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tensor_train.py
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tensor_train.py
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# -*- coding: utf-8 -*-
import scikit_tt.utils as utl
import time as _time
import numpy as np
from scipy import linalg
from typing import List, Tuple, Union, Optional
class TT(object):
"""
Tensor train class
Tensor trains [1]_ are defined in terms of different attributes. That is, a tensor train with order ``d`` is
given by a list of 4-dimensional tensors
``[cores[0] , ..., cores[d-1]]``,
where ``cores[i]`` is an ndarray with dimensions
``ranks[i] x row_dims[i] x col_dims[i] x ranks[i+1]``.
There is no distinguish between tensor trains and tensor trains operators, i.e. a classical tensor train is
represented by cores with column dimensions equal to 1.
An instance of the tensor train class can be initialized either from a list of cores, i.e. ``t = TT(cores)``
where ``cores`` is a list as described above, or from a full tensor representation, i.e. ``t = TT(x)`` where
``x`` is an ndarray with dimensions
``row_dims[0] x ... x row_dims[-1] x col_dims[0] x ... x col_dims[-1]``.
In the latter case, the tensor is decomposed into the TT format. For more information on the implemented tensor
operations, we refer to [2]_.
Attributes
----------
order : int
Order of the tensor train
row_dims : list[int]
List of the row dimensions of the tensor train
col_dims : list[int]
List of the column dimensions of the tensor train
ranks : list[int]
List of the ranks of the tensor train
cores : list[np.ndarray]
List of the cores of the tensor train
Methods
-------
print(t)
String representation of tensor trains
+
Sum of two tensor trains
-
Difference of two tensor trains
*
Multiplication of tensor trains and scalars
@/dot(t,u)
Multiplication of two tensor trains
tensordot
Index contraction between two tensortrains
rank_tensordot
Index contraction between TT and matrix along the rank-dimension
concatenate
Concatenate cores of two TT
transpose
Transpose of a tensor train
rank_transpose
Rank-transpose of a tensor train
conj
Complex conjugate of a tensor train
isoperator
Check is given tensor train is an operator
copy
Deep copy of a tensor train
element
Element of t at given indices
full
Convert tensor train to full format
matricize
Matricization of a tensor train
ortho_left
Left-orthonormalization of a tensor train
ortho_right
Right-orthonormalization of a tensor train
ortho
Left- and right-orthonormalization of a tensor train
norm
Norm of a tensor train
tt2qtt
Conversion from TT format into QTT format
qtt2tt
Conversion from QTT format into TT format
svd
Computation of a global SVD of a tensor train
pinv
Computation of the pseudoinverse of a tensor train
diag
Construction of diagonal MPO from MPS
squeeze
Squeeze TT decomposition
zeros
Tensor train filled with zeros
ones
Tensor train filled with ones
eye
Identity tensor train
unit
Canonical unit tensor
rand
Random tensor train
canonical
Full-rank tensor train consisting of tensor products of the canonical basis
uniform
Uniformly distributed tensor train
residual_error
Compute the residual error ||A@x-b|| in TT format.
References
----------
.. [1] I. V. Oseledets, "Tensor-Train Decomposition", SIAM Journal on Scientific Computing 33 (5), 2011
.. [2] P. Gelß. "The Tensor-Train Format and Its Applications: Modeling and Analysis of Chemical Reaction
Networks, Catalytic Processes, Fluid Flows, and Brownian Dynamics", Freie Universität Berlin, 2017
Examples
--------
Construct tensor train from list of cores:
>>> import numpy as np
>>> from scikit_tt.tensor_train import TT
>>>
>>> cores = [np.random.rand(1, 2, 3, 4), np.random.rand(4, 3, 2, 1)]
>>> t = TT(cores)
>>> print(t)
>>> ...
Construct tensor train from ndarray:
>>> import numpy as np
>>> from scikit_tt.tensor_train import TT
>>>
>>> x = np.random.rand(1, 2, 3, 4, 5, 6)
>>> t = TT(x)
>>> print(t)
>>> ...
