def tensordot_matrix_product_except_dim(self, tensor2, matrices, dim):
'''
Particular type of contraction.
Compute a special contraction of two tensors self, tensor2, a list of
matrices matrices and a particular dimension dim. Note that dim must
be a scalar, while matrices must be a list array with self.order
elements.
Parameters
----------
tensor2 : FullTensor
The second tensor of the contraction.
matrices : list
The list of matrices of the contraction.
dim : int
The excluded dimension.
Returns
-------
numpy.ndarray
The result of the contraction.
'''
assert isinstance(matrices, list), 'matrices should be a list.'
assert len(matrices) == self.order, \
'len(matrices) must be self.order.'
dims = tensap.fast_setdiff(np.arange(self.order), dim)
matrices = [matrices[i] for i in dims]
tmp = tensor2.tensor_matrix_product(matrices, dims)
tmp = self.tensordot(tmp, dims, dims)
return tmp
def matricize(self, dims1, dims2=None):
'''
Return the matricization of the tensor.
Parameters
----------
dims1 : list or numpy.ndarray
The dimensions of the tensor corresponding to the first dimension
of the matricization.
dims2 : list or numpy.ndarray, optional
The dimensions of the tensor corresponding to the first dimension
of the matricization. The default is None, for which they are
deduced from dims1.
Returns
-------
FullTensor
The matricization of the tensor.
'''
dims1 = np.atleast_1d(dims1)
if dims1.size == 1 and dims1 == -1:
dims1 = np.array([self.order - 1])
if dims2 is None:
dims2 = tensap.fast_setdiff(np.arange(self.order), dims1)
else:
dims2 = np.atleast_1d(dims2)
shape1 = [self.shape[i] for i in dims1]
shape2 = [self.shape[i] for i in dims2]
tensor = FullTensor(self)
tensor = tensor.transpose(np.concatenate((dims1, dims2)))
tensor = tensor.reshape([np.prod(shape1), np.prod(shape2)])
return FullTensor(tensor)
def eval(self, x):
'''
Evaluate the CompositionalModelFunction at points x.
Parameters
----------
x : list or numpy.ndarray
The points at which the function is to be evaluated.
Returns
-------
numpy.ndarray
The evaluations of the function at points x.
'''
x = np.atleast_2d(x)
tree = self.tree
z = np.empty(tree.nb_nodes, dtype=object)
for nu in range(self.dim):
z[tree.dim2ind[nu] - 1] = x[:, nu]
for level in np.arange(np.max(tree.level), -1, -1):
nodes = tree.nodes_with_level(level)
for nod in tensap.fast_setdiff(nodes, tree.dim2ind):
ch = tree.children(nod)
z[nod - 1] = self.fun[nod - 1](*z[ch - 1])
return z[tree.root - 1]
def cat(self, tensor2, dims=None):
'''
Concatenate the tensors.
Concatenates self and tensor2 in a tensor z such that:
z(i_1 ,..., i_d) = x(i_1, ..., i_d) if i_k <= sz[k]-1 for k in dims,
z(i_1, ..., i_d) = y(i_1-sz[0], ..., i_d-sz[d-1]) if i_k >= sz[k]
for k in dims,
z(i_1, ..., i_d) = 0 otherwise, with sz = self.shape and
dims = range(self.order) if not provided.
Parameters
----------
tensor2 : FullTensor
The second tensor to be concatenaed.
dims : list or numpy.ndarray, optional
The dimensions of the concatenation. The default is None,
indicating all the dimensions.
Returns
-------
data : FullTensor
The concatenated tensors.
'''
assert self.order == tensor2.order, \
'The orders of the tensors must be equal.'
tensor1 = FullTensor(self)
order = self.order
shape1 = np.atleast_1d(self.shape)
shape2 = np.atleast_1d(tensor2.shape)
if dims is None:
dims = range(self.order)
dims = np.atleast_1d(dims)
dims_not = tensap.fast_setdiff(np.arange(order), dims)
assert np.all([a == b for a, b in zip(shape1[dims_not],
shape2[dims_not])]), \
'The dimensions of the tensors are not compatible.'
if dims.size == 1:
data = np.concatenate([tensor1.data, tensor2.data], dims[0])
else:
shape_out = np.array(shape1)
shape_out[dims] = shape1[dims] + shape2[dims]
padding = np.transpose([[0] * order, shape_out - shape1])
data = np.pad(tensor1.data, padding)
padding = np.transpose([shape_out - shape2, [0] * order])
data += np.pad(tensor2.data, padding)
return FullTensor(data)
def tensor_matrix_product(self, matrices, dims=None):
'''
Contract a tensor with matrices.
