Re-compose a Tensor after tensor factorization

Question

I am trying to decompose a 3D matrix using python library scikit-tensor. I managed to decompose my Tensor (with dimensions 100x50x5) into three matrices. My question is how can I compose the initial matrix again using the decomposed matrix produced with Tensor factorization? I want to check if the decomposition has any meaning. My code is the following:

import logging
from scipy.io.matlab import loadmat
from sktensor import dtensor, cp_als
import numpy as np

//Set logging to DEBUG to see CP-ALS information
logging.basicConfig(level=logging.DEBUG)
T = np.ones((400, 50))
T = dtensor(T)
P, fit, itr, exectimes = cp_als(T, 10, init='random')
// how can I re-compose the Matrix T? TA = np.dot(P.U[0], P.U[1].T)

I am using the canonical decomposition as provided from the scikit-tensor library function cp_als. Also what is the expected dimensionality of the decomposed matrices?

Iatrogenic

cp.py says: $A\approx\sum_{r=1}^{rank} \\vec{u}_r^{(1)} \outer \cdots \outer \\vec{u}_r^{(N)}$. Have you tried that? This should be identical to "P.totensor()" – Fitted

@Fitted you mean the lines 145 and 146 of cp.py? – Iatrogenic

Answer 1

The CP product of, for example, 4 matrices

can be expressed using Einstein notation as

or in numpy as

numpy.einsum('az,bz,cz,dz -> abcd', A, B, C, D)

so in your case you would use

numpy.einsum('az,bz->ab', P.U[0], P.U[1])

or, in your 3-matrix case

numpy.einsum('az,bz,cz->abc', P.U[0], P.U[1], P.U[2])

sktensor.ktensor.ktensor also have a method totensor() that does exactly this:

np.allclose(np.einsum('az,bz->ab', P.U[0], P.U[1]), P.totensor())
>>> True

Traditional

Hey Nils thanks for the reply. Are you sure about this? I tried to re-compose the matrix and the result was not close to the intial tensor. – Iatrogenic

Well, it depends on how nicely you can decompose the tensor in the first place. If your epsilon is big the two will be noticeably different. – Traditional

What are the parameters except the number of latent dimension I can handle in the case of cp_als algorithm? – Iatrogenic

I don't understand your question, sorry. – Traditional

I am trying to figure out what are the parameters I can tune during the decomposition except the tensor rank. – Iatrogenic

The parameters are in the docstring. – Traditional

Does this answer your question? – Traditional

Answer 2

See an explanation of CP here. You may also use tensorlearn package to rebuild the tensor.

Maurilla