---
title: "7. Discretized Non-negative Tucker Decomposition (`dNTD`)"
author:
- name: Koki Tsuyuzaki
  affiliation: Laboratory for Bioinformatics Research,
    RIKEN Center for Biosystems Dynamics Research
  email: k.t.the-answer@hotmail.co.jp
date: "`r Sys.Date()`"
bibliography: bibliography.bib
package: dcTensor
output: rmarkdown::html_vignette
vignette: |
  %\VignetteIndexEntry{7. Discretized Non-negative Tucker Decomposition (`dNTD`)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

# Introduction

In this vignette, we consider approximating a non-negative tensor as a product of binary or non-negative low-rank matrices (a.k.a., factor matrices).

Test data is available from `toyModel`.

```{r data, echo=TRUE}
library("dcTensor")
X <- dcTensor::toyModel("dNTF")
```

You will see that there are four blocks in the data tensor as follows.

```{r data2, echo=TRUE, fig.height=8, fig.width=8}
suppressMessages(library("nnTensor"))
plotTensor3D(X)
```

# Binary Tensor Factorization (BTF)

Here, we introduce a non-negative tensor decomposition method, non-negative Tucker decomposition (NTD [@ntd; @amari]). The difference with the NTF is that different ranks can be specified for factor matrices such as $A_1$ ($J1 \times N$), $A_2$ ($J2 \times M$), and $A_3$ ($J3 \times L$) and that the core tensor can have non-negative values in the non-diagonal elements. This degree of freedom will show that NTD fits the second set of data well. For the details, see `NTD` function of [nnTensor](https://cran.r-project.org/package=nnTensor) package.

## Basic Usage

In BTF by NTD, rank parameters are needed to be set in advance. Other settings such as the number of iterations (`num.iter`) or factorization algorithm (`algorithm`) are also available. For the details of arguments of dNTD, see `?dNTD`. After the calculation, various objects are returned by `dNTD`. BTF is achieved by specifying the binary regularization parameter as a large value like the below:

```{r bmf, echo=TRUE}
set.seed(123456)
out_dNTD <- dNTD(X, Bin_A=c(1e+6, 1e+6, 1e+6), thr=1e-20, rank=c(4,4,4))
str(out_dNTD, 2)
```

After the calculation, various objects are returned by `dNTD`.

The reconstruction error (`RecError`) and relative error (`RelChange`, the amount of change from the reconstruction error in the previous step) can be used to diagnose whether the calculation is converged or not.

```{r conv_bmf, echo=TRUE, fig.height=4, fig.width=8}
layout(t(1:2))
plot(log10(out_dNTD$RecError[-1]), type="b", main="Reconstruction Error")
plot(log10(out_dNTD$RelChange[-1]), type="b", main="Relative Change")
```

The product of core tensor $S$ and factor matrices $A_{k}$ shows whether the original data is well-recovered by `dNTD`.

```{r rec_bmf, echo=TRUE, fig.height=4, fig.width=8}
recX <- recTensor(out_dNTD$S, out_dNTD$A)
layout(t(1:2))
plotTensor3D(X)
plotTensor3D(recX, thr=0)
```

The histograms of $A_{k}$s show that $A_{k}$ looks binary.

```{r a_bmf, echo=TRUE, fig.height=4, fig.width=8}
layout(t(1:3))
hist(out_dNTD$A[[1]], main="A1", breaks=100)
hist(out_dNTD$A[[2]], main="A2", breaks=100)
hist(out_dNTD$A[[3]], main="A3", breaks=100)
```

# Semi-Binary Tensor Factorization (SBTF)

Here, we define this formalization as semi-binary tensor factorization (SBTF). SBTF can capture discrete patterns from non-negative matrices.

To demonstrate SBMF, next we use a non-negative tensor from the `nnTensor` package. You will see that there are four blocks in the data tensor as follows.

```{r data3, echo=TRUE, fig.height=8, fig.width=8}
X2 <- nnTensor::toyModel("CP")
plotTensor3D(X2)
```

## Basic Usage

In SBTF by NTD, rank parameters are needed to be set in advance. Other settings such as the number of iterations (`num.iter`) or factorization algorithm (`algorithm`) are also available. For the details of arguments of dNTF, see `?dNTF`. After the calculation, various objects are returned by `dNTF`. SBTF is achieved by specifying the binary regularization parameter as a large value like the below:

```{r sbmf, echo=TRUE}
set.seed(123456)
out_dNTD2 <- dNTD(X2, Bin_A=c(1e-10, 1e-10, 1e+5), rank=c(4,4,4))
str(out_dNTD2, 2)
```

`RecError` and `RelChange` can be used to diagnose whether the calculation is converged or not.

```{r conv_sbmf, echo=TRUE, fig.height=4, fig.width=8}
layout(t(1:2))
plot(log10(out_dNTD2$RecError[-1]), type="b", main="Reconstruction Error")
plot(log10(out_dNTD2$RelChange[-1]), type="b", main="Relative Change")
```

The product of core tensor $S$ and factor matrices $A_{k}$ shows whether the original data is well-recovered by `dNTD`.

```{r rec_sbmf, echo=TRUE, fig.height=4, fig.width=8}
recX <- recTensor(out_dNTD2$S, out_dNTD2$A)
layout(t(1:2))
plotTensor3D(X2)
plotTensor3D(recX, thr=0)
```

The histograms of $A_{k}$s show that $A_{k}$ looks binary.

```{r a_sbmf, echo=TRUE, fig.height=4, fig.width=8}
layout(t(1:3))
hist(out_dNTD2$A[[1]], main="A1", breaks=100)
hist(out_dNTD2$A[[2]], main="A2", breaks=100)
hist(out_dNTD2$A[[3]], main="A3", breaks=100)
```

# Session Information {.unnumbered}

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

# References