tf.Tensor: shape=(3, 3), dtype=float32, numpy= 
array([[ 1.,  3.,  0.], 
       [ 7., -1.,  4.], 
       [ 0.,  8.,  2.]], dtype=float32)> 
operator.shape 
TensorShape([3, 3]) 

Scalar Tensor output.

operator.log_abs_determinant() 
<tf.Tensor: shape=(), dtype=float32, numpy=4.3307333> 

Create a [2, 3] batch of 4 x 4 linear operators.

diagonals = tf.random.normal(shape=[2, 3, 3, 4]) 
operator = tf.linalg.LinearOperatorTridiag( 
  diagonals, 
  diagonals_format='compact') 

Create a shape [2, 1, 4, 2] vector. Note that this shape is compatible since the batch dimensions, [2, 1], are broadcast to operator.batch_shape = [2, 3].

y = tf.random.normal(shape=[2, 1, 4, 2]) 
x = operator.solve(y) 
x 
<tf.Tensor: shape=(2, 3, 4, 2), dtype=float32, numpy=..., 
dtype=float32)> 

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 =   [C1,...,Cc] + [N, R],
and [C1,...,Cc] broadcasts with [B1,...,Bb].

Performance

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

  • operator.matmul(x) will take O(N * R) time.
  • operator.solve(x) will take O(N * R) time.

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.

Args:

  • diagonals: Tensor or list of Tensors depending on diagonals_format.

    If diagonals_format=sequence, this is a list of three Tensor's each with shape [B1, ..., Bb, N], b >= 0, N >= 0, representing the superdiagonal, diagonal and subdiagonal in that order. Note the superdiagonal is padded with an element in the last position, and the subdiagonal is padded with an element in the front.

    If diagonals_format=matrix this is a [B1, ... Bb, N, N] shaped Tensor representing the full tridiagonal matrix.

    If diagonals_format=compact this is a [B1, ... Bb, 3, N] shaped Tensor with the second to last dimension indexing the superdiagonal, diagonal and subdiagonal in that order. Note the superdiagonal is padded with an element in the last position, and the subdiagonal is padded with an element in the front.

    In every case, these Tensors are all floating dtype.

  • diagonals_format: one of matrix, sequence, or compact. Default is compact.

  • is_non_singular: Expect that this operator is non-singular.

  • is_self_adjoint: Expect that this operator is equal to its hermitian transpose. If diag.dtype is real, this is auto-set to True.

  • 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]

  • diagonals

  • diagonals_format

  • 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_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.
  • ValueError: If diag.dtype is real, and is_self_adjoint is not True.

Methods

__matmul__

View source

__matmul__(
    other
)

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.

cond

View source

cond(
    name='cond'
)

Returns the condition number of this linear operator.

Args:

  • name: A name for this Op.

Returns:

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

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.

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.