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_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 :math:`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 that connects to all of the `left_edges`.
Its underlying tensor is :math:`R^*`
right_node:
A new node that connects to all of the `right_edges`.
Its underlying tensor is :math:`Q^*`
Raises:
AttributeError: If `node` has no backend attribute
"""
if not hasattr(node, 'backend'):
raise AttributeError('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
transp_tensor = node.tensor_from_edge_order(left_edges + right_edges)
r, q = backend.rq_decomposition(transp_tensor, len(left_edges))
left_node = Node(
r, name=left_name, axis_names=left_axis_names, backend=backend)
left_axes_order = [
edge.axis1 if edge.node1 is node else edge.axis2 for edge in left_edges
]
for i, edge in enumerate(left_edges):
left_node.add_edge(edge, i)
edge.update_axis(left_axes_order[i], node, i, left_node)
right_node = Node(
q, name=right_name, axis_names=right_axis_names, backend=backend)
right_axes_order = [
edge.axis1 if edge.node1 is node else edge.axis2 for edge in right_edges
]
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(right_axes_order[i], 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
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,
relative: Optional[bool] = False,
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 :math:`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 `relative` is set `True` then `max_truncation_err` is understood
relative to the largest singular value.
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.
relative: Multiply `max_truncation_err` with the largest singular value.
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 (:math:`U`) and the new central node (:math:`S`).
If `None`, a name will be generated automatically.
right_edge_name: The name of the new right `Edge` connecting
the new central node (:math:`S`) and the new right node (:math:`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.
Raises:
AttributeError: If `node` has no backend attribute
"""
if not hasattr(node, 'backend'):
raise AttributeError('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
transp_tensor = node.tensor_from_edge_order(left_edges + right_edges)
u, s, vh, trun_vals = backend.svd_decomposition(
transp_tensor,
len(left_edges),
max_singular_values,
max_truncation_err,
relative=relative)
left_node = Node(
u, name=left_name, axis_names=left_axis_names, backend=backend)
singular_values_node = Node(
backend.diag(s),
name=middle_name,
axis_names=center_axis_names,
backend=backend)
right_node = Node(
vh, name=right_name, axis_names=right_axis_names, backend=backend)
left_axes_order = [
edge.axis1 if edge.node1 is node else edge.axis2 for edge in left_edges
]
for i, edge in enumerate(left_edges):
left_node.add_edge(edge, i)
edge.update_axis(left_axes_order[i], node, i, left_node)
right_axes_order = [
edge.axis1 if edge.node1 is node else edge.axis2 for edge in right_edges
]
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(right_axes_order[i], 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)
node.fresh_edges(node.axis_names)
return left_node, singular_values_node, right_node, trun_vals
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,
relative: Optional[bool] = False,
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 :math:`M` be the matrix created by flattening `left_edges` and
`right_edges` into 2 axes.
Let :math:`U S V^* = M` be the SVD 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 `relative` is set `True` then `max_truncation_err` is understood
relative to the largest singular value.
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.
relative: Multiply `max_truncation_err` with the largest singular value.
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.
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.
Raises:
AttributeError: If `node` has no backend attribute
"""
if not hasattr(node, 'backend'):
raise AttributeError('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
transp_tensor = node.tensor_from_edge_order(left_edges + right_edges)
u, s, vh, trun_vals = backend.svd_decomposition(
transp_tensor,
len(left_edges),
max_singular_values,
max_truncation_err,
relative=relative)
sqrt_s = backend.sqrt(s)
u_s = backend.broadcast_right_multiplication(u, sqrt_s)
vh_s = backend.broadcast_left_multiplication(sqrt_s, vh)
left_node = Node(
u_s, name=left_name, axis_names=left_axis_names, backend=backend)
left_axes_order = [
edge.axis1 if edge.node1 is node else edge.axis2 for edge in left_edges
]
for i, edge in enumerate(left_edges):
left_node.add_edge(edge, i)
edge.update_axis(left_axes_order[i], node, i, left_node)
right_node = Node(
vh_s, name=right_name, axis_names=right_axis_names, backend=backend)
right_axes_order = [
edge.axis1 if edge.node1 is node else edge.axis2 for edge in right_edges
]
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(right_axes_order[i], 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