---
title: "6. Discretized Non-negative Tensor Factorization (`dNTF`)"
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{6. Discretized Non-negative Tensor Factorization (`dNTF`)}
  %\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)

To decompose a binary tensor ($\mathcal{X}$), non-negative CP decomposition (a.k.a. non-negative tensor factorization; NTF [@ntf; @amari]) can be applied. NTF appoximates $\mathcal{X}$ ($N \times M \times L$) as the mode-product of a core tensor $S$ ($J \times J \times J$) and factor matrices $A_1$ ($J \times N$), $A_2$ ($J \times M$), and $A_3$ ($J \times L$).

$$
\mathcal{X} \approx \mathcal{S} \times_{1} A_1 \times_{2} A_2 \times_{3} A_3\ \mathrm{s.t.}\ \mathcal{S} \geq 0, A_{k} \geq 0\ (k=1 \ldots 3)
$$

Note that \times_{k} is the mode-$k$ product [@amari] and the core tensor $S$ has non-negative values only in the diagonal element. For the details, see `NTF` function of [nnTensor](https://cran.r-project.org/package=nnTensor) package.


## Basic Usage

In BTF, a rank parameter $J$ ($\leq \min(N, M)$) is 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`. BTF is achieved by specifying the binary regularization parameter as a large value like the below:

```{r bmf, echo=TRUE, fig.height=4, fig.width=8}
set.seed(123456)
out_dNTF <- dNTF(X, Bin_A=c(1e+2, 1e+2, 1e+2), algorithm="KL", rank=4)
str(out_dNTF, 2)
```

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_dNTF$RecError[-1]), type="b", main="Reconstruction Error")
plot(log10(out_dNTF$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 `dNTF`.

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

The histograms of $A_{k}$s show that all the factor matrices $A_{k}$ looks binary.

```{r a_bmf, echo=TRUE, fig.height=4, fig.width=8}
layout(t(1:3))
hist(out_dNTF$A[[1]], main="A1", breaks=100)
hist(out_dNTF$A[[2]], main="A2", breaks=100)
hist(out_dNTF$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, a rank parameter $J$ ($\leq \min(N, M)$) is 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, fig.height=4, fig.width=8}
set.seed(123456)
out_dNTF2 <- dNTF(X2, Bin_A=c(1e+5, 1e+5, 1e-10), algorithm="KL", rank=4)
str(out_dNTF2, 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_dNTF2$RecError[-1]), type="b", main="Reconstruction Error")
plot(log10(out_dNTF2$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 `dNTF`.

```{r rec_sbmf, echo=TRUE, fig.height=4, fig.width=8}
recX <- recTensor(out_dNTF2$S, out_dNTF2$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_dNTF2$A[[1]], main="A1", breaks=100)
hist(out_dNTF2$A[[2]], main="A2", breaks=100)
hist(out_dNTF2$A[[3]], main="A3", breaks=100)
```

# Session Information {.unnumbered}

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

# References