# Source code for tensorly.tt_matrix

```"""Module for matrices in the TT format"""

import tensorly as tl

# Note how tt_matrix_to_tensor is implemented in tenalg to allow for more efficient implementations
# (e.g. using the einsum backend)
from .tenalg import _tt_matrix_to_tensor as tt_matrix_to_tensor

from ._factorized_tensor import FactorizedTensor
import numpy as np

def validate_tt_matrix_rank(tensorized_shape, rank='same'):
"""Returns the rank of a TT-Matrix Decomposition

Parameters
----------
tensor_shape : tupe
shape of the tensorized matrix to decompose
rank : {'same', float, tuple, int}, default is same
way to determine the rank, by default 'same'
if 'same': rank is computed to keep the number of parameters (at most) the same
if float, computes a rank so as to keep rank percent of the original number of parameters
if int or tuple, just returns rank
constant_rank : bool, default is False
* if True, the *same* rank will be chosen for each modes
* if False (default), the rank of each mode will be proportional to the corresponding tensor_shape

*used only if rank == 'same' or 0 < rank <= 1*

rounding = {'round', 'floor', 'ceil'}

Returns
-------
rank : int tuple
rank of the decomposition
"""

n_dim = len(tensorized_shape) // 2

if n_dim*2 != len(tensorized_shape):
msg = (f'The order of the give tensorized shape is not a multiple of 2.'
'However, there should be as many dimensions for the left side (number of rows)'
' as of the right side (number of columns). '
' For instance, to convert a matrix of size (8, 9) to the TT-format, '
' it can be tensorized to (2, 4, 3, 3) but NOT to (2, 2, 2, 3, 3).')
raise ValueError(msg)

left_shape = tensorized_shape[:n_dim]
right_shape = tensorized_shape[n_dim:]

full_shape = tuple(i*o for i, o in zip(left_shape, right_shape))
return tl.tt_tensor.validate_tt_rank(full_shape, rank)

def _tt_matrix_n_param(tensorized_shape, rank):
"""Number of parameters of a TT-Matrix decomposition for a given `rank` and full `tensor_shape`.

Parameters
----------
tensorized_shape : int tuple
shape of the full tensorized matrix to decompose (or approximate)

rank : tuple
rank of the TT-Matrix decomposition

Returns
-------
n_params : int
Number of parameters of a TT-Matrix decomposition of rank `rank` of a full tensor of shape `tensor_shape`
"""
n_dim = len(tensorized_shape) // 2

if n_dim*2 != len(tensorized_shape):
msg = (f'The order of the give tensorized shape is not a multiple of 2.'
'However, there should be as many dimensions for the left side (number of rows)'
' as of the right side (number of columns). '
' For instance, to convert a matrix of size (8, 9) to the TT-format, '
' it can be tensorized to (2, 4, 3, 3) but NOT to (2, 2, 2, 3, 3).')
