def _matmul(self, x, adjoint=False, adjoint_arg=False): arg_dim = -1 if adjoint_arg else -2 block_dimensions = (self._block_range_dimensions() if adjoint else self._block_domain_dimensions()) blockwise_arg = linear_operator_util.arg_is_blockwise( block_dimensions, x, arg_dim) if blockwise_arg: split_x = x else: split_dim = -1 if adjoint_arg else -2 # Split input by rows normally, and otherwise columns. split_x = linear_operator_util.split_arg_into_blocks( self._block_domain_dimensions(), self._block_domain_dimension_tensors, x, axis=split_dim) result_list = [] for index, operator in enumerate(self.operators): result_list += [ operator.matmul(split_x[index], adjoint=adjoint, adjoint_arg=adjoint_arg) ] if blockwise_arg: return result_list result_list = linear_operator_util.broadcast_matrix_batch_dims( result_list) return array_ops.concat(result_list, axis=-2)
def _test_matmul_base(self, use_placeholder, shapes_info, dtype, adjoint, adjoint_arg, blockwise_arg, with_batch): # If batch dimensions are omitted, but there are # no batch dimensions for the linear operator, then # skip the test case. This is already checked with # with_batch=True. if not with_batch and len(shapes_info.shape) <= 2: return with self.session(graph=ops.Graph()) as sess: sess.graph.seed = random_seed.DEFAULT_GRAPH_SEED operator, mat = self.operator_and_matrix( shapes_info, dtype, use_placeholder=use_placeholder) x = self.make_x(operator, adjoint=adjoint, with_batch=with_batch) # If adjoint_arg, compute A X^H^H = A X. if adjoint_arg: op_matmul = operator.matmul(linalg.adjoint(x), adjoint=adjoint, adjoint_arg=adjoint_arg) else: op_matmul = operator.matmul(x, adjoint=adjoint) mat_matmul = math_ops.matmul(mat, x, adjoint_a=adjoint) if not use_placeholder: self.assertAllEqual(op_matmul.shape, mat_matmul.shape) # If the operator is blockwise, test both blockwise `x` and `Tensor` `x`; # else test only `Tensor` `x`. In both cases, evaluate all results in a # single `sess.run` call to avoid re-sampling the random `x` in graph mode. if blockwise_arg and len(operator.operators) > 1: # pylint: disable=protected-access block_dimensions = (operator._block_range_dimensions() if adjoint else operator._block_domain_dimensions()) block_dimensions_fn = (operator._block_range_dimension_tensors if adjoint else operator._block_domain_dimension_tensors) # pylint: enable=protected-access split_x = linear_operator_util.split_arg_into_blocks( block_dimensions, block_dimensions_fn, x, axis=-2) if adjoint_arg: split_x = [linalg.adjoint(y) for y in split_x] split_matmul = operator.matmul(split_x, adjoint=adjoint, adjoint_arg=adjoint_arg) self.assertEqual(len(split_matmul), len(operator.operators)) split_matmul = linear_operator_util.broadcast_matrix_batch_dims( split_matmul) fused_block_matmul = array_ops.concat(split_matmul, axis=-2) op_matmul_v, mat_matmul_v, fused_block_matmul_v = sess.run( [op_matmul, mat_matmul, fused_block_matmul]) # Check that the operator applied to blockwise input gives the same result # as matrix multiplication. self.assertAC(fused_block_matmul_v, mat_matmul_v) else: op_matmul_v, mat_matmul_v = sess.run([op_matmul, mat_matmul]) # Check that the operator applied to a `Tensor` gives the same result as # matrix multiplication. self.assertAC(op_matmul_v, mat_matmul_v)
def _test_solve_base(self, use_placeholder, shapes_info, dtype, adjoint, adjoint_arg, blockwise_arg, with_batch): # If batch dimensions are omitted, but there are # no batch dimensions for the linear operator, then # skip the test case. This is already checked with # with_batch=True. if not with_batch and len(shapes_info.shape) <= 2: return with self.session(graph=ops.Graph()) as sess: sess.graph.seed = random_seed.DEFAULT_GRAPH_SEED operator, mat = self.operator_and_matrix( shapes_info, dtype, use_placeholder=use_placeholder) rhs = self.make_rhs(operator, adjoint=adjoint, with_batch=with_batch) # If adjoint_arg, solve A X = (rhs^H)^H = rhs. if