The second dimension of the matrix matrices[k] is contracted with the
k-th dimension of self, with the indices k given in dims (if provided).
Parameters
----------
matrices : numpy.ndarray or list of numpy.ndarray
The matrices to use in the product.
dims : list or numpy.ndarray, optional
Indices of the contractions. The default is None, indicating all
the dimensions.
Returns
-------
FullTensor
The tensor after the contractions with the matrices.
'''
if dims is None:
assert isinstance(matrices, (list, np.ndarray)), \
'matrices should be a list or a numpy.ndarray.'
assert len(matrices) == self.order, \
'len(matrices) must be self.order.'
dims = range(self.order)
else:
dims = np.atleast_1d(dims)
if not isinstance(matrices, list):
matrices = [matrices]
assert len(matrices) == dims.size, \
'len(matrices) must be equal to dims.size.'
# Numpy implementation
# matrices = [np.array(x) for x in matrices]
# data = self.numpy()
# if self.order == 1:
# data = np.matmul(matrices[0], data)
# else:
# k = 0
# for dim in np.nditer(dims):
# perm_dims = np.concatenate(([dim], np.arange(dim),
# np.arange(dim+1, self.order)))
# data = np.transpose(data, perm_dims)
# shape0 = np.array(data.shape)
# data = np.reshape(data, [shape0[0], np.prod(shape0[1:])])
# data = np.matmul(matrices[k], data)
# shape0[0] = matrices[k].shape[0]
# data = np.reshape(data, shape0)
# data = np.transpose(data, np.argsort(perm_dims))
# k += 1
# return FullTensor(data)
tensor = FullTensor(self)
matrices = [FullTensor(x) for x in matrices]
for i, dim in enumerate(dims):
index = np.concatenate(
(tensap.fast_setdiff(np.arange(tensor.order), dim), [dim]))
tensor = tensor.tensordot(matrices[i], dim, 1).itranspose(index)
return tensor
def eval_at_indices(self, indices, dims=None):
'''
Evaluate the tensor at indices.
If dims is None, return
s(k) = x(indices(k, 1), indices(k, 2), ..., indices(k, d)),
1 <= k <= self.shape[0].
If dims is not None, return a partial evaluation: up to a permutation
(placing the dimensions dims on the left), return
s(k, i_1, ..., i_d') = x(indices(k, 1), indices(k, 2), ...,
indices(k, M), i_1, ..., i_d'),
1 <= k <= self.shape[0], with M = dims.size and d' = self.order - M.
Parameters
----------
indices : list of numpy.ndarray
The indices of the tensor.
dims : list of numpy.ndarray, optional
The dimensions associated with the indices. The default is None,
indicating that indices refers to all the dimensions.
Returns
-------
evaluations : numpy.ndarray or FullTensor
The evaluations of the tensor.
'''
indices = np.atleast_2d(indices)
if dims is None:
dims = np.arange(self.order)
else:
dims = np.atleast_1d(dims)
if indices.shape[1] != dims.size:
indices = np.transpose(indices)
assert dims.size == indices.shape[1], \
'Wrong size of multi-indices.'
sort_ind = np.argsort(dims)
dims = dims[sort_ind]
indices = indices[:, sort_ind]
assert dims.size == indices.shape[1], 'Wrong size of multi-indices.'
if dims.size == self.order:
data = self
evaluations = np.array([data[tuple(i)] for i in indices.tolist()])
elif dims.size == 1:
ind = [':'] * self.order
ind[dims[0]] = np.ravel(indices).tolist()
evaluations = self.sub_tensor(*ind)
else:
no_dims = tensap.fast_setdiff(np.arange(self.order), dims)
indices = np.ravel_multi_index(np.transpose(indices),
[self.shape[i] for i in dims])
evaluations = self.matricize(dims).sub_tensor(indices, ':')
evaluations = evaluations.reshape([indices.size] +
[self.shape[i] for i in no_dims])
left_dims = np.arange(dims[0])
evaluations = evaluations.transpose(
np.concatenate((np.arange(1, left_dims.size + 1), [0],
np.arange(left_dims.size + 1,
self.order - dims.size + 1))))
return evaluations
def hsvd(self, tensor, tree=None, is_active_node=None):
'''
Compute the truncated svd in tree-based tensor format of tensor.