raise ValueError(msg)

left_shape = tensorized_shape[:n_dim]
right_shape = tensorized_shape[n_dim:]

factor_params = []
for i, (ls, rs) in enumerate(zip(left_shape, right_shape)):
factor_params.append(rank[i]*ls*rs*rank[i+1])

return np.sum(factor_params)

def tt_matrix_to_matrix(tt_matrix):
"""Reconstruct the original matrix that was tensorized and compressed in the TT-Matrix format

Re-assembles 'factors', which represent a tensor in TT-Matrix format
into the corresponding matrix

Parameters
----------
factors: list of 4D-arrays
TT-Matrix factors (known as core) of shape (rank_k, left_dim_k, right_dim_k, rank_{k+1})

Returns
-------
output_matrix: 2D-array
matrix whose TT-Matrix decomposition was given by 'factors'
"""
in_shape = tuple(c.shape for c in tt_matrix)
return tl.reshape(tt_matrix_to_tensor(tt_matrix), (np.prod(in_shape), -1))

[docs]def tt_matrix_to_unfolded(tt_matrix, mode):
"""Returns the unfolding matrix of a tensor given in TT-Matrix format

Reassembles a full tensor from 'factors' and returns its unfolding matrix
with mode given by 'mode'

Parameters
----------
factors : list of 3D-arrays
TT-Matrix factors
mode : int
unfolding matrix to be computed along this mode

Returns
-------
2-D array
unfolding matrix at mode given by 'mode'
"""
return tl.unfold(tt_matrix_to_tensor(tt_matrix), mode)

[docs]def tt_matrix_to_vec(tt_matrix):
"""Returns the tensor defined by its TT-Matrix format ('factors') into
its vectorized format

Parameters
----------
factors : list of 3D-arrays
TT factors

Returns
-------
1-D array
format of tensor defined by 'factors'
"""
return tl.tensor_to_vec(tt_matrix_to_tensor(tt_matrix))

def _validate_tt_matrix(tt_tensor):
factors = tt_tensor
n_factors = len(factors)

if n_factors < 1:
raise ValueError('A Tensor-Train (MPS) tensor should be composed of at least one factor.'
'However, {} factor was given.'.format(n_factors))

rank = []
left_shape = []
right_shape = []
for index, factor in enumerate(factors):
current_rank, current_left_shape, current_right_shape, next_rank = tl.shape(factor)

# Check that factors are third order tensors
if not tl.ndim(factor)==4:
raise ValueError('A TTMatrix expresses a tensor as fourth order factors (tt-cores).\n'
'However, tl.ndim(factors[{}]) = {}'.format(
index, tl.ndim(factor)))
# Consecutive factors should have matching ranks
if index and tl.shape(factors[index - 1])[-1] != current_rank:
raise ValueError('Consecutive factors should have matching ranks\n'
' -- e.g. tl.shape(factors)[-1]) == tl.shape(factors))\n'
'However, tl.shape(factor[{}])[-1] == {} but'
' tl.shape(factor[{}]) == {} '.format(
index - 1, tl.shape(factors[index - 1])[-1], index, current_rank))
# Check for boundary conditions
if (index == 0) and current_rank != 1:
raise ValueError('Boundary conditions dictate factor.shape == 1.'
'However, got factor.shape = {}.'.format(
current_rank))
if (index == n_factors - 1) and next_rank != 1:
raise ValueError('Boundary conditions dictate factor[-1].shape == 1.'
'However, got factor[{}].shape = {}.'.format(
n_factors, next_rank))

left_shape.append(current_left_shape)
right_shape.append(current_right_shape)

rank.append(current_rank)

# Add last rank (boundary condition)
rank.append(next_rank)

return tuple(left_shape) + tuple(right_shape), tuple(rank)

class TTMatrix(FactorizedTensor):
def __init__(self, factors, inplace=False):
super().__init__()

# Will raise an error if invalid
shape, rank = _validate_tt_matrix(factors)

self.shape = tuple(shape)
self.order = len(self.shape) // 2
self.left_shape = self.shape[:self.order]
self.right_shape = self.shape[self.order:]
self.rank = tuple(rank)
self.factors = factors

def __getitem__(self, index):
return self.factors[index]

def __setitem__(self, index, value):
self.factors[index] = value

def __iter__(self):
for index in range(len(self)):
yield self[index]

def __len__(self):
return len(self.factors)

def __repr__(self):
message = (f'factors list : rank-{self.rank} TT-Matrix of tensorized shape {self.shape}'
f' corresponding to a matrix of size {np.prod(self.left_shape)} x {np.prod(self.right_shape)}')
return message

def to_tensor(self):
return tt_matrix_to_tensor(self)

def to_matrix(self):
return tt_matrix_to_matrix(self)

def to_unfolding(self, mode):
return tt_matrix_to_unfolded(self, mode)

def to_vec(self):
return tt_matrix_to_vec(self)
```