init_fn3.Rmd

---
title: "Comparing initialization functions (part II)"
author: "Jason Willwerscheid"
date: "8/28/2018"
output:
  workflowr::wflow_html
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```


## Introduction

Here I return to the question of which initialization function is best, and in which cases. I run some simple experiments on simulated datasets of various dimensions and with or without missing data.

## Results

I simulate data from both a null model and a rank-one model with $p \in \{1000, 10000\}$ and with $n$ ranging from 10 to 1000. I either retain all of the data or I delete 20% of entries. I initialize using `"udv_si"`, `"udv_si_svd"`, and, when there is no missing data, `"udv_svd"`.

I pre-run the code [below](#code) and load the results from file.

```{r results}
all_res <- readRDS("./data/init_fn3/all_res.rds")
 
plot_results <- function(res, n, main, colors) {
  colors = colors[c("udv_si_svd", "udv_si", "udv_svd")]
  plot(log10(n), log10(res[["udv_si_svd"]]), type='l', col=colors[1],
       xlab = "log10(n)", ylab = "time elapsed (log10 s)",
       ylim = log10(c(min(unlist(res)), max(unlist(res)))),
       main = main)
  lines(log10(n), log10(res[["udv_si"]]), col=colors[2])
  if (length(res) == 3) {
    lines(log10(n), log10(res[["udv_svd"]]), col=colors[3])
  }
  
  legend.txt <- c("udv_si_svd", "udv_si", "udv_svd")
  legend("bottomright", legend.txt[1:length(res)], lty=1,
         col=colors[1:length(res)])
}

ns <- c(10, 25, 50, 100, 250, 500, 1000)
colors = RColorBrewer::brewer.pal(3, "Dark2")[1:3]
names(colors) = c("udv_svd", "udv_si", "udv_si_svd")

plot_results(all_res$null_noNA_p1000, ns, 
             "Null data (no missing), p = 1000", colors)
plot_results(all_res$null_noNA_p10000, ns, 
             "Null data (no missing), p = 10000", colors)

plot_results(all_res$r1_noNA_p1000, ns, 
             "Rank-one data (no missing), p = 1000", colors)
plot_results(all_res$r1_noNA_p10000, ns, 
             "Rank-one data (no missing), p = 10000", colors)

plot_results(all_res$null_missing_p1000, ns, 
             "Null data (with missing), p = 1000", colors)
plot_results(all_res$null_missing_p10000, ns, 
             "Null data (with missing), p = 10000", colors)

plot_results(all_res$r1_missing_p1000, ns, 
             "Rank-one data (with missing), p = 1000", colors)
plot_results(all_res$r1_missing_p10000, ns, 
             "Rank-one data (with missing), p = 10000", colors)
```

## Conclusions

The current default `init_fn = "udv_si"` is sensible. In cases with missing data, `"udv_si"` almost always beats `"udv_si_svd"`; the only exceptions are for small $n$, but in such cases initialization is very fast anyway. 

When there is no missing data, then `"udv_svd"` is the fastest method for small $n$, but `"udv_si"` is again the fastest method when $n$ becomes large. (Interestingly, the relative speeds do not seem to depend on the larger dimension $p$.)

It would be possible to programmatically set `init_fn` based on $n$ (or more precisely, based on the smaller of $n$ and $p$), but I don't think it's worth the trouble, since `"udv_si_svd"` seems to consistently be the fastest (or nearly fastest) method when the speed of initialization actually becomes an issue.

## Code

```{r code, code=readLines("../code/init_fn3.R"), eval=FALSE}
```