adjoint_arg: op_solve = operator.solve(linalg.adjoint(rhs), adjoint=adjoint, adjoint_arg=adjoint_arg) else: op_solve = operator.solve(rhs, adjoint=adjoint, adjoint_arg=adjoint_arg) mat_solve = linear_operator_util.matrix_solve_with_broadcast( mat, rhs, adjoint=adjoint) if not use_placeholder: self.assertAllEqual(op_solve.shape, mat_solve.shape) # If the operator is blockwise, test both blockwise rhs and `Tensor` rhs; # else test only `Tensor` rhs. In both cases, evaluate all results in a # single `sess.run` call to avoid re-sampling the random rhs in graph mode. if blockwise_arg and len(operator.operators) > 1: split_rhs = linear_operator_util.split_arg_into_blocks( operator._block_domain_dimensions(), # pylint: disable=protected-access operator._block_domain_dimension_tensors, # pylint: disable=protected-access rhs, axis=-2) if adjoint_arg: split_rhs = [linalg.adjoint(y) for y in split_rhs] split_solve = operator.solve(split_rhs, adjoint=adjoint, adjoint_arg=adjoint_arg) self.assertEqual(len(split_solve), len(operator.operators)) split_solve = linear_operator_util.broadcast_matrix_batch_dims( split_solve) fused_block_solve = array_ops.concat(split_solve, axis=-2) op_solve_v, mat_solve_v, fused_block_solve_v = sess.run( [op_solve, mat_solve, fused_block_solve]) # Check that the operator and matrix give the same solution when the rhs # is blockwise. self.assertAC(mat_solve_v, fused_block_solve_v) else: op_solve_v, mat_solve_v = sess.run([op_solve, mat_solve]) # Check that the operator and matrix give the same solution when the rhs is # a `Tensor`. self.assertAC(op_solve_v, mat_solve_v)
def solve(self, 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. Given the blockwise `n + 1`-by-`n + 1` linear operator: op = [[A_00 0 ... 0 ... 0], [A_10 A_11 ... 0 ... 0], ... [A_k0 A_k1 ... A_kk ... 0], ... [A_n0 A_n1 ... A_nk ... A_nn]] we find `x = op.solve(y)` by observing that `y_k = A_k0.matmul(x_0) + A_k1.matmul(x_1) + ... + A_kk.matmul(x_k)` and therefore `x_k = A_kk.solve(y_k - A_k0.matmul(x_0) - ... - A_k(k-1).matmul(x_(k-1)))` where `x_k` and `y_k` are the `k`th blocks obtained by decomposing `x` and `y` along their appropriate axes. We first solve `x_0 = A_00.solve(y_0)`. Proceeding inductively, we solve for `x_k`, `k = 1..n`, given `x_0..x_(k-1)`. The adjoint case is solved similarly, beginning with `x_n = A_nn.solve(y_n, adjoint=True)` and proceeding backwards. 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 ``` Args: rhs: `Tensor` with same `dtype` as this operator and compatible shape, or a list of `Tensor`s. `Tensor`s are treated like a [batch] matrices 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. """ if self.is_non_singular is False: raise NotImplementedError( "Exact solve not implemented for an operator that is expected to " "be singular.") if self.is_square is False: raise NotImplementedError( "Exact solve not implemented for an operator that is expected to " "not be square.") if isinstance(rhs, linear_operator.LinearOperator): left_operator = self.adjoint() if adjoint else self right_operator = rhs.adjoint() if adjoint_arg else rhs if (right_operator.range_dimension is not None and left_operator.domain_dimension is not None and right_operator.range_dimension != left_operator.domain_dimension): raise ValueError( "Operators are incompatible. Expected `rhs` to have dimension" " {} but got {}.".format(left_operator.domain_dimension, right_operator.range_dimension)) with self._name_scope(name): # pylint: disable=not-callable return linear_operator_algebra.solve(left_operator, right_operator) with self._name_scope(name): # pylint: disable=not-callable block_dimensions = (self._block_domain_dimensions() if adjoint else self._block_range_dimensions()) arg_dim = -1 if adjoint_arg else -2 blockwise_arg = linear_operator_util.arg_is_blockwise( block_dimensions, rhs, arg_dim) if blockwise_arg: for i, block in enumerate(rhs): if not