"""
def __init__(self, x: Union[List[np.ndarray], np.ndarray],
threshold: float=0,
max_rank: int=np.infty,
progress: bool=False,
string: str=None):
"""
Parameters
----------
x : list[np.ndarray] or np.ndarray
either a list[TT] cores or a full tensor
threshold : float, optional
threshold for reduced SVD decompositions, default is 0
max_rank : int, optional
maximum rank of the left-orthonormalized tensor train, default is np.infty
Raises
------
TypeError
if x is neither a list of ndarray nor a single ndarray
ValueError
if list elements of x are not 4-dimensional tensors or shapes do not match
ValueError
if number of dimensions of the ndarray x is not a multiple of 2
"""
# initialize from list of cores
if isinstance(x, list):
# check if orders of list elements are correct
if np.all([x[i].ndim == 4 for i in range(len(x))]):
# check if ranks are correct
if np.all([x[i].shape[3] == x[i + 1].shape[0] for i in range(len(x) - 1)]):
# define order, row dimensions, column dimensions, ranks, and cores
self.order = len(x)
self.row_dims = [x[i].shape[1] for i in range(self.order)]
self.col_dims = [x[i].shape[2] for i in range(self.order)]
self.ranks = [x[i].shape[0] for i in range(self.order)] + [x[-1].shape[3]]
self.cores = x
# rank reduction
if threshold != 0 or max_rank != np.infty:
self.ortho(threshold=threshold, max_rank=max_rank)
else:
raise ValueError('Shapes of list elements do not match.')
else:
raise ValueError('List elements must be 4-dimensional arrays.')
# initialize from full array
elif isinstance(x, np.ndarray):
# check if order of ndarray is a multiple of 2
if np.mod(x.ndim, 2) == 0:
# show progress
if string is None:
string = 'HOSVD'
start_time = utl.progress(string, 0, show=progress)
# define order, row dimensions, column dimensions, ranks, and cores
order = len(x.shape) // 2
row_dims = x.shape[:order]
col_dims = x.shape[order:]
ranks = [1] * (order + 1)
cores = []
# permute dimensions, e.g., for order = 4: p = [0, 4, 1, 5, 2, 6, 3, 7]
p = [order * j + i for i in range(order) for j in range(2)]
y = np.transpose(x, p).copy()
# decompose the full tensor
for i in range(order - 1):
# reshape residual tensor
m = ranks[i] * row_dims[i] * col_dims[i]
n = np.prod(row_dims[i + 1:]) * np.prod(col_dims[i + 1:])
y = np.reshape(y, [m, n])
# apply SVD in order to isolate modes
[u, s, v] = linalg.svd(y, full_matrices=False)
# rank reduction
if threshold != 0:
indices = np.where(s / s[0] > threshold)[0]
u = u[:, indices]
s = s[indices]
v = v[indices, :]
if max_rank != np.infty:
u = u[:, :np.minimum(u.shape[1], max_rank)]
s = s[:np.minimum(s.shape[0], max_rank)]
v = v[:np.minimum(v.shape[0], max_rank), :]
# define new TT core
ranks[i + 1] = u.shape[1]
cores.append(np.reshape(u, [ranks[i], row_dims[i], col_dims[i], ranks[i + 1]]))
# set new residual tensor
y = np.diag(s).dot(v)
# show progress
utl.progress(string, 100 * (i + 1) / order, cpu_time=_time.time() - start_time, show=progress)
# define last TT core
cores.append(np.reshape(y, [ranks[-2], row_dims[-1], col_dims[-1], 1]))
# initialize tensor train
self.__init__(cores)
# show progress
utl.progress(string, 100, cpu_time=_time.time() - start_time, show=progress)
else:
raise ValueError('Number of dimensions must be a multiple of 2.')
else:
raise TypeError('Parameter must be either a list of cores or an ndarray.')
def __repr__(self):
"""
String representation of tensor trains
Print the attributes of a given tensor train.
"""
return ('\n'
'Tensor train with order = {d}, \n'
' row_dims = {m}, \n'
' col_dims = {n}, \n'
' ranks = {r}'.format(d=self.order, m=self.row_dims, n=self.col_dims, r=self.ranks))
def __add__(self, tt_add: 'TT') -> 'TT':
"""
Sum of two tensor trains.
Add two given tensor trains with same row and column dimensions.
