treeflow.tf_util.linear_operator_upper_triangular module

class treeflow.tf_util.linear_operator_upper_triangular.LinearOperatorUpperTriangular(triu, is_non_singular=None, is_self_adjoint=None, is_positive_definite=None, is_square=None, name='LinearOperatorUpperTriangular')

Bases: LinearOperator

property triu: The upper triangular matrix defining this operator.

property H: LinearOperator

Returns the adjoint of the current LinearOperator.

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

Parameters:: name – A name for this Op.
Returns:: LinearOperator which represents the adjoint of this LinearOperator.

add_to_tensor(x, name='add_to_tensor')

Add matrix represented by this operator to x. Equivalent to A + x.

Parameters:

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(name: str = 'adjoint') → LinearOperator

Returns the adjoint of the current LinearOperator.

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

Parameters:: name – A name for this Op.
Returns:: LinearOperator which represents the adjoint of this LinearOperator.

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 `

Parameters:

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(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.

Parameters:

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(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.

Parameters:

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.

property 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]

Returns:: TensorShape, statically determined, may be undefined.

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].

Parameters:: name – A name for this Op.
Returns:: int32 Tensor

cholesky(name: str = 'cholesky') → LinearOperator

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.

Parameters:: 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(name='cond')

Returns the condition number of this linear operator.

Parameters:: name – A name for this Op.
Returns:: Shape [B1,…,Bb] Tensor of same dtype as self.

determinant(name='det')

Determinant for every batch member.

Parameters:: 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(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.] ```

Parameters:: name – A name for this Op.
Returns:: A Tensor of same dtype as self.
Return type:: diag_part

property 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.

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.

Parameters:: name – A name for this Op.
Returns:: int32 Tensor

property dtype: The DType of Tensor`s handled by this `LinearOperator.

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.

Parameters:: name – A name for this Op.
Returns:: Shape [B1,…,Bb, N] Tensor of same dtype as self.

property graph_parents

List of graph dependencies of this LinearOperator. (deprecated)

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

inverse(name: str = 'inverse') → LinearOperator

Returns the Inverse of this LinearOperator.

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

Parameters:: 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.

property is_non_singular

property is_positive_definite

property is_self_adjoint

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

log_abs_determinant(name='log_abs_det')

Log absolute value of determinant for every batch member.

Parameters:: 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(x, adjoint=False, adjoint_arg=False, name='matmul')

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

```python # 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] ```

Parameters:

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(x, adjoint=False, name='matvec')

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

```python # Make an operator acting like batch matrix 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] ```

Parameters:

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.

property name: Name prepended to all ops created by this LinearOperator.

property name_scope: Returns a tf.name_scope instance for this class.

property non_trainable_variables

Sequence of non-trainable variables owned by this module and its submodules.

Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don’t expect the return value to change.

Returns:: A sequence of variables for the current module (sorted by attribute name) followed by variables from all submodules recursively (breadth first).

property parameters: Dictionary of parameters used to instantiate this LinearOperator.

property 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.

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.

Parameters:: name – A name for this Op.
Returns:: int32 Tensor

property 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.

Returns:: TensorShape, statically determined, may be undefined.

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).

Parameters:: name – A name for this Op.
Returns:: int32 Tensor

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:

```python # 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 ```

Parameters:

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(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:

```python # 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 ```

Parameters:

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.

property submodules

Sequence of all sub-modules.

Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on).

>>> a = tf.Module()
>>> b = tf.Module()
>>> c = tf.Module()
>>> a.b = b
>>> b.c = c
>>> list(a.submodules) == [b, c]
True
>>> list(b.submodules) == [c]
True
>>> list(c.submodules) == []
True

Returns:: A sequence of all submodules.

property 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.

Parameters:: name – A name for this Op.
Returns:: Python integer, or None if the tensor rank is undefined.

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.

Parameters:: 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.

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.

Parameters:: name – A name for this Op.
Returns:: Shape [B1,…,Bb] Tensor of same dtype as self.

property trainable_variables

Sequence of trainable variables owned by this module and its submodules.

Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don’t expect the return value to change.

Returns:: A sequence of variables for the current module (sorted by attribute name) followed by variables from all submodules recursively (breadth first).

property variables

Sequence of variables owned by this module and its submodules.

Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don’t expect the return value to change.

Returns:: A sequence of variables for the current module (sorted by attribute name) followed by variables from all submodules recursively (breadth first).

classmethod with_name_scope(method)

Decorator to automatically enter the module name scope.

>>> class MyModule(tf.Module):
...   @tf.Module.with_name_scope
...   def __call__(self, x):
...     if not hasattr(self, 'w'):
...       self.w = tf.Variable(tf.random.normal([x.shape[1], 3]))
...     return tf.matmul(x, self.w)

Using the above module would produce `tf.Variable`s and `tf.Tensor`s whose names included the module name:

>>> mod = MyModule()
>>> mod(tf.ones([1, 2]))
<tf.Tensor: shape=(1, 3), dtype=float32, numpy=..., dtype=float32)>
>>> mod.w
<tf.Variable 'my_module/Variable:0' shape=(2, 3) dtype=float32,
numpy=..., dtype=float32)>

Parameters:: method – The method to wrap.
Returns:: The original method wrapped such that it enters the module’s name scope.