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LinearOperators into a blockwise lower-triangular matrix.
Compat aliases for migration
See Migration guide for more details.
tf.linalg.LinearOperatorBlockLowerTriangular( operators, is_non_singular=None, is_self_adjoint=None, is_positive_definite=None, is_square=None, name='LinearOperatorBlockLowerTriangular' )
This operator is initialized with a nested list of linear operators, which
are combined into a new
LinearOperator whose underlying matrix
representation is square and has each operator on or below the main diagonal,
and zero's elsewhere. Each element of the outer list is a list of
LinearOperators corresponding to a row-partition of the blockwise structure.
The number of
LinearOperators in row-partion
i must be equal to
For example, a blockwise
3 x 3
initialized with the list
[[op_00], [op_10, op_11], [op_20, op_21, op_22]],
i < 3, j <= i, are
LinearOperator instances. The
LinearOperatorBlockLowerTriangular behaves as the following blockwise
0 represents appropriately-sized [batch] matrices of zeros:
[[op_00, 0, 0], [op_10, op_11, 0], [op_20, op_21, op_22]]
op_jj on the diagonal is required to represent a square matrix, and
hence will have shape
batch_shape_j + [M_j, M_j].
LinearOperators in row
j of the blockwise structure must have
range_dimension equal to that of
LinearOperators in column
j must have
equal to that of
op_jj on the diagonal has shape
batch_shape_j + [M_j, M_j], then
the combined operator has shape
broadcast_batch_shape + [sum M_j, sum M_j],
broadcast_batch_shape is the mutual broadcast of
j = 0, 1, ..., J, assuming the intermediate batch shapes broadcast.
Even if the combined shape is well defined, the combined operator's
methods may fail due to lack of broadcasting ability in the defining
For example, to create a 4 x 4 linear operator combined of three 2 x 2 operators:
operator_0 = tf.linalg.LinearOperatorFullMatrix([[1., 2.], [3., 4.]]) operator_1 = tf.linalg.LinearOperatorFullMatrix([[1., 0.], [0., 1.]]) operator_2 = tf.linalg.LinearOperatorLowerTriangular([[5., 6.], [7., 8]]) operator = LinearOperatorBlockLowerTriangular( ... [[operator_0], [operator_1, operator_2]])
<tf.Tensor: shape=(4, 4), dtype=float32, numpy=
array([[1., 2., 0., 0.],
[3., 4., 0., 0.],
[1., 0., 5., 0.],
[0., 1., 7., 8.]], dtype=float32)>
<tf.Tensor: shape=(), dtype=float32, numpy=4.3820267>
x0 = [[1., 6.], [-3., 4.]]
x1 = [[0., 2.], [4., 0.]]
x = tf.concat([x0, x1], 0) # Shape [2, 4] Tensor
<tf.Tensor: shape=(4, 2), dtype=float32, numpy=
[ 1., 16.],
[29., 18.]], dtype=float32)>
matmul is equivalent to:
tf.concat([operator_0.matmul(x0), ... operator_1.matmul(x0) + operator_2.matmul(x1)], axis=0)
This operator acts on [batch] matrix with compatible shape.
x is a batch matrix with compatible shape for
operator.shape = [B1,...,Bb] + [M, N], with b >= 0 x.shape = [B1,...,Bb] + [N, R], with R >= 0.
Create a [2, 3] batch of 4 x 4 linear operators:
matrix_44 = tf.random.normal(shape=[2, 3, 4, 4]) operator_44 = tf.linalg.LinearOperatorFullMatrix(matrix_44)
Create a [1, 3] batch of 5 x 4 linear operators:
matrix_54 = tf.random.normal(shape=[1, 3, 5, 4]) operator_54 = tf.linalg.LinearOperatorFullMatrix(matrix_54)
Create a [1, 3] batch of 5 x 5 linear operators:
matrix_55 = tf.random.normal(shape=[1, 3, 5, 5]) operator_55 = tf.linalg.LinearOperatorFullMatrix(matrix_55)
Combine to create a [2, 3] batch of 9 x 9 operators:
operator_99 = LinearOperatorBlockLowerTriangular( ... [[operator_44], [operator_54, operator_55]]) operator_99.shape TensorShape([2, 3, 9, 9])
Create a shape [2, 1, 9] batch of vectors and apply the operator to it.
x = tf.random.normal(shape=[2, 1, 9]) y = operator_99.matvec(x) y.shape TensorShape([2, 3, 9])
Create a blockwise list of vectors and apply the operator to it. A blockwise list is returned.
x4 = tf.random.normal(shape=[2, 1, 4]) x5 = tf.random.normal(shape=[2, 3, 5]) y_blockwise = operator_99.matvec([x4, x5]) y_blockwise.shape TensorShape([2, 3, 4]) y_blockwise.shape TensorShape([2, 3, 5])
operator is a
LinearOperatorBlockLowerTriangular consisting of
D column-partitions, such that the total number of
N = D * (D + 1) // 2.
operator.matmulhas complexity equal to the sum of the
matmulcomplexities of the individual operators.
operator.solvehas complexity equal to the sum of the
solvecomplexities of the operators on the diagonal and the
matmulcomplexities of the operators off the diagonal.
operator.determinanthas complexity equal to the sum of the
determinantcomplexities of the operators on the diagonal.
Matrix property hints
LinearOperator is initialized with boolean flags of the form
X = non_singular, self_adjoint, positive_definite, square.
These have the following meaning:
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.
is_X == False, callers should expect the operator to not have
is_X == None(the default), callers should have no expectation either way.
Iterable of iterables of |