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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,