Parameters
----------
tt_add : TT
tensor train which is added to self
Returns
-------
TT
sum of tt_add and self
Raises
------
TypeError
if tt_add is not an instance of the TT class
ValueError
if dimensions of both tensor trains do not match
"""
if isinstance(tt_add, TT):
# check if row and column dimension are equal
if self.row_dims == tt_add.row_dims and self.col_dims == tt_add.col_dims:
# define order, ranks, and cores
order = self.order
ranks = [1] + [self.ranks[i] + tt_add.ranks[i] for i in range(1, order)] + [1]
cores = []
# construct cores
for i in range(order):
# set core to zero array
if np.iscomplexobj(self.cores[i]) or np.iscomplexobj(tt_add.cores[i]):
cores.append(
np.zeros([ranks[i], self.row_dims[i], self.col_dims[i], ranks[i + 1]], dtype=complex))
else:
cores.append(np.zeros([ranks[i], self.row_dims[i], self.col_dims[i], ranks[i + 1]]))
# insert core of self
cores[i][0:self.ranks[i], :, :, 0:self.ranks[i + 1]] = self.cores[i]
# insert core of tt_add
r_1 = ranks[i] - tt_add.ranks[i]
r_2 = ranks[i]
r_3 = ranks[i + 1] - tt_add.ranks[i + 1]
r_4 = ranks[i + 1]
cores[i][r_1:r_2, :, :, r_3:r_4] = tt_add.cores[i]
# define tt_sum
tt_sum = TT(cores)
return tt_sum
else:
raise ValueError('Tensor trains must have the same dimensions')
else:
raise TypeError('Unsupported parameter.')
def __sub__(self, tt_sub: 'TT') -> 'TT':
"""
Difference of two tensor trains.
Subtract two given tensor trains.
Parameters
----------
tt_sub : TT
tensor train which is subtracted from self
Returns
-------
TT
difference of tt_add and self
"""
# define difference in terms of addition and left-multiplication
tt_diff = self + (-1) * tt_sub.copy()
return tt_diff
def __mul__(self, scalar: Union[int, float, complex]) -> 'TT':
"""
Left-multiplication of tensor trains and scalars.
Parameters
----------
scalar : int or float or complex
scalar value for the left-multiplication
Returns
-------
TT
product of scalar and self
Raises
------
TypeError
if scalar is neither int nor float nor complex
"""
# copy self
tt_prod = self.copy()
# check if scalar is int, float, or complex
if isinstance(scalar, (int, float, complex)):
# multiply first core by scalar
tt_prod.cores[0] = scalar * tt_prod.cores[0]
else:
raise TypeError('Unsupported parameter.')
return tt_prod
def __rmul__(self, scalar: float) -> 'TT':
"""
Right-multiplication of tensor trains and scalars.
Parameters
----------
scalar : float
scalar value for the right-multiplication
Returns
-------
TT
product of self and scalar
"""
# define product in terms of left-multiplication
tt_prod = self.copy() * scalar
return tt_prod
def __matmul__(self, tt_mul: 'TT') -> 'TT':
"""
Multiplication of tensor trains.
For Python 3.5 and higher, use the operator, i.e. T @ U = T.__matmul__(T,U). Otherwise you can use T.dot(U) or
TT.dot(T,U).
Parameters
----------
tt_mul : TT
tensor train which is multiplied with self
Returns
-------
TT
product of self and tt_mul
Raises
------
TypeError
if tt_mul is not an instance of the TT class
ValueError
if column dimensions of self do not match row dimensions of tt_mul
"""
def core_multiplication(core_1: np.ndarray, core_2: np.ndarray):
"""
Multiplies two 4-dimensional cores of the following shapes:
(r1 x m x n x r2) (s1 x n x p x s2)
Returns:
Product U of cores T and S of shape (r1 * s1 x m x p x r2 * s2)
"""
# Prepare cores for matrix multiplication
c1_row = np.arange(core_1.shape[ 0], dtype = np.intp)[:, None]
c1_col = np.arange(core_1.shape[-1], dtype = np.intp)[None, :]
c2_row = np.arange(core_2.shape[ 0], dtype = np.intp)[:, None]
c2_col = np.arange(core_2.shape[-1], dtype = np.intp)[None, :]
# Index and broadcast accordingly
core1_broad = core_1[c1_row, :, :, c1_col][:, None, :, None, :, :]
core2_broad = core_2[c2_row, :, :, c2_col][None, :, None, :, :, :]
contraction = core1_broad @ core2_broad
reshape_contraction = contraction.reshape(
core_1.shape[ 0] * core_2.shape[ 0],
core_1.shape[-1] * core_2.shape[-1],
core_1.shape[ 1],
core_2.shape[ 2]
)
result = reshape_contraction.transpose(0, 2, 3, 1)
return result
if isinstance(tt_mul, TT):
# check if dimensions match
if self.col_dims == tt_mul.row_dims:
# multiply TT cores
cores = [core_multiplication(self.cores[i], tt_mul.cores[i]) for i in range(self.order)]
# define product tensor
tt_prod = TT(cores)
# set tt_prod to scalar if all dimensions are equal to 1
if np.prod(tt_prod.row_dims) == 1 and np.prod(tt_prod.col_dims) == 1:
tt_prod = tt_prod.element([0] * tt_prod.order * 2)
return tt_prod
else:
raise ValueError('Dimensions do not match.')