isinstance(block, linear_operator.LinearOperator): block = ops.convert_to_tensor_v2_with_dispatch(block) self._check_input_dtype(block) block_dimensions[i].assert_is_compatible_with( block.shape[arg_dim]) rhs[i] = block if adjoint_arg: split_rhs = [linalg.adjoint(y) for y in rhs] else: split_rhs = rhs else: rhs = ops.convert_to_tensor_v2_with_dispatch(rhs, name="rhs") self._check_input_dtype(rhs) op_dimension = (self.domain_dimension if adjoint else self.range_dimension) op_dimension.assert_is_compatible_with(rhs.shape[arg_dim]) rhs = linalg.adjoint(rhs) if adjoint_arg else rhs split_rhs = linear_operator_util.split_arg_into_blocks( self._block_domain_dimensions(), self._block_domain_dimension_tensors, rhs, axis=-2) solution_list = [] if adjoint: # For an adjoint blockwise lower-triangular linear operator, the system # must be solved bottom to top. Iterate backwards over rows of the # adjoint (i.e. columns of the non-adjoint operator). for index in reversed(range(len(self.operators))): y = split_rhs[index] # Iterate top to bottom over the operators in the off-diagonal portion # of the column-partition (i.e. row-partition of the adjoint), apply # the operator to the respective block of the solution found in # previous iterations, and subtract the result from the `rhs` block. # For example,let `A`, `B`, and `D` be the linear operators in the top # row-partition of the adjoint of # `LinearOperatorBlockLowerTriangular([[A], [B, C], [D, E, F]])`, # and `x_1` and `x_2` be blocks of the solution found in previous # iterations of the outer loop. The following loop (when `index == 0`) # expresses # `Ax_0 + Bx_1 + Dx_2 = y_0` as `Ax_0 = y_0*`, where # `y_0* = y_0 - Bx_1 - Dx_2`. for j in reversed(range(index + 1, len(self.operators))): y = y - self.operators[j][index].matmul( solution_list[len(self.operators) - 1 - j], adjoint=adjoint) # Continuing the example above, solve `Ax_0 = y_0*` for `x_0`. solution_list.append(self._diagonal_operators[index].solve( y, adjoint=adjoint)) solution_list.reverse() else: # Iterate top to bottom over the row-partitions. for row, y in zip(self.operators, split_rhs): # Iterate left to right over the operators in the off-diagonal portion # of the row-partition, apply the operator to the block of the # solution found in previous iterations, and subtract the result from # the `rhs` block. For example, let `D`, `E`, and `F` be the linear # operators in the bottom row-partition of # `LinearOperatorBlockLowerTriangular([[A], [B, C], [D, E, F]])` and # `x_0` and `x_1` be blocks of the solution found in previous # iterations of the outer loop. The following loop # (when `index == 2`), expresses # `Dx_0 + Ex_1 + Fx_2 = y_2` as `Fx_2 = y_2*`, where # `y_2* = y_2 - D_x0 - Ex_1`. for i, operator in enumerate(row[:-1]): y = y - operator.matmul(solution_list[i], adjoint=adjoint) # Continuing the example above, solve `Fx_2 = y_2*` for `x_2`. solution_list.append(row[-1].solve(y, adjoint=adjoint)) if blockwise_arg: return solution_list solution_list = linear_operator_util.broadcast_matrix_batch_dims( solution_list) return array_ops.concat(solution_list, axis=-2)
def _matmul(self, x, adjoint=False, adjoint_arg=False): arg_dim = -1 if adjoint_arg else -2 block_dimensions = (self._block_range_dimensions() if adjoint else self._block_domain_dimensions()) blockwise_arg = linear_operator_util.arg_is_blockwise( block_dimensions, x, arg_dim) if blockwise_arg: split_x = x else: split_dim = -1 if adjoint_arg else -2 # Split input by columns if adjoint_arg is True, else rows split_x = linear_operator_util.split_arg_into_blocks( self._block_domain_dimensions(), self._block_domain_dimension_tensors, x, axis=split_dim) result_list = [] # Iterate over row-partitions (i.e. column-partitions of the adjoint). if adjoint: for index in range(len(self.operators)): # Begin with the operator on the diagonal and apply it to the # respective `rhs` block. result = self.operators[index][index].matmul( split_x[index], adjoint=adjoint, adjoint_arg=adjoint_arg) # Iterate top to bottom over the operators in the remainder of the # column-partition (i.e. left to right over the row-partition of the # adjoint), apply the operator to the respective `rhs` block and # accumulate the sum. For example, given the # `LinearOperatorBlockLowerTriangular`: # # op = [[A, 0, 0], # [B, C, 0], # [D, E, F]] # # if `index = 1`, the following loop calculates: # `y_1 = (C.matmul(x_1, adjoint=adjoint) + # E.matmul(x_2, adjoint=adjoint)`, # where `x_1` and `x_2` are splits of `x`. for j in range(index + 1, len(self.operators)): result += self.operators[j][index].matmul( split_x[j], adjoint=adjoint, adjoint_arg=adjoint_arg) result_list.append(result) else: for row in self.operators: # Begin with the left-most operator in the row-partition and apply it # to the first `rhs` block. result = row[0].matmul(split_x[0], adjoint=adjoint, adjoint_arg=adjoint_arg) # Iterate left to right over the operators in the remainder of the row # partition, apply the operator to the respective `rhs` block, and # accumulate the sum. for j, operator in enumerate(row[1:]): result += operator.matmul(split_x[j + 1], adjoint=adjoint, adjoint_arg=adjoint_arg) result_list.append(result) if blockwise_arg: return result_list result_list = linear_operator_util.broadcast_matrix_batch_dims( result_list) return array_ops.concat(result_list, axis=-2)
def solve(self, 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 ``` Args: rhs: `Tensor` with same `dtype` as this operator and compatible shape, or a list of `Tensor`s (for blockwise operators). `Tensor`s are treated like a [batch] matrices 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. """ if self.is_non_singular is False: raise NotImplementedError( "Exact solve not implemented for an operator that is expected to " "be singular.") if self.is_square is False: raise NotImplementedError( "Exact solve not implemented for an operator that is expected to " "not be square.") if isinstance(rhs, linear_operator.LinearOperator): left_operator = self.adjoint() if adjoint else self right_operator = rhs.adjoint() if adjoint_arg else rhs if (right_operator.range_dimension is not None and left_operator.domain_dimension is not None and right_operator.range_dimension != left_operator.domain_dimension): raise ValueError( "Operators are incompatible. Expected `rhs` to have dimension" " {} but got {}.".format(left_operator.domain_dimension, right_operator.range_dimension)) with self._name_scope(name): return linear_operator_algebra.solve(left_operator, right_operator) with self._name_scope(name): block_dimensions = (self._block_domain_dimensions() if adjoint else self._block_range_dimensions()) arg_dim = -1 if adjoint_arg else -2 blockwise_arg = linear_operator_util.arg_is_blockwise( block_dimensions, rhs, arg_dim) if blockwise_arg: split_rhs = rhs for i, block in enumerate(split_rhs): if not isinstance(block, linear_operator.LinearOperator): block = ops.convert_to_tensor_v2_with_dispatch(block) self._check_input_dtype(block) block_dimensions[i].assert_is_compatible_with( block.shape[arg_dim]) split_rhs[i] = block else: rhs = ops.convert_to_tensor_v2_with_dispatch(rhs, name="rhs") self._check_input_dtype(rhs) op_dimension = (self.domain_dimension if adjoint else self.range_dimension) op_dimension.assert_is_compatible_with(rhs.shape[arg_dim]) split_dim = -1 if adjoint_arg else -2 # Split input by rows normally, and otherwise columns. split_rhs = linear_operator_util.split_arg_into_blocks( self._block_domain_dimensions(), self._block_domain_dimension_tensors, rhs, axis=split_dim) solution_list = [] for index, operator in enumerate(self.operators): solution_list += [ operator.solve(split_rhs[index], adjoint=adjoint, adjoint_arg=adjoint_arg) ] if blockwise_arg: return solution_list solution_list = linear_operator_util.broadcast_matrix_batch_dims( solution_list) return array_ops.concat(solution_list, axis=-2)