CVE/README.md

# TODOs
Doc:
- [x] Stiefel (instead of Stiefl)
- [x] Return value description (`@returs`)
- [x] DESCRIPTION
    - [x] Maintainer
    - [x] Author
    - [x] Volume
    - [x] Description (from Paper) and Ref.
- [x] Ref paper in doc
- [ ] Data set descriptions and augmentations.
- [x] Demonstration of the `Logger` function usage (Demo file or so, ...)
- [ ] Update Paper (to new version / version consistent with current code!)

Methods to be implemented:
- [x] simple
- [x] weighted
- [x] momentum
- [x] weighted with momentum

Performance:
- [x] Pure C implementation.
- [NOT Feasible] Stochastic Version
- [NOT Feasible] Gradient Approximations (using Algebraic Software for alternative Loss function formulations and gradient optimizations)
- [NOT Sufficient] Alternative Kernels for reducing samples
- [ ] (To Be further investigated) "Kronecker" optimization

Features (functions):
- [x] Initial `V.init` parameter (only ONE try, ignore number of `attempts` parameter)
- [x] `basis.cve` list of estimated `B`s (with `k` supplied, only `B`)
- [x] `directions.cve` Projected `X` given `k`
- [x] `predict.cve` using `mars` for predicting responses given new data.
- [x] `predict.dim.cve` Cross-validation or `aov` (in stats package) or "elbow" estimation
- [x] `plot.elbow`
- [x] `summary`

Changes:
- [x] New `estimate.bandwidth` implementation.
    (h = 2 * (tr(\Sigma) / p) * (6/5 * n^(-1 / (4 + k)))^2,
    \Sigma = 1/n * (X-mean)'(X-mean))

# Development
## Build and install.