else:
raise TypeError('Unsupported argument.')
def dot(self, tt_mul: 'TT') -> 'TT':
"""
Multiplication of tensor trains.
Alias for TT.__matmul__().
Parameters
----------
tt_mul : TT
tensor train which is multiplied with self
Returns
-------
tt_prod : TT or float
product of self and tt_mul
"""
tt_prod = self.__matmul__(tt_mul)
return tt_prod
def tensordot(self, other: 'TT',
num_axes: int,
mode: str='last-first',
overwrite: bool=False) -> 'TT':
"""
Computes index contraction between self and other.
The axes for contraction have to be the last or the first axes of self and other. Thus, there are 4 modes of
operation: 'last-first', 'last-last', 'first-last' and 'first-first'. The sequence of not contracted cores of
self is always maintained (cf. the 2nd example below).
For saving memory, you can choose to overwrite self with the tensordot.
Parameters
----------
other : TT
num_axes : int
number of axes that should be contracted
mode : {'last-first', 'last-last', 'first-last', 'first-first'}, optional
location of the axes for contraction on self-other
overwrite : bool, optional
whether to overwrite self or not
Returns
-------
TT
tensordot(self, other)
Examples
--------
Example for mode='last-first':
>>> import scikit_tt.tensor_train as tt
>>> t = tt.ones([1, 2, 3, 4], [5, 6, 7, 8], ranks=[1, 2, 4, 3, 1])
>>> u = tt.ones([3, 4, 5], [7, 8, 2], ranks=[1, 7, 8, 1])
>>> t.tensordot(u, 2)
Tensor train with order = 3,
row_dims = [1, 2, 5],
col_dims = [5, 6, 2],
ranks = [1, 2, 8, 1]
Example for mode='last-last'
>>> t = tt.ones([2, 3, 4, 5], [1, 1, 1, 1], ranks=3)
>>> u = tt.ones([7, 6, 4, 5], [1, 1, 1, 1], ranks=2)
>>> t.tensordot(u, 2, mode='last-last')
Tensor train with order = 4,
row_dims = [2, 3, 6, 7],
col_dims = [1, 1, 1, 1],
ranks = [1, 3, 2, 2, 1]
As you can see, the sequence of not contracted cores of t (2, 3) is maintained. The sequence of the not
contracted cores of u (7, 6) is reversed and the cores rank-transposed to fit together.
"""
# define first and last core indices for contraction
if mode == 'last-first':
first_idx_self = self.order - num_axes
first_idx_other = 0
last_idx_self = self.order - 1
last_idx_other = num_axes - 1
elif mode == 'first-last':
first_idx_self = 0
first_idx_other = other.order - num_axes
last_idx_self = num_axes - 1
last_idx_other = other.order - 1
elif mode == 'first-first':
first_idx_self = 0
first_idx_other = 0
last_idx_other = num_axes - 1
last_idx_self = num_axes - 1
elif mode == 'last-last':
first_idx_self = self.order - num_axes
first_idx_other = other.order - num_axes
last_idx_self = self.order - 1
last_idx_other = other.order - 1
else:
raise ValueError('unknown mode')
# check dimensions for contraction
if num_axes > self.order or num_axes > other.order:
raise ValueError('num_axes is too big')
if self.row_dims[first_idx_self:last_idx_self + 1] != other.row_dims[first_idx_other:last_idx_other + 1] or \
self.col_dims[first_idx_self:last_idx_self + 1] != other.col_dims[first_idx_other:last_idx_other + 1]:
raise ValueError('axes do not match')
# check if the needed ranks are 1
if mode == 'last-first':
if self.ranks[-1] != 1 or other.ranks[0] != 1:
raise ValueError('last rank of self and first rank of other have to be 1')
elif mode == 'last-last':
if self.ranks[-1] != 1 or other.ranks[-1] != 1:
raise ValueError('last rank of self and last rank of other have to be 1')
elif mode == 'first-last':
if self.ranks[0] != 1 or other.ranks[-1] != 1:
raise ValueError('first rank of self and last rank of other have to be 1')
else: # mode == 'first-first':
if self.ranks[0] != 1 or other.ranks[0] != 1:
raise ValueError('first rank of self and first rank of other have to be 1')
# copy self
if overwrite is False:
tdot = ones([1], [1], 1) # placeholder
else:
tdot = self
# calculate the contraction
M = np.tensordot(self.cores[first_idx_self], other.cores[first_idx_other], axes=([1, 2], [1, 2]))
