_cmtf_als.py
``````import warnings
import tensorly as tl
from ..tenalg import khatri_rao
from ..cp_tensor import CPTensor, validate_cp_rank, cp_to_tensor, cp_normalize
from ._cp import initialize_cp

# Authors: Isabell Lehmann <isabell.lehmann94@outlook.de>

def coupled_matrix_tensor_3d_factorization(
tensor_3d,
matrix,
rank,
init="svd",
n_iter_max=100,
tol=1e-6,
normalize_factors=False,
):
"""
Calculates a coupled matrix and tensor factorization of 3rd order tensor and matrix which are
coupled in first mode.

Assume you have tensor_3d = [[lambda; A, B, C]] and matrix = [[gamma; A, V]], which are
coupled in 1st mode. With coupled matrix and tensor factorization (CTMF), the normalized
factor matrices A, B, C for the CP decomposition of X, the normalized matrix V and the
weights lambda_ and gamma are found. This implementation only works for a coupling in the
first mode.

Solution is found via alternating least squares (ALS) as described in Figure 5 of
@article{acar2011all,
title={All-at-once optimization for coupled matrix and tensor factorizations},
author={Acar, Evrim and Kolda, Tamara G and Dunlavy, Daniel M},
journal={arXiv preprint arXiv:1105.3422},
year={2011}
}

Notes
-----
In the paper, the columns of the factor matrices are not normalized and therefore weights are
not included in the algorithm.

Parameters
----------
tensor_3d : tl.tensor or CP tensor
3rd order tensor X = [[A, B, C]]
matrix : tl.tensor or CP tensor
matrix that is coupled with tensor in first mode: Y = [[A, V]]
rank : int
rank for CP decomposition of X
tol : float, optional
(Default: 1e-6) Relative reconstruction error tolerance. The
algorithm is considered to have found the global minimum when the
reconstruction error is less than `tol`.

Returns
-------
tensor_3d_pred : CPTensor
tensor_3d_pred = [[lambda; A,B,C]]
matrix_pred : CPTensor
matrix_pred = [[gamma; A,V]]
rec_errors : list
contains the reconstruction error of each iteration:
error = 1 / 2 * | X - [[ lambda_; A, B, C ]] | ^ 2 + 1 / 2 * | Y - [[ gamma; A, V ]] | ^ 2

Examples
--------
A = tl.tensor([[1, 2], [3, 4]])
B = tl.tensor([[1, 0], [0, 2]])
C = tl.tensor([[2, 0], [0, 1]])
V = tl.tensor([[2, 0], [0, 1]])
R = 2

X = (None, [A, B, C])
Y = (None, [A, V])

tensor_3d_pred, matrix_pred = cmtf_als_for_third_order_tensor(X, Y, R)

"""
rank = validate_cp_rank(tl.shape(tensor_3d), rank=rank)

# initialize values
tensor_cp = initialize_cp(tensor_3d, rank, init=init)
# the coupled factor should be initialized with the concatenated dataset
coupled_unfold = tl.concatenate((tl.unfold(tensor_3d, 0), matrix), axis=1)
coupled_init = initialize_cp(coupled_unfold, rank, init=init)
tensor_cp.factors = coupled_init.factors

rec_errors = []

# alternating least squares
# note that the order of the khatri rao product is reversed since tl.unfold has another order
# than assumed in paper
for iteration in range(n_iter_max):
V = tl.transpose(tl.lstsq(tensor_cp.factors, matrix))

# Loop over modes of the tensor
# We want to solve for mode 0 last, since the coupled factor matrix is most influential and SVD gave us a good approximation
for ii in reversed(range(tl.ndim(tensor_3d))):
kr = khatri_rao(tensor_cp.factors, skip_matrix=ii)
unfolded = tl.unfold(tensor_3d, ii)

# If we are at the coupled mode, concat the matrix
if ii == 0:
kr = tl.concatenate((kr, V), axis=0)
unfolded = tl.concatenate((unfolded, matrix), axis=1)

tensor_cp.factors[ii] = tl.transpose(
tl.lstsq(kr, tl.transpose(unfolded))
)

error_new = (
tl.norm(tensor_3d - cp_to_tensor(tensor_cp)) ** 2
+ tl.norm(matrix - cp_to_tensor((None, [tensor_cp.factors, V]))) ** 2
)

if iteration > 0 and (
tl.abs(error_new - error_old) / error_old <= tol or error_new < tol
):
break

error_old = error_new
rec_errors.append(error_new)

if iteration == n_iter_max - 1:
warnings.warn("Reached maximum iteration number without convergence.")

matrix_pred = CPTensor((None, [tensor_cp.factors, V]))

if normalize_factors:
tensor_cp = cp_normalize(tensor_cp)
matrix_pred = cp_normalize(matrix_pred)

return tensor_cp, matrix_pred, rec_errors
``````