tf.contrib.linalg.LinearOperatorIdentity

class tf.contrib.linalg.LinearOperatorIdentity

See the guide: Linear Algebra (contrib) > LinearOperator

LinearOperator acting like a [batch] square identity matrix.

This operator acts like a [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 an N x N matrix. This matrix A is not materialized, but for purposes of broadcasting this shape will be relevant.

LinearOperatorIdentity is initialized with num_rows, and optionally batch_shape, and dtype arguments. If batch_shape is None, this operator efficiently passes through all arguments. If batch_shape is provided, broadcasting may occur, which will require making copies.

# Create a 2 x 2 identity matrix.
operator = LinearOperatorIdentity(num_rows=2, dtype=tf.float32)

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

operator.shape
==> [2, 2]

operator.log_determinant()
==> 0.

x = ... Shape [2, 4] Tensor
operator.apply(x)
==> Shape [2, 4] Tensor, same as 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].
# This broadcast does NOT require copying data, since we can infer that y
# will be passed through without changing shape.  We are always able to infer
# this if the operator has no batch_shape.
x = operator.solve(y)
==> Shape [3, 2, 4] Tensor, same as y.

# Create a 2-batch of 2x2 identity matrices
operator = LinearOperatorIdentity(num_rows=2, batch_shape=[2])
operator.to_dense()
==> [[[1., 0.]
      [0., 1.]],
     [[1., 0.]
      [0., 1.]]]

# Here, even though the operator has a batch shape, the input is the same as
# the output, so x can be passed through without a copy.  The operator is able
# to detect that no broadcast is necessary because both x and the operator
# have statically defined shape.
x = ... Shape [2, 2, 3]
operator.apply(x)
==> Shape [2, 2, 3] Tensor, same as x

# Here the operator and x have different batch_shape, and are broadcast.
# This requires a copy, since the output is different size than the input.
x = ... Shape [1, 2, 3]
operator.apply(x)
==> Shape [2, 2, 3] Tensor, equal to [x, x]

Shape compatibility

This operator acts on [batch] matrix with compatible shape. x is a batch matrix with compatible shape for apply 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

If batch_shape initialization arg is None:

• operator.apply(x) is O(1)
• operator.solve(x) is O(1)
• operator.determinant() is O(1)

If batch_shape initialization arg is provided, and static checks cannot rule out the need to broadcast:

• operator.apply(x) is O(D1*...*Dd*N*R)
• operator.solve(x) is O(D1*...*Dd*N*R)
• operator.determinant() is O(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. 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__(num_rows, batch_shape=None, dtype=None, is_non_singular=True, is_self_adjoint=True, is_positive_definite=True, assert_proper_shapes=False, name='LinearOperatorIdentity')

Initialize a LinearOperatorIdentity.

The LinearOperatorIdentity is initialized with arguments defining dtype and shape.

This operator is able to broadcast the leading (batch) dimensions, which sometimes requires copying data. If batch_shape is None, the operator can take arguments of any batch shape without copying. See examples.

Args:

• num_rows: Scalar non-negative integer Tensor. Number of rows in the corresponding identity matrix.
• batch_shape: Optional 1-D integer Tensor. The shape of the leading dimensions. If None, this operator has no leading dimensions.
• dtype: Data type of the matrix that this operator represents.
• 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.
• 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

Raises:

• ValueError: If num_rows is determined statically to be non-scalar, or negative.
• ValueError: If batch_shape is determined statically to not be 1-D, or negative.
• ValueError: If any of the following is not True: {is_self_adjoint, is_non_singular, is_positive_definite}.

add_to_tensor(mat, name='add_to_tensor')

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.

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_identity.py.