# M.shape (r_0, r_1, s_0, s_1)
for i in range(1, num_axes):
M_new = np.tensordot(self.cores[first_idx_self + i], other.cores[first_idx_other + i],
axes=([1, 2], [1, 2]))
# M_new.shape (r_{i-1}, r_i, s_{i-1}, s_i)
M = np.tensordot(M, M_new, axes=([1, 3], [0, 2])) # shape (r_0, s_0, r_i, s_i)
M = np.transpose(M, [0, 2, 1, 3]) # shape (r_0, r_i, s_0, s_i)
if mode == 'last-first':
M = M[:, 0, 0, :] # shape (r_0, s_{num_axes})
elif mode == 'last-last':
M = M[:, 0, :, 0] # shape (r_0, s_0)
elif mode == 'first-last':
M = M[0, :, :, 0] # shape (r_{num_axes}, s_0)
else: # first-first
M = M[0, :, 0, :] # shape (r_{num_axes}, s_{num_axes})
# build the cores
if num_axes == self.order and num_axes == other.order: # complete contraction over both
tdot.cores = [M[:, np.newaxis, np.newaxis, :]]
elif num_axes == self.order: # complete contraction over self -> merge M into other
tdot.cores = []
if mode == 'last-first':
tdot.cores.append(np.tensordot(M, other.cores[last_idx_other + 1], axes=([1], [0])))
tdot.cores.extend(other.cores[last_idx_other + 2:]) # append the remaining cores of other
elif mode == 'last-last':
tdot.cores.append(np.tensordot(other.cores[first_idx_other - 1], M, axes=([3], [1])))
tdot.cores.extend(other.cores[:first_idx_other - 1][::-1]) # append the remaining cores of other
for i in range(len(tdot.cores)): # they need to be rank-transposed
tdot.cores[i] = np.transpose(tdot.cores[i], [3, 1, 2, 0])
elif mode == 'first-last':
tdot.cores.append(np.tensordot(other.cores[first_idx_other - 1], M, axes=([3], [1])))
tdot.cores = other.cores[:first_idx_other - 1] + tdot.cores # append the remaining cores of other
else: # mode = 'first-first', merge M into last core of other
tdot.cores.append(np.tensordot(M, other.cores[last_idx_other + 1], axes=([1], [0])))
tdot.cores = other.cores[last_idx_other + 2:][::-1] + tdot.cores # append the remaining cores of other
for i in range(len(tdot.cores)): # they need to be rank-transposed
tdot.cores[i] = np.transpose(tdot.cores[i], [3, 1, 2, 0])
else: # merge M into tdot
if mode == 'last-first':
tdot.cores = self.cores[:first_idx_self] + other.cores[last_idx_other + 1:]
tdot.cores[first_idx_self - 1] = np.tensordot(tdot.cores[first_idx_self - 1], M, axes=([3], [0]))
elif mode == 'last-last':
tdot.cores = self.cores[:first_idx_self]
tdot.cores[first_idx_self - 1] = np.tensordot(tdot.cores[first_idx_self - 1], M, axes=([3], [0]))
tdot.cores.extend(other.cores[:first_idx_other][::-1]) # append the remaining cores of other
for i in range(first_idx_self, len(tdot.cores)): # they need to be rank-transposed
tdot.cores[i] = np.transpose(tdot.cores[i], [3, 1, 2, 0])
elif mode == 'first-last':
tdot.cores = self.cores[last_idx_self + 1:]
tdot.cores[0] = np.tensordot(M, tdot.cores[0], axes=([0], [0]))
tdot.cores = other.cores[:first_idx_other] + tdot.cores
else: # first-first
tdot.cores = self.cores[last_idx_self + 1:]
tdot.cores[0] = np.tensordot(M, tdot.cores[0], axes=([0], [0]))
tdot.cores = other.cores[last_idx_other + 1:][::-1] + tdot.cores
for i in range(other.order - num_axes): # they need to be rank-transposed
tdot.cores[i] = np.transpose(tdot.cores[i], [3, 1, 2, 0])
# define new order, row dimensions, column dimensions and ranks
tdot.order = len(tdot.cores)
tdot.row_dims = [tdot.cores[i].shape[1] for i in range(tdot.order)]
tdot.col_dims = [tdot.cores[i].shape[2] for i in range(tdot.order)]
tdot.ranks = [tdot.cores[i].shape[0] for i in range(tdot.order)] + [tdot.cores[-1].shape[3]]
return tdot
def rank_tensordot(self, matrix: np.ndarray,
mode: str='last',
overwrite: bool=False) -> 'TT':
"""
Return index contraction between self and a 2D-array matrix along the first/last rank axis of the first/last
core of self. Thus, this method is only useful in the unusual case where self.ranks[0] or self.ranks[-1] != 1.
