Date: 2023-04-28

# 3. Multilinear Independent Component Analysis (MultilinearICA)

#### Koki Tsuyuzaki

#### 2023-04-28

# Introduction

In this vignette we consider approximating a tensor as a product of multiple low-rank matrices (a.k.a., factor matrices) and a core tensor.

Test data is available from `toyModel`.

    library("iTensor")
    library("nnTensor")

    ## Warning: no DISPLAY variable so Tk is not available

    ##
    ## Attaching package: 'nnTensor'

    ## The following object is masked from 'package:iTensor':
    ##
    ##     toyModel

    data <- nnTensor::toyModel("CP")
    str(data, 2)

    ## Formal class 'Tensor' [package "rTensor"] with 3 slots
    ##   ..@ num_modes: int 3
    ##   ..@ modes    : int [1:3] 30 30 30
    ##   ..@ data     : num [1:30, 1:30, 1:30] 113 81 120 95 102 2 4 2 0 1 ...

You will see that there are four small blocks in the diagonal direction of the data tensor.

    plotTensor3D(data)

# Multilinear Independent Component Analysis (MultilinearICA)

There are so many tensor decomposition algorithms but here we consider that each factor matrix is extracted by Independent Component Analysis (ICA). This is called Multilinear Independent Component Analysis (MultilinearICA (Vasilescu 2005)).

`MultilinearICA` can be performed as follows.

    out <- MultilinearICA(data, Js=c(4,4,4), algorithm="FastICA")

The rank for each factor matrix can be set as `Js` and the decomposition algorithm can be easily switched by `algorithm`.

By using `recTensor` of `nnTensor`, user can easily reconstruct the data from core tensor and factor matrices as follows.

    rec_data <- recTensor(out$S, out$As)
    plotTensor3D(rec_data)

# Session Information

    ## R version 4.3.0 (2023-04-21)
    ## Platform: x86_64-pc-linux-gnu (64-bit)
    ## Running under: Debian GNU/Linux bookworm/sid
    ##
    ## Matrix products: default
    ## BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
    ## LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.21.so;  LAPACK version 3.11.0
    ##
    ## locale:
    ##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    ##  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8
    ##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
    ##  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C
    ##  [9] LC_ADDRESS=C               LC_TELEPHONE=C
    ## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
    ##
    ## time zone: Etc/UTC
    ## tzcode source: system (glibc)
    ##
    ## attached base packages:
    ## [1] stats     graphics  grDevices utils     datasets  methods   base
    ##
    ## other attached packages:
    ## [1] nnTensor_1.1.13 mixOmics_6.24.0 ggplot2_3.4.2   lattice_0.21-8
    ## [5] MASS_7.3-59     iTensor_1.0.2
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] dotCall64_1.0-2     spam_2.9-1          gtable_0.3.3
    ##  [4] ellipse_0.4.5       xfun_0.39           bslib_0.4.2
    ##  [7] tagcloud_0.6        ggrepel_0.9.3       vctrs_0.6.2
    ## [10] tools_4.3.0         generics_0.1.3      parallel_4.3.0
    ## [13] tibble_3.2.1        fansi_1.0.4         highr_0.10
    ## [16] rARPACK_0.11-0      pkgconfig_2.0.3     Matrix_1.5-4
    ## [19] RColorBrewer_1.1-3  lifecycle_1.0.3     groupICA_0.1.1
    ## [22] compiler_4.3.0      stringr_1.5.0       fields_14.1
    ## [25] munsell_0.5.0       codetools_0.2-19    misc3d_0.9-1
    ## [28] einsum_0.1.0        htmltools_0.5.5     maps_3.4.1
    ## [31] sass_0.4.5          yaml_2.3.7          pillar_1.9.0
    ## [34] jquerylib_0.1.4     tidyr_1.3.0         BiocParallel_1.34.0
    ## [37] cachem_1.0.7        viridis_0.6.2       nlme_3.1-162
    ## [40] RSpectra_0.16-1     tidyselect_1.2.0    digest_0.6.31
    ## [43] stringi_1.7.12      dplyr_1.1.2         reshape2_1.4.4
    ## [46] purrr_1.0.1         splines_4.3.0       fastmap_1.1.1
    ## [49] grid_4.3.0          colorspace_2.1-0    cli_3.6.1
    ## [52] rTensor_1.4.8       magrittr_2.0.3      utf8_1.2.3
    ## [55] corpcor_1.6.10      withr_2.5.0         scales_1.2.1
    ## [58] plot3D_1.4          rmarkdown_2.21      matrixStats_0.63.0
    ## [61] igraph_1.4.2        gridExtra_2.3       evaluate_0.20
    ## [64] knitr_1.42          tcltk_4.3.0         viridisLite_0.4.1
    ## [67] mgcv_1.8-42         jointDiag_0.4       rlang_1.1.0
    ## [70] Rcpp_1.0.10         glue_1.6.2          geigen_2.3
    ## [73] jsonlite_1.8.4      R6_2.5.1            plyr_1.8.8

# References

Vasilescu, M. A. O. et al. 2005. “Multilinear Independent Component Analysis.” IEEE CVPR 2005.