def split_node_rq( self, node: network_components.BaseNode, left_edges: List[network_components.Edge], right_edges: List[network_components.Edge], left_name: Optional[Text] = None, right_name: Optional[Text] = None, ) -> Tuple[network_components.BaseNode, network_components.BaseNode]: """Split a `Node` using RQ (reversed QR) decomposition Let M be the matrix created by flattening left_edges and right_edges into 2 axes. Let :math:`QR = M^*` be the QR Decomposition of :math:`M^*`. This will split the network into 2 nodes. The left node's tensor will be :math:`R^*` (a lower triangular matrix) and the right node's tensor will be :math:`Q^*` (an orthonormal matrix) Args: node: The node you want to split. left_edges: The edges you want connected to the new left node. right_edges: The edges you want connected to the new right node. left_name: The name of the new left node. If `None`, a name will be generated automatically. right_name: The name of the new right node. If `None`, a name will be generated automatically. Returns: A tuple containing: left_node: A new node created that connects to all of the `left_edges`. Its underlying tensor is :math:`Q` right_node: A new node created that connects to all of the `right_edges`. Its underlying tensor is :math:`R` """ node.reorder_edges(left_edges + right_edges) q, r = self.backend.qr_decomposition(node.tensor, len(left_edges)) left_node = self.add_node(q, name=left_name) for i, edge in enumerate(left_edges): left_node.add_edge(edge, i) edge.update_axis(i, node, i, left_node) right_node = self.add_node(r, name=right_name) for i, edge in enumerate(right_edges): # i + 1 to account for the new edge. right_node.add_edge(edge, i + 1) edge.update_axis(i + len(left_edges), node, i + 1, right_node) self.connect(left_node[-1], right_node[0]) self.nodes_set.remove(node) return left_node, right_node
def split_node_full_svd( node: BaseNode, left_edges: List[Edge], right_edges: List[Edge], max_singular_values: Optional[int] = None, max_truncation_err: Optional[float] = None, left_name: Optional[Text] = None, middle_name: Optional[Text] = None, right_name: Optional[Text] = None, left_edge_name: Optional[Text] = None, right_edge_name: Optional[Text] = None, ) -> Tuple[BaseNode, BaseNode, BaseNode, Tensor]: """Split a node by doing a full singular value decomposition. Let M be the matrix created by flattening left_edges and right_edges into 2 axes. Let :math:`U S V^* = M` be the Singular Value Decomposition of :math:`M`. The left most node will be :math:`U` tensor of the SVD, the middle node is the diagonal matrix of the singular values, ordered largest to smallest, and the right most node will be the :math:`V*` tensor of the SVD. The singular value decomposition is truncated if `max_singular_values` or `max_truncation_err` is not `None`. The truncation error is the 2-norm of the vector of truncated singular values. If only `max_truncation_err` is set, as many singular values will be truncated as possible while maintaining: `norm(truncated_singular_values) <= max_truncation_err`. If only `max_singular_values` is set, the number of singular values kept will be `min(max_singular_values, number_of_singular_values)`, so that `max(0, number_of_singular_values - max_singular_values)` are truncated. If both `max_truncation_err` and `max_singular_values` are set, `max_singular_values` takes priority: The truncation error may be larger than `max_truncation_err` if required to satisfy `max_singular_values`. Args: node: The node you want to split. left_edges: The edges you want connected to the new left node. right_edges: The edges you want connected to the new right node. max_singular_values: The maximum number of singular values to keep. max_truncation_err: The maximum allowed truncation error. left_name: The name of the new left node. If None, a name will be generated automatically. middle_name: The name of the new center node. If None, a name will be generated automatically. right_name: The name of the new right node. If None, a name will be generated automatically. left_edge_name: The name of the new left `Edge` connecting the new left node (`U`) and the new central node (`S`). If `None`, a name will be generated