For example, these type of TT's appear as an output of the TT.svd method.
Parameters
----------
matrix: np.ndarray
2D array
mode: string
one of the following: 'last', 'first'
overwrite: bool
whether to overwrite self or not, default is False
Returns
-------
tdot : TT
tensordot of self and matrix along the first/last rank axis of the first/last core of self
"""
if len(matrix.shape) != 2:
raise ValueError('argument matrix has to be 2D-Array')
# copy self
if overwrite is False:
tdot = self.copy()
else:
tdot = self
if mode == 'last':
if tdot.ranks[-1] != matrix.shape[0]:
raise ValueError('dimensions do not match')
tdot.cores[-1] = np.tensordot(tdot.cores[-1], matrix, axes=([3], [0]))
elif mode == 'first':
if tdot.ranks[0] != matrix.shape[1]:
raise ValueError('dimensions do not match')
tdot.cores[0] = np.tensordot(matrix, tdot.cores[0], axes=([1], [0]))
else:
raise ValueError('unknown mode')
tdot.ranks = [tdot.cores[i].shape[0] for i in range(tdot.order)] + [tdot.cores[-1].shape[3]]
return tdot
def concatenate(self, other: Union['TT', List[np.ndarray]],
overwrite: bool=False) -> 'TT':
"""
Expand the list of cores of self by appending more cores.
If other is a TT, concatenate the cores of self and the cores of other.
If other is a list of cores, the cores are appended to self.cores.
For example, this method can be used to reconstruct the original tensor from u,s,v from TT.svd.
Parameters
----------
other : TT or list[np.ndarray]
overwrite : bool, optional
Returns
-------
TT
"""
# copy self
if overwrite is False:
tt = self.copy()
else:
tt = self
if isinstance(other, TT):
if tt.ranks[-1] != other.ranks[0]:
raise ValueError('ranks do not match!')
tt.cores.extend(other.cores)
elif isinstance(other, list):
# check if orders of list elements are correct
if not np.all([other[i].ndim == 4 for i in range(len(other))]):
raise ValueError('list elements must be 4-dimensional arrays')
# check if ranks are correct
if not np.all([other[i].shape[3] == other[i + 1].shape[0] for i in range(len(other) - 1)]):
raise ValueError('List elements must be 4-dimensional arrays.')
if tt.ranks[-1] != other[0].shape[0]:
raise ValueError('ranks do not match!')
tt.cores.extend(other)
tt.order = len(tt.cores)
tt.row_dims = [tt.cores[i].shape[1] for i in range(tt.order)]
tt.col_dims = [tt.cores[i].shape[2] for i in range(tt.order)]
tt.ranks = [tt.cores[i].shape[0] for i in range(tt.order)] + [tt.cores[-1].shape[3]]
return tt
def transpose(self,
cores: Optional[List[int]]=None,
conjugate: bool=False,
overwrite: bool=False) -> 'TT':
"""
Transpose of tensor trains.
