---
title: "2. Discretized Singular Value Decomposition (`dSVD`)"
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{2. Discretized Singular Value Decomposition (`dSVD`)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

# Introduction

In this vignette, we consider approximating a matrix as a product of two low-rank matrices (a.k.a., factor matrices).

Test data is available from `toyModel`.

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

You will see that there are five blocks in the data matrix as follows.

```{r data2, echo=TRUE, fig.height=4, fig.width=5}
suppressMessages(library("fields"))
image.plot(X, main="Original Data", legend.mar=8)
```

# Semi-Ternary Matrix Factorization (STMF)

Here, we introduce the ternary regularization to take {-1,0,1} values in $U$ as below:

$$
X \approx U V' \ \mathrm{s.t.}\ U \in \{-1,0,1\},
$$
where $X$ ($N \times M$) is a data matrix, $U$ ($N \times J$) is a ternary score matrix, and $V$ ($M \times J$) is a loading matrix. In `dcTensor` package, the object function is optimized by combining gradient-descent algorithm [@svd] and ternary regularization.

## Basic Usage

In STMF, 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`) are also available. For the details of arguments of dSVD, see `?dSVD`. After the calculation, various objects are returned by `dSVD`. STMF is achieved by specifying the ternary regularization parameter as a large value like the below:

```{r stmf, echo=TRUE}
set.seed(123456)
out_STMF <- dSVD(X, Ter_U=1E+10, J=5)
str(out_STMF, 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_stmf, echo=TRUE, fig.height=4, fig.width=8}
layout(t(1:2))
plot(log10(out_STMF$RecError[-1]), type="b", main="Reconstruction Error")
plot(log10(out_STMF$RelChange[-1]), type="b", main="Relative Change")
```

The product of $U$ and $V$ shows whether the original data is well-recovered by `dSVD`.

```{r rec_stmf, echo=TRUE, fig.height=4, fig.width=8}
recX <- out_STMF$U %*% t(out_STMF$V)
layout(t(1:2))
image.plot(X, main="Original Data", legend.mar=8)
image.plot(recX, main="Reconstructed Data (STMF)", legend.mar=8)
```

The histograms of $U$ and $V$ show that $U$ looks ternary but $V$ does not.

```{r u_v5, echo=TRUE, fig.height=4, fig.width=8}
layout(t(1:2))
hist(out_STMF$U, breaks=100)
hist(out_STMF$V, breaks=100)
```

# Session Information {.unnumbered}

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

# References