---
title: "4. Discretized Joint Non-negative Matrix Factrozation (`djNMF`)"
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{4. Discretized Joint Non-negative Matrix Factrozation (`djNMF`)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

# Introduction

In this vignette, we consider approximating non-negative multiple matrices 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")
suppressMessages(library("nnTensor"))
X <- nnTensor::toyModel("siNMF_Hard")
```

You will see that there are some blocks in the data matrices as follows.

```{r data2, echo=TRUE, fig.height=2.7, fig.width=8}
suppressMessages(library("fields"))
layout(t(1:3))
image.plot(X[[1]], main="X1", legend.mar=8)
image.plot(X[[2]], main="X2", legend.mar=8)
image.plot(X[[3]], main="X3", legend.mar=8)
```

# Semi-Binary Simultaneous Matrix Factorization (SBSMF)

Here, we consider the approximation of $K$ binary data matrices $X_{k}$ ($N \times M_{k}$) as the matrix product of $W$ ($N \times J$) and $V_{k}$ (J \times $M_{k}$):

$$
X_{k} \approx (W + V_{k}) H_{k} \ \mathrm{s.t.}\ W,V_{k},H_{k} \in \{0,1\}
$$

This is the combination of binary matrix factorization (BMF [@bmf]) and joint non-negative matrix decomposition (jNMF [@jnmf; @amari]), which is implemented by adding binary regularization against jNMF. See also `jNMF` function of [nnTensor](https://cran.r-project.org/package=nnTensor) package.

## Basic Usage

In SBSMF, 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 djNMF, see `?djNMF`. After the calculation, various objects are returned by `djNMF`. SBSMF is achieved by specifying the binary regularization parameter as a large value like the below:

```{r sbmf, echo=TRUE}
set.seed(123456)
out_djNMF <- djNMF(X, Bin_W=1E-1, J=4)
str(out_djNMF, 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_sbmf, echo=TRUE, fig.height=4, fig.width=8}
layout(t(1:2))
plot(log10(out_djNMF$RecError[-1]), type="b", main="Reconstruction Error")
plot(log10(out_djNMF$RelChange[-1]), type="b", main="Relative Change")
```

The products of $W$ and $H_{k}$s show whether the original data matrices are well-recovered by `djNMF`.

```{r rec_sbmf, echo=TRUE, fig.height=8, fig.width=8}
recX1 <- lapply(seq_along(X), function(x){
  out_djNMF$W %*% t(out_djNMF$H[[x]])
})
recX2 <- lapply(seq_along(X), function(x){
  out_djNMF$V[[x]] %*% t(out_djNMF$H[[x]])
})
layout(rbind(1:3, 4:6, 7:9))
image.plot(X[[1]], legend.mar=8, main="X1")
image.plot(X[[2]], legend.mar=8, main="X2")
image.plot(X[[3]], legend.mar=8, main="X3")
image.plot(recX1[[1]], legend.mar=8, main="Reconstructed X1 (Common Factor)")
image.plot(recX1[[2]], legend.mar=8, main="Reconstructed X2 (Common Factor)")
image.plot(recX1[[3]], legend.mar=8, main="Reconstructed X3 (Common Factor)")
image.plot(recX2[[1]], legend.mar=8, main="Reconstructed X1 (Specific Factor)")
image.plot(recX2[[2]], legend.mar=8, main="Reconstructed X2 (Specific Factor)")
image.plot(recX2[[3]], legend.mar=8, main="Reconstructed X3 (Specific Factor)")
```

The histogram of $W$ shows that the factor matrix $W$ looks binary.

```{r w_h_sbmf, echo=TRUE, fig.height=4, fig.width=8}
layout(rbind(1:4, 5:8))
hist(out_djNMF$W, main="W", breaks=100)
hist(out_djNMF$H[[1]], main="H1", breaks=100)
hist(out_djNMF$H[[2]], main="H2", breaks=100)
hist(out_djNMF$H[[3]], main="H3", breaks=100)
hist(out_djNMF$V[[1]], main="V1", breaks=100)
hist(out_djNMF$V[[2]], main="V2", breaks=100)
hist(out_djNMF$V[[3]], main="V3", breaks=100)
```

# Session Information {.unnumbered}

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

# References