---
title: "Introduction to frequentdirections"
author: "Shinichi Takayanagi"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Introduction to frequentdirections}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  cache=TRUE,
  fig.height=6,
  fig.width=8
)
```

### Download example data

Here, we use [MNIST package](https://github.com/stillmatic/MNIST) developped by  [\@stillmatic](https://github.com/stillmatic) as sample data.

You can install this package like the following:
```{r, eval=FALSE}
devtools::install_github("stillmatic/MNIST")
```

### Load data

Once you install `stillmatic/MNIST`, MNIST data is exported as `MNIST::mnist_train`.

Example the number `8`
```{r plot-example-image}
MNIST::show_digit(MNIST::mnist_train[770,])
```

### Sampling

There are 60,000 records in the data, it is little bit too much data for usual SVD (for usual PC).

That's why we would like to do sampling here.

```{r}
df <- MNIST::mnist_train[sample(seq_len(nrow(MNIST::mnist_train)), size=10^4), ]
```

### Plot SVD
Plot the original data on the first and second singular vector plane.
```{r plot-svd}
# Last column is y column
x <- as.matrix(df[, -ncol(df)])/255
y <- df$y
frequentdirections::plot_svd(x, y)
```

### Matrix Sketching
#### l = 8 case
```{r frequentdirections-8}
eps <- 10^(-8)
# 10000 x 256 -> 8 * 256 matrix
b <- frequentdirections::sketching(x, 8, eps)
frequentdirections::plot_svd(x, y, b)
```

#### l = 32 case
```{r frequentdirections-32}
# 10000 x 256 -> 32 * 256 matrix
b <- frequentdirections::sketching(x, 32, eps)
frequentdirections::plot_svd(x, y, b)
```

#### l = 128 case
```{r frequentdirections-128}
# 10000 x 256 -> 128 * 256 matrix
b <- frequentdirections::sketching(x, 128, eps)
frequentdirections::plot_svd(x, y, b)
```

This result is almost the same with the original data SVD expression.

That's why we can think that the original data is expressed with only `128` rows.