automatically. right_edge_name: The name of the new right `Edge` connecting the new central node (`S`) and the new right node (`V*`). If `None`, a name will be generated automatically. Returns: A tuple containing: left_node: A new node created that connects to all of the `left_edges`. Its underlying tensor is :math:`U` singular_values_node: A new node that has 2 edges connecting `left_node` and `right_node`. Its underlying tensor is :math:`S` right_node: A new node created that connects to all of the `right_edges`. Its underlying tensor is :math:`V^*` truncated_singular_values: The vector of truncated singular values. """ if not hasattr(node, 'backend'): raise TypeError('Node {} of type {} has no `backend`'.format( node, type(node))) if node.axis_names and left_edge_name and right_edge_name: left_axis_names = [] right_axis_names = [right_edge_name] for edge in left_edges: left_axis_names.append(node.axis_names[edge.axis1] if edge.node1 is node else node.axis_names[edge.axis2]) for edge in right_edges: right_axis_names.append(node.axis_names[edge.axis1] if edge.node1 is node else node.axis_names[edge.axis2]) left_axis_names.append(left_edge_name) center_axis_names = [left_edge_name, right_edge_name] else: left_axis_names = None center_axis_names = None right_axis_names = None backend = node.backend node.reorder_edges(left_edges + right_edges) u, s, vh, trun_vals = backend.svd_decomposition(node.tensor, len(left_edges), max_singular_values, max_truncation_err) left_node = Node(u, name=left_name, axis_names=left_axis_names, backend=backend.name) singular_values_node = Node(backend.diag(s), name=middle_name, axis_names=center_axis_names, backend=backend.name) right_node = Node(vh, name=right_name, axis_names=right_axis_names, backend=backend.name) for i, edge in enumerate(left_edges): left_node.add_edge(edge, i) edge.update_axis(i, node, i, left_node) for i, edge in enumerate(right_edges): # i + 1 to account for the new edge. right_node.add_edge(edge, i + 1) edge.update_axis(i + len(left_edges), node, i + 1, right_node) connect(left_node.edges[-1], singular_values_node.edges[0], name=left_edge_name) connect(singular_values_node.edges[1], right_node.edges[0], name=right_edge_name) return left_node, singular_values_node, right_node, trun_vals
def split_node_rq( node: BaseNode, left_edges: List[Edge], right_edges: List[Edge], left_name: Optional[Text] = None, right_name: Optional[Text] = None, edge_name: Optional[Text] = None, ) -> Tuple[BaseNode, BaseNode]: """Split a `Node` using RQ (reversed QR) decomposition Let M be the matrix created by flattening left_edges and right_edges into 2 axes. Let :math:`QR = M^*` be the QR Decomposition of :math:`M^*`. This will split the network into 2 nodes. The left node's tensor will be :math:`R^*` (a lower triangular matrix) and the right node's tensor will be :math:`Q^*` (an orthonormal matrix) Args: node: The node you want to split. left_edges: The edges you want connected to the new left node. right_edges: The edges you want connected to the new right node. left_name: The name of the new left node. If `None`, a name will be generated automatically. right_name: The name of the new right node. If `None`, a name will be generated automatically. edge_name: The name of the new `Edge` connecting the new left and right node. If `None`, a name will be generated automatically. Returns: A tuple containing: left_node: A new node created that connects to all of the `left_edges`. Its underlying tensor is :math:`Q` right_node: A new node created that connects to all of the `right_edges`. Its underlying tensor is :math:`R` """ if not hasattr(node, 'backend'): raise TypeError('Node {} of type {} has no `backend`'.format( node, type(node))) if node.axis_names and edge_name: left_axis_names = [] right_axis_names = [edge_name] for edge in left_edges: left_axis_names.append(node.axis_names[edge.axis1] if edge.node1 is node else node.axis_names[edge.axis2]) for edge in right_edges: right_axis_names.append(node.axis_names[edge.axis1] if edge.node1 is node else node.axis_names[edge.axis2]) left_axis_names.append(edge_name) else: left_axis_names = None right_axis_names = None backend = node.backend node.reorder_edges(left_edges + right_edges) r, q = backend.rq_decomposition(node.tensor, len(left_edges)) left_node = Node(r, name=left_name, axis_names=left_axis_names, backend=backend.name) for i, edge in enumerate(left_edges): left_node.add_edge(edge, i) edge.update_axis(i, node, i, left_node) right_node = Node(q, name=right_name, axis_names=right_axis_names, backend=backend.name) for i, edge in enumerate(right_edges): # i + 1 to account for the new edge. right_node.add_edge(edge, i + 1) edge.update_axis(i + len(left_edges), node, i + 1, right_node) connect(left_node.edges[-1], right_node.edges[0], name=edge_name) return left_node, right_node