Parameters
----------
tensor : tensap.FullTensor or tensap.TreeBasedTensor
The tensor to truncate.
tree : tensap.DimensionTree, optional
The tree of the output tree-based tensor. The default is None,
indicating if tensor is a tensap.TreeBasedTensor to take
tensor.tree.
is_active_node : numpy.ndarray, optional
Logical array indicating if the nodes are active.. The default is
None, indicating if tensor is a tensap.TreeBasedTensor to take
tensor.is_active_node.
Raises
------
ValueError
If the wrong value of the atttribude _hsvd_type is provided.
NotImplementedError
If the method is not implemented for the format.
Returns
-------
out : tensap.TreeBasedTensor
The truncated tensor in tree-based tensor format.
'''
if isinstance(tensor, tensap.TreeBasedTensor):
if tree is not None or is_active_node is not None:
warnings.warn('The provided tree and/or is_active_node '
'are not taken into account when x is a '
'tensap.TreeBasedTensor.')
is_active_node = tensor.is_active_node
tree = tensor.tree
elif is_active_node is None:
is_active_node = np.full(tree.nb_nodes, True)
max_rank = np.atleast_1d(self.max_rank)
if max_rank.size == 1:
max_rank = np.repeat(max_rank, tree.nb_nodes)
max_rank[tree.root-1] = 1
local_tol = self.tolerance / np.sqrt(
np.count_nonzero(is_active_node)-1)
if isinstance(tensor, tensap.FullTensor):
root_rank_greater_than_one = tensor.order == len(tree.dim2ind)+1
tensors = np.empty(tree.nb_nodes, dtype=object)
shape = np.array(tensor.shape)
nodes_x = tree.dim2ind
ranks = np.ones(tree.nb_nodes, dtype=int)
for level in np.arange(np.max(tree.level), 0, -1):
for nod in tree.nodes_with_level(level):
if is_active_node[nod-1]:
if tree.is_leaf[nod-1]:
rep = np.nonzero(nod == nodes_x)[0][0]
else:
children = tree.children(nod)
rep = [np.nonzero(np.isin(nodes_x, x))[0][0]
for x in children]
rep_c = tensap.fast_setdiff(np.arange(nodes_x.size),
rep)
if root_rank_greater_than_one:
rep_c = np.concatenate((rep_c, [tensor.order-1]))
self.max_rank = max_rank[nod-1]
tmp = self.trunc_svd(tensor.matricize(rep).numpy(),
local_tol)
tensors[nod-1] = tmp.space[0]
ranks[nod-1] = tensors[nod-1].shape[1]
shape_loc = np.hstack((shape[rep], ranks[nod-1]))
tensors[nod-1] = tensap.FullTensor(tensors[nod-1],
shape=shape_loc)
tmp = np.matmul(tmp.space[1],
np.diag(tmp.core.data))
shape = np.hstack((shape[rep_c], ranks[nod-1]))
tensor = tensap.FullTensor(tmp, shape=shape)
if root_rank_greater_than_one:
perm = np.concatenate((np.arange(tensor.order-2),
[tensor.order-1],
[tensor.order-2]))
tensor = tensor.transpose(perm)
shape = shape[perm]
rep_c = rep_c[:-1]
nodes_x = np.hstack((nodes_x[rep_c], nod))
else:
tensors[nod-1] = []
root_ch = tree.children(tree.root)
rep = [np.nonzero(np.isin(nodes_x, x))[0][0] for x in root_ch]
