Multilinear tensors

using TensorDec

We consider a multi-linear tensor of size 3 x 5 x 4, which is sum of r=4 tensor products of the random column vectors of the matrices A0, B0, C0with weights w0:

r=4
w0 = rand(r)
A0 = rand(3,r)
B0 = rand(5,r)
C0 = rand(4,r)

T0 = tensor(w0, A0, B0, C0)

3×5×4 Array{Float64, 3}:
[:, :, 1] =
 0.316184  0.345023   0.382037  0.36165   0.486526
 0.404335  0.276629   0.434605  0.344533  0.509244
 0.111832  0.0641309  0.111185  0.103334  0.11392

[:, :, 2] =
 0.353912  0.447882  0.428276  0.478536  0.565683
 0.491082  0.387773  0.529839  0.517911  0.582768
 0.175713  0.117616  0.188383  0.200218  0.146661

[:, :, 3] =
 0.234157  0.189894   0.277909  0.200739  0.326495
 0.333178  0.184865   0.373708  0.241155  0.376656
 0.104873  0.0597654  0.117054  0.100222  0.088638

[:, :, 4] =
 0.372136  0.0808872  0.301894   0.187236   0.398308
 0.499487  0.093278   0.399356   0.240435   0.524856
 0.126637  0.0208347  0.0864272  0.0717313  0.127397

We compute its decomposition:

w, A, B, C = decompose(T0);

We obtain a decomposition of rank 4 with weights:

w

4-element Vector{Float64}:
 0.33792668673343457
 0.7408084924294192
 0.9605854249810668
 1.0267280992544108

The r=4 vectors of norm 1 of the first components of the decomposition are the columns of the matrix A:

A

3×4 Matrix{Float64}:
 0.0808627  0.576234  0.590579  0.831589
 0.744569   0.778708  0.794381  0.548351
 0.66263    0.248132  0.142043  0.0881477

The r=4 vectors of norm 1 of the second components are the columns of the matrix B:

B

5×4 Matrix{Float64}:
 0.414404  0.624532   0.503332   -0.136551
 0.334042  0.0598829  0.108848   -0.615309
 0.571347  0.233124   0.623258   -0.335895
 0.607836  0.469743   0.0629459  -0.52911
 0.144163  0.575646   0.585147   -0.458221

The r=4 vectors of norm 1 of the third components are the columns of the matrix C:

C

4×4 adjoint(::Matrix{Float64}) with eltype Float64:
 0.273397   0.39815   0.489947  -0.575025
 0.84564    0.521266  0.407014  -0.764628
 0.458126   0.232774  0.467259  -0.287248
 0.0163813  0.718035  0.613156  -0.0466739

It corresponds to the tensor $\sum_{i=1}^{r} w_i \, A[:,i] \otimes B[:,i] \otimes C[:,i]$ for $i \in 1:r$:

T = tensor(w, A, B, C)

3×5×4 Array{Float64, 3}:
[:, :, 1] =
 0.316184  0.345023   0.382037  0.36165   0.486526
 0.404335  0.276629   0.434605  0.344533  0.509244
 0.111832  0.0641309  0.111185  0.103334  0.11392

[:, :, 2] =
 0.353912  0.447882  0.428276  0.478536  0.565683
 0.491082  0.387773  0.529839  0.517911  0.582768
 0.175713  0.117616  0.188383  0.200218  0.146661

[:, :, 3] =
 0.234157  0.189894   0.277909  0.200739  0.326495
 0.333178  0.184865   0.373708  0.241155  0.376656
 0.104873  0.0597654  0.117054  0.100222  0.088638

[:, :, 4] =
 0.372136  0.0808872  0.301894   0.187236   0.398308
 0.499487  0.093278   0.399356   0.240435   0.524856
 0.126637  0.0208347  0.0864272  0.0717313  0.127397

We compute the $L^2$ norm of the difference between $T$ and $T_0$:

using LinearAlgebra
norm(T-T0)

5.414241769776905e-15