def split_node( node: BaseNode, left_edges: List[Edge], right_edges: List[Edge], max_singular_values: Optional[int] = None, max_truncation_err: Optional[float] = None, left_name: Optional[Text] = None, right_name: Optional[Text] = None, edge_name: Optional[Text] = None, ) -> Tuple[BaseNode, BaseNode, Tensor]: """Split a `Node` using Singular Value Decomposition. Let M be the matrix created by flattening left_edges and right_edges into 2 axes. Let :math:`U S V^* = M` be the Singular Value Decomposition of :math:`M`. This will split the network into 2 nodes. The left node's tensor will be :math:`U \\sqrt{S}` and the right node's tensor will be :math:`\\sqrt{S} V^*` where :math:`V^*` is the adjoint of :math:`V`. The singular value decomposition is truncated if `max_singular_values` or `max_truncation_err` is not `None`. The truncation error is the 2-norm of the vector of truncated singular values. If only `max_truncation_err` is set, as many singular values will be truncated as possible while maintaining: `norm(truncated_singular_values) <= max_truncation_err`. If only `max_singular_values` is set, the number of singular values kept will be `min(max_singular_values, number_of_singular_values)`, so that `max(0, number_of_singular_values - max_singular_values)` are truncated. If both `max_truncation_err` and `max_singular_values` are set, `max_singular_values` takes priority: The truncation error may be larger than `max_truncation_err` if required to satisfy `max_singular_values`. Args: node: The node you want to split. left_edges: The edges you want connected to the new left node. right_edges: The edges you want connected to the new right node. max_singular_values: The maximum number of singular values to keep. max_truncation_err: The maximum allowed truncation error. left_name: The name of the new left node. If `None`, a name will be generated automatically. right_name: The name of the new right node. If `None`, a name will be genenerated automatically. edge_name: The name of the new `Edge` connecting the new left and right node. If `None`, a name will be generated automatically. The new axis will get the same name as the edge. Returns: A tuple containing: left_node: A new node created that connects to all of the `left_edges`. Its underlying tensor is :math:`U \\sqrt{S}` right_node: A new node created that connects to all of the `right_edges`. Its underlying tensor is :math:`\\sqrt{S} V^*` truncated_singular_values: The vector of truncated singular values. """ if not hasattr(node, 'backend'): raise TypeError('Node {} of type {} has no `backend`'.format( node, type(node))) if node.axis_names and edge_name: left_axis_names = [] right_axis_names = [edge_name] for edge in left_edges: left_axis_names.append(node.axis_names[edge.axis1] if edge.node1 is node else node.axis_names[edge.axis2]) for edge in right_edges: right_axis_names.append(node.axis_names[edge.axis1] if edge.node1 is node else node.axis_names[edge.axis2]) left_axis_names.append(edge_name) else: left_axis_names = None right_axis_names = None backend = node.backend node.reorder_edges(left_edges + right_edges) u, s, vh, trun_vals = backend.svd_decomposition(node.tensor, len(left_edges), max_singular_values, max_truncation_err) sqrt_s = backend.sqrt(s) u_s = u * sqrt_s # We have to do this since we are doing element-wise multiplication against # the first axis of vh. If we don't, it's possible one of the other axes of # vh will be the same size as sqrt_s and would multiply across that axis # instead, which is bad. sqrt_s_broadcast_shape = backend.concat( [backend.shape(sqrt_s), [1] * (len(vh.shape) - 1)], axis=-1) vh_s = vh * backend.reshape(sqrt_s, sqrt_s_broadcast_shape) left_node = Node(u_s, name=left_name, axis_names=left_axis_names, backend=backend.name) for i, edge in enumerate(left_edges): left_node.add_edge(edge, i) edge.update_axis(i, node, i, left_node) right_node = Node(vh_s, name=right_name, axis_names=right_axis_names, backend=backend.name) for i, edge in enumerate(right_edges): # i + 1 to account for the new edge. right_node.add_edge(edge, i + 1) edge.update_axis(i + len(left_edges), node, i + 1, right_node) connect(left_node.edges[-1], right_node.edges[0], name=edge_name) node.fresh_edges(node.axis_names) return left_node, right_node, trun_vals
def split_node_full_svd(self, node: network_components.BaseNode, left_edges: List[network_components.Edge], right_edges: List[network_components.Edge], max_singular_values: Optional[int] = None, max_truncation_err: Optional[float] = None, left_name: Optional[Text] = None, middle_name: Optional[Text] = None, right_name: Optional[Text] = None ) -> Tuple[network_components.BaseNode, network_components.BaseNode, network_components.BaseNode, Tensor]: """Split a node by doing a full singular value decomposition. Let M be the matrix created by flattening left_edges and right_edges into 2 axes. Let :math:`U S V^* = M` be the Singular Value Decomposition of :math:`M`. The left most node will be :math:`U` tensor of the SVD, the middle node is the diagonal matrix of the singular values, ordered largest to smallest, and the right most node will be the :math:`V*` tensor of the SVD. The singular value decomposition is truncated if `max_singular_values` or `max_truncation_err` is not `None`. The truncation error is the 2-norm of the vector of truncated singular values. If only `max_truncation_err` is set, as many singular values will be truncated as possible while maintaining: `norm(truncated_singular_values) <= max_truncation_err`. If only `max_singular_values` is set, the number of singular values kept will be `min(max_singular_values, number_of_singular_values)`, so that `max(0, number_of_singular_values - max_singular_values)` are truncated. If both `max_truncation_err` and `max_singular_values` are set, `max_singular_values` takes priority: The truncation error may be larger than `max_truncation_err` if required to satisfy `max_singular_values`. Args: node: The node you want to split. left_edges: The edges you want connected to the new left node. right_edges: The edges you want connected to the new right node. max_singular_values: The maximum number of singular values to keep. max_truncation_err: The maximum allowed truncation error. left_name: The name of the new left node. If None, a name will be generated automatically. middle_name: The name of the new center node. If None, a name will be generated automatically. right_name: The name of the new right node. If None, a name will be generated automatically. Returns: A tuple containing: left_node: A new node created that connects to all of the `left_edges`. Its underlying tensor is :math:`U` singular_values_node: A new node that has 2 edges connecting `left_node` and `right_node`. Its underlying tensor is :math:`S` right_node: A new node created that connects to all of the `right_edges`. Its underlying tensor is :math:`V^*` truncated_singular_values: The vector of truncated singular values. """ node.reorder_edges(left_edges + right_edges) u, s, vh, trun_vals = self.backend.svd_decomposition( node.tensor, len(left_edges), max_singular_values, max_truncation_err) left_node = self.add_node(u, name=left_name) singular_values_node = self.add_node(self.backend.diag(s), name=middle_name) right_node = self.add_node(vh, name=right_name) for i, edge in enumerate(left_edges): left_node.add_edge(edge, i) edge.update_axis(i, node, i, left_node) for i, edge in enumerate(right_edges): # i + 1 to account for the new edge. right_node.add_edge(edge, i + 1) edge.update_axis(i + len(left_edges), node, i + 1, right_node) self.connect(left_node[-1], singular_values_node[0]) self.connect(singular_values_node[1], right_node[0]) self.nodes_set.remove(node) return left_node, singular_values_node, right_node, trun_vals