if root_rank_greater_than_one:
rep = np.concatenate((rep, [tensor.order-1]))
tensors[tree.root-1] = tensor.transpose(rep)
out = tensap.TreeBasedTensor(tensors, tree)
elif isinstance(tensor, tensap.TreeBasedTensor):
if self._hsvd_type == 1:
out = tensor.orth()
gram = out.gramians()[0]
mat = np.empty(gram.shape, dtype=object)
shape = np.zeros(gram.shape)
for nod in range(gram.size):
# Truncation of the Gramian in trace norm for a control
# of Frobenius norm of the tensor
if gram[nod] is not None:
self.max_rank = max_rank[nod]
tmp = self.trunc_svd(gram[nod], local_tol ** 2)
shape[nod] = tmp.core.shape[0]
mat[nod] = np.transpose(tmp.space[0])
# Interior nodes without the root
for level in np.arange(1, np.max(tree.level)):
nod_level = tensap.fast_setdiff(
tree.nodes_with_level(level),
np.nonzero(tree.is_leaf)[0]+1)
for nod in tree.nodes_indices[nod_level-1]:
order = out.tensors[nod-1].order
out.tensors[nod-1] = \
out.tensors[nod-1].tensor_matrix_product(
mat[nod-1], order-1)
parent = tree.parent(nod)
ch_nb = tree.child_number(nod)
out.tensors[parent-1] = \
out.tensors[parent-1].tensor_matrix_product(
mat[nod-1], ch_nb-1)
# Leaves
for nod in tree.dim2ind:
if out.is_active_node[nod-1]:
order = out.tensors[nod-1].order
out.tensors[nod-1] = \
out.tensors[nod-1].tensor_matrix_product(
mat[nod-1], order-1)
parent = tree.parent(nod)
ch_nb = tree.child_number(nod)
out.tensors[parent-1] = \
out.tensors[parent-1].tensor_matrix_product(
mat[nod-1], ch_nb-1)
# Update the shape
out = out.update_attributes()
out.is_orth = False
elif self._hsvd_type == 2:
out = tensor.orth()
gram = out.gramians()[0]
for level in np.arange(np.max(tree.level), 0, -1):
for nod in tensap.fast_intersect(
tree.nodes_with_level(level), out.active_nodes):
# Truncation of the Gramian in trace norm for a control
# of Frobenius norm of the tensor
self.max_rank = max_rank[nod-1]
tmp = self.trunc_svd(gram[nod-1], local_tol ** 2)
tmp = np.transpose(tmp.space[0])
order = out.tensors[nod-1].order
out.tensors[nod-1] = \
out.tensors[nod-1].tensor_matrix_product(tmp,
order-1)
parent = tree.parent(nod)
ch_nb = tree.child_number(nod)
out.tensors[parent-1] = out.tensors[parent-1].\
tensor_matrix_product(tmp, ch_nb-1)
out = out.update_attributes()
out.is_orth = True
out.orth_node = tree.root
else:
raise ValueError('Wrong value of _hsvd_type.')
else:
raise NotImplementedError('Method not implemented.')
return out
def parameter_gradient_eval_dmrg(self,
alpha,
x=None,
dmrg_type='dmrg',
*args):
if self.evaluated_bases:
bases_eval = self.bases
elif x is not None:
bases_eval = self.bases.eval(x)
else:
raise ValueError('Must provide the evaluation points or the ' +
'bases evaluations.')
dims = np.arange(self.tensor.order)
if isinstance(self.tensor, tensap.TreeBasedTensor):
