##### https://github.com/tensorly/tensorly

tt_matrix.py

```
"""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[1] for c in tt_matrix)
return tl.reshape(tt_matrix_to_tensor(tt_matrix), (np.prod(in_shape), -1))
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)
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."
f"However, {n_factors} factor was given."
)
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"
f"However, tl.ndim(factors[{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[0])[-1]) == tl.shape(factors[1])[0])\n"
f"However, tl.shape(factor[{index-1}])[-1] == {tl.shape(factors[index - 1])[-1]} but"
f" tl.shape(factor[{index}])[0] == {current_rank} "
)
# Check for boundary conditions
if (index == 0) and current_rank != 1:
raise ValueError(
"Boundary conditions dictate factor[0].shape[0] == 1."
f"However, got factor[0].shape[0] = {current_rank}."
)
if (index == n_factors - 1) and next_rank != 1:
raise ValueError(
"Boundary conditions dictate factor[-1].shape[2] == 1."
f"However, got factor[{n_factors}].shape[2] = {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)
```