---
title: "3. Multilinear Independent Component Analysis (MultilinearICA)"
author:
- name: Koki Tsuyuzaki
  affiliation: Department of Artificial Intelligence Medicine,
    Graduate School of Medicine, Chiba University
  email: k.t.the-answer@hotmail.co.jp
date: "`r Sys.Date()`"
bibliography: bibliography.bib
package: iTensor
output: rmarkdown::html_vignette
vignette: |
  %\VignetteIndexEntry{3. Multilinear Independent Component Analysis (MultilinearICA)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

# 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`.

```{r data, echo=TRUE}
library("iTensor")
library("nnTensor")
data <- nnTensor::toyModel("CP")
str(data, 2)
```

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

```{r data2, echo=TRUE, fig.height=6, fig.width=6}
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 [@multilinearica]).

`MultilinearICA` can be performed as follows.

```{r multilinearica, echo=TRUE}
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.

```{r plot_multilinearica, echo=TRUE, fig.height=6, fig.width=6}
rec_data <- recTensor(out$S, out$As)
plotTensor3D(rec_data)
```

# Session Information {.unnumbered}

```{r sessionInfo, echo=FALSE}
sessionInfo()
```

# References