# Compute fH, the TimesMatrixEvalDiag of f with bases_eval in all
# the dimensions except the ones associated with alpha (if alpha
# is a leaf node) or with the inactive children of alpha (if
# alpha is an internal node). The tensor fH is used to compute
# the gradient of f with respect to f.tensor.tensors[alpha-1].
tree = self.tensor.tree
if tree.is_leaf[alpha - 1]:
dims = dims[tree.dim2ind != alpha]
else:
children = tree.children(alpha)
ind = tensap.fast_intersect(
tree.dim2ind, children[np.logical_not(
self.tensor.is_active_node[children - 1])])
dims = dims[np.logical_not(np.isin(tree.dim2ind, ind))]
fH = self.tensor.tensor_matrix_product(
[bases_eval[x] for x in dims], dims)
else:
if alpha <= self.tensor.order:
dims = np.delete(dims, alpha - 1)
fH = self.tensor.tensor_matrix_product(
[bases_eval[x] for x in dims], dims)
grad, g_alpha, g_gamma = \
fH.parameter_gradient_eval_diag_dmrg(alpha, bases_eval)
if isinstance(self.tensor, tensap.TreeBasedTensor):
# If the order of the children has been modified in grad, compute
# the inverse permutation.
ch = tree.children(alpha)
if ch.size == 0:
perm_1 = np.array([0])
else:
perm_1 = np.argsort(
np.concatenate(
(np.atleast_1d(ch[fH.is_active_node[ch - 1]]),
np.atleast_1d(ch[np.logical_not(
fH.is_active_node[ch - 1])]))))
gamma = tree.parent(alpha)
ch = tensap.fast_setdiff(tree.children(gamma), alpha)
perm_1b = np.argsort(
np.concatenate((np.atleast_1d(ch[fH.is_active_node[ch - 1]]),
np.atleast_1d(ch[np.logical_not(
fH.is_active_node[ch - 1])]))))
if dmrg_type == 'dmrg':
perm_1 = np.concatenate((perm_1, perm_1.size + perm_1b))
perm_2 = []
if alpha != tree.root and gamma != tree.root:
perm_2 = [
fH.tensors[alpha - 1].order +
fH.tensors[gamma - 1].order - 2
]
perm_3 = []
if alpha != tree.root and self.tensor.ranks[tree.root - 1] > 1:
perm_3 = [grad.order - 1]
grad = grad.transpose(
np.concatenate(
([0], perm_1 + 1, perm_2, perm_3)).astype(int))
elif dmrg_type == 'dmrg_low_rank':
g_alpha = g_alpha.transpose(np.concatenate(([0], perm_1 + 1)))
perm_2 = []
if gamma != tree.root:
# TODO Checks
perm_2 = [fH.tensors[gamma - 1].order - 1]
perm_3 = []
if alpha != tree.root and self.tensor.ranks[tree.root - 1] > 1:
perm_3 = [grad.order - 1]
g_gamma = g_gamma.transpose(
np.concatenate(
([0], perm_1b + 1, perm_2, perm_3)).astype(int))
grad = [g_alpha, g_gamma]
else:
raise ValueError('Wrong DMRG type.')
return grad
def parameter_gradient_eval(self, alpha, x=None, *args):
'''
Compute the gradient of the function with respect to its alpha-th
parameter, evaluated at some points.
Parameters
----------
alpha : int
The number of the parameter with respect to which compute the
gradient of self.
x : list or numpy.ndarray, optional
The points at which the gradient is to be evaluated. The default is
None, indicating to use self.bases if self.evaluated_bases is True.
Raises
------
ValueError
If no input points are provided.
Returns
-------
grad : Tensor
The gradient of the function with respect to its alpha-th
parameter, evaluated at some points.
'''
if self.evaluated_bases:
bases_eval = self.bases
elif x is not None:
bases_eval = self.bases.eval(x)
else:
raise ValueError('Must provide the evaluation points or the ' +
'bases evaluations.')
dims = np.arange(self.tensor.order)
if isinstance(self.tensor, tensap.TreeBasedTensor):
# Compute fH, the TimesMatrixEvalDiag of f with bases_eval in all
# the dimensions except the ones associated with alpha (if alpha
# is a leaf node) or with the inactive children of alpha (if
# alpha is an internal node). The tensor fH is used to compute
# the gradient of f with respect to f.tensor.tensors[alpha-1].
tree = self.tensor.tree
if tree.is_leaf[alpha - 1]:
dims = dims[tree.dim2ind != alpha]
else:
children = tree.children(alpha)
ind = tensap.fast_intersect(
tree.dim2ind, children[np.logical_not(
self.tensor.is_active_node[children - 1])])
dims = dims[np.logical_not(np.isin(tree.dim2ind, ind))]
if np.all(self.tensor.is_active_node):
fH = self.tensor.tensor_matrix_product(
[bases_eval[x] for x in dims], dims)
else:
remaining_dims = np.arange(self.tensor.order)
tensors = np.array(self.tensor.tensors)
dim2ind = np.array(tree.dim2ind)
for leaf in tensap.fast_intersect(tree.dim2ind[dims],
self.tensor.active_nodes):
dims = tensap.fast_setdiff(
dims,
np.nonzero(tree.dim2ind == leaf)[0][0])
tensors[leaf-1] = self.tensor.tensors[leaf-1].\
tensor_matrix_product(bases_eval[
np.nonzero(tree.dim2ind == leaf)[0][0]], 0)
for pa in np.unique(
tree.parent(
tensap.fast_setdiff(tree.dim2ind[dims],
self.tensor.active_nodes))):
ind = tensap.fast_intersect(tree.dim2ind[dims],
tree.children(pa))
ind = tensap.fast_setdiff(ind, self.tensor.active_nodes)
dims_loc = np.array(
[np.nonzero(x == tree.dim2ind)[0][0] for x in ind])
if len(ind) > 1:
tensors[pa-1] = self.tensor.tensors[pa-1].\
tensor_matrix_product_eval_diag([bases_eval[x] for
x in dims_loc],
tree.child_number(
ind)-1)
remaining_dims = tensap.fast_setdiff(
remaining_dims, dims_loc[1:])
if np.all(
np.logical_not(self.tensor.is_active_node[
tree.children(pa) - 1])):
dim2ind[dims_loc[0]] = tree.parent(
tree.dim2ind[dims_loc[0]])
else:
dims = tensap.fast_setdiff(dims, dims_loc[0])
dim2ind[dims_loc[1:]] = 0
perm = np.concatenate(
([tree.child_number(ind[0]) - 1],
tensap.fast_setdiff(
np.arange(tensors[pa - 1].order),
tree.child_number(ind[0]) - 1)))
tensors[pa - 1] = tensors[pa - 1].itranspose(perm)
elif len(ind) == 1:
dims = dims[dims != dims_loc]
tensors[pa-1] = self.tensor.tensors[pa-1].\
tensor_matrix_product([bases_eval[x] for
x in dims_loc],
tree.child_number(ind)-1)
dim2ind[dims_loc] = tree.dim2ind[dims_loc]
keep_ind = tensap.fast_setdiff(np.arange(tree.nb_nodes),
tree.dim2ind[dims] - 1)
adj_mat = tree.adjacency_matrix[np.ix_(keep_ind, keep_ind)]
dim2ind = dim2ind[dim2ind != 0]
ind = np.zeros(tree.nb_nodes)
ind[tensap.fast_setdiff(np.arange(tree.nb_nodes),
keep_ind)] = 1
ind = np.cumsum(ind).astype(int)
dim2ind -= ind[dim2ind - 1]
alpha = alpha - ind[alpha - 1]
tree = tensap.DimensionTree(dim2ind, adj_mat)
fH = tensap.TreeBasedTensor(tensors[keep_ind], tree)
fH = fH.remove_unique_children()
bases_eval = [bases_eval[x] for x in remaining_dims]
else:
if alpha <= self.tensor.order:
dims = np.delete(dims, alpha - 1)
fH = self.tensor.tensor_matrix_product(
[bases_eval[x] for x in dims], dims)
grad = fH.parameter_gradient_eval_diag(alpha, bases_eval)
if isinstance(self.tensor, tensap.TreeBasedTensor) and \
not tree.is_leaf[alpha-1]:
# If the order of the children has been modified in grad, compute
# the inverse permutation.
ch = tree.children(alpha)
perm_1 = np.argsort(
np.concatenate((np.atleast_1d(ch[fH.is_active_node[ch - 1]]),
np.atleast_1d(ch[np.logical_not(
fH.is_active_node[ch - 1])]))))
if alpha == tree.root:
perm_2 = []
else:
perm_2 = [fH.tensors[alpha - 1].order]
if alpha != tree.root and self.tensor.ranks[tree.root - 1] > 1:
perm_3 = [grad.order - 1]
else:
perm_3 = []
grad = grad.transpose(
np.concatenate(([0], perm_1 + 1, perm_2, perm_3)).astype(int))
return grad