Combines LinearOperators into a blockwise lower-triangular matrix.

Inherits From: LinearOperator

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 i.

For example, a blockwise 3 x 3 LinearOperatorBlockLowerTriangular is initialized with the list [[op_00], [op_10, op_11], [op_20, op_21, op_22]], where the op_ij, i < 3, j <= i, are LinearOperator instances. The LinearOperatorBlockLowerTriangular behaves as the following blockwise matrix, where 0 represents appropriately-sized [batch] matrices of zeros:

[[op_00,     0,     0],
 [op_10, op_11,     0],
 [op_20, op_21, op_22]]

Each 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 op_jj, and LinearOperators in column j must have domain_dimension equal to that of op_jj.

If each 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], where broadcast_batch_shape is the mutual broadcast of batch_shape_j, 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 operators' methods.

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)>
TensorShape([4, 4])
<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=
array([[-5., 14.],
       [-9., 34.],
       [ 1., 16.],
       [29., 18.]], dtype=float32)>

The above matmul is equivalent to:

tf.concat([operator_0.matmul(x0), ... operator_1.matmul(x0) + operator_2.matmul(x1)], axis=0)

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] + [M, N],  with b >= 0
x.shape =        [B1,...,Bb] + [N, R],  with R >= 0.

For example:

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[0].shape TensorShape([2, 3, 4]) y_blockwise[1].shape TensorShape([2, 3, 5])


Suppose operator is a LinearOperatorBlockLowerTriangular consisting of D row-partitions and D column-partitions, such that the total number of operators is N = D * (D + 1) // 2.

  • operator.matmul has complexity equal to the sum of the matmul complexities of the individual operators.
  • operator.solve has complexity equal to the sum of the solve complexities of the operators on the diagonal and the matmul complexities of the operators off the diagonal.
  • operator.determinant has complexity equal to the sum of the determinant complexities of the operators on the diagonal.

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.

operators Iterable of iterables of LinearOperator objects, each with the same dtype. Each element of operators corresponds to a row- partition, in top-to-bottom order. The operators in each row-partition are filled in left-to-right. For example, operators = [[op_0], [op_1, op_2], [op_3, op_4, op_5]] creates a LinearOperatorBlockLowerTriangular with full block structure [[op_0, 0, 0], [op_1, op_2, 0], [op_3, op_4, op_5]]. The number of operators in the ith row must be equal to i, such that each operator falls on or below the diagonal of the blockwise structure. LinearOperators that fall on the diagonal (the last elements of each row) must be square. The other LinearOperators must have domain dimension equal to the domain dimension of the LinearOperators in the same column-partition, and range dimension equal to the range dimension of the LinearOperators in the same row-partition.
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:
is_square Expect that this operator acts like square [batch] matrices. This will raise a ValueError if set to False.
name A name for this LinearOperator.

TypeError If all operators do not have the same dtype.
ValueError If operators is empty, contains an erroneous number of elements, or contains operators with incompatible shapes.

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]

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)