Parameters
----------
cores : list[int], optional
cores which should be transposed, if cores=None (default), all cores are transposed
conjugate : bool, optional
whether to compute the conjugate transpose, default is False
overwrite : bool, optional
whether to overwrite self or not, default is False
Returns
-------
TT
transpose of self
Examples
--------
>>> import scikit_tt.tensor_train as tt
>>> t = tt.ones([1, 2, 3], [4, 5, 6], ranks=[1, 7, 8, 1])
>>> t.transpose()
Tensor train with order = 3,
row_dims = [4, 5, 6],
col_dims = [1, 2, 3],
ranks = [1, 7, 8, 1]
>>> t.transpose(cores=[0, 1])
Tensor train with order = 3,
row_dims = [4, 5, 3],
col_dims = [1, 2, 6],
ranks = [1, 7, 8, 1]
"""
# define list of core numbers
if cores is None:
cores = np.arange(0, self.order)
# copy self
if overwrite is False:
tt_transpose = self.copy()
else:
tt_transpose = self
for i in range(self.order):
if np.isin(i, cores):
# permute second and third dimension of each core
tt_transpose.cores[i] = np.transpose(tt_transpose.cores[i], [0, 2, 1, 3])
# interchange row and column dimensions
row_dim = tt_transpose.row_dims[i]
col_dim = tt_transpose.col_dims[i]
tt_transpose.row_dims[i] = col_dim
tt_transpose.col_dims[i] = row_dim
if conjugate:
tt_transpose.cores[i] = np.conj(tt_transpose.cores[i])
return tt_transpose
def rank_transpose(self, overwrite: bool=False) -> 'TT':
"""
Computes the rank-transposed of self.
The rank-transposed has the same cores as self but in reversed order. To fit together,
every core needs to be transposed with respect to its ranks.
Parameters
----------
overwrite : bool, optional
whether to overwrite self or not, default is False
Returns
-------
TT
rank-transpose of self
Examples
--------
>>> import scikit_tt.tensor_train as tt
>>> t = tt.ones([1, 2, 3], [4, 5, 6], ranks=[1, 7, 8, 1])
>>> t.rank_transpose()
Tensor train with order = 3,
row_dims = [3, 2, 1],
col_dims = [6, 5, 4],
ranks = [1, 8, 7, 1]
"""
# copy self
if overwrite is False:
tt_transpose = self.copy()
else:
tt_transpose = self
tt_transpose.cores.reverse()
for i in range(len(tt_transpose.cores)):
tt_transpose.cores[i] = np.transpose(tt_transpose.cores[i], [3, 1, 2, 0])
tt_transpose.row_dims.reverse()
tt_transpose.col_dims.reverse()
tt_transpose.ranks.reverse()
return tt_transpose
def conj(self, overwrite: bool=False) -> 'TT':
"""
Complex conjugate of tensor trains.
Parameters
----------
overwrite : bool, optional
whether to overwrite self or not, default is False
Returns
-------
TT
complex conjugate of self
"""
# copy self
if overwrite is False:
tt_conj = self.copy()
else:
tt_conj = self
# conjugate each core
for i in range(self.order):
tt_conj.cores[i] = np.conj(tt_conj.cores[i])
return tt_conj
def isoperator(self) -> bool:
"""
Operator check.
Returns
-------
bool
true if self is a TT operator
"""
# check if all row dimensions or column dimensions of self are equal to 1
op_bool = not (all([i == 1 for i in self.row_dims]) or all([i == 1 for i in self.col_dims]))
return op_bool
def copy(self) -> 'TT':
"""
Deep copy of tensor trains.
Returns
-------
TT
deep copy of self
"""
# copy TT cores
cores = [self.cores[i].copy() for i in range(self.order)]
# define copied version of self
tt_copy = TT(cores)
return tt_copy
def element(self, indices: List[int]) -> float:
"""
Single element of tensor trains.
Parameters
----------
indices : list[int]
indices of a single entry of self ([x_1, ..., x_d, y_1, ..., y_d])
Returns
-------
float
single entry of self
Raises
------
TypeError
if indices is not a list[int]
ValueError
if length of indices does not match the order of self
IndexError
if one or more indices are out of range
"""
if isinstance(indices, list):
# check is all indices are ints
if np.all([isinstance(indices[i], (int, np.int32, np.int64)) for i in range(len(indices))]):
# check if length of indices is correct
if len(indices) == 2 * self.order:
# check if indices are in range
if np.all([indices[i] >= 0 for i in range(2 * self.order)]) and \
np.all([indices[i] < self.row_dims[i] for i in range(self.order)]) and \