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tf.linalg.LinearOperatorScaledIdentity

LinearOperator acting like a scaled [batch] identity matrix A = c I.

This operator acts like a scaled [batch] identity 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 a scaled version of the N x N identity matrix.

LinearOperatorIdentity is initialized with num_rows, and a multiplier (a Tensor) of shape [B1,...,Bb]. N is set to num_rows, and the multiplier determines the scale for each batch member.

# Create a 2 x 2 scaled identity matrix.
operator = LinearOperatorIdentity(num_rows=2, multiplier=3.)

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

operator.shape
==> [2, 2]

operator.log_abs_determinant()
==> 2 * Log

x = ... Shape [2, 4] Tensor
operator.matmul(x)
==> 3 * x

y = tf.random.normal(shape=[3, 2, 4])
# Note that y.shape is compatible with operator.shape because operator.shape
# is broadcast to [3, 2, 2].
x = operator.solve(y)
==> 3 * x

# Create a 2-batch of 2x2 identity matrices
operator = LinearOperatorIdentity(num_rows=2, multiplier=5.)
operator.to_dense()
==> [[[5., 0.]
[0., 5.]],
[[5., 0.]
[0., 5.]]]

x = ... Shape [2, 2, 3]
operator.matmul(x)
==> 5 * x

# Here the operator and x have different batch_shape, and are broadcast.
x = ... Shape [1, 2, 3]
operator.matmul(x)
==> 5 * x

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] to [D1,...,Dd]

Performance

• operator.matmul(x) is O(D1*...*Dd*N*R)
• operator.solve(x) is O(D1*...*Dd*N*R)
• operator.determinant() is O(D1*...*Dd)

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.

num_rows Scalar non-negative integer Tensor. Number of rows in the corresponding identity matrix.
multiplier Tensor of shape [B1,...,Bb], or [] (a scalar).
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.
assert_proper_shapes Python bool. If False, only perform static checks that initialization and method arguments have proper shape. If True, and static checks are inconclusive, add asserts to the graph.
name A name for this LinearOperator

ValueError If num_rows is determined statically to be non-scalar, or negative.

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.get_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.
is_non_singular

is_positive_definite

is_square Return True/False depending on if this operator is square.
multiplier The [batch] scalar Tensor, c in cI.
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.get_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.

Methods

View source

Add matrix represented by this operator to mat. Equiv to I + mat.

Args
mat Tensor with same dtype and shape broadcastable to self.
name A name to give this Op.

Returns
A Tensor with broadcast shape and same dtype as self.

View source

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

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

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.

View source

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

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

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 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

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

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

inverse

View source

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 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

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

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

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 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 (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

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

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

Return a dense (batch) matrix representing this operator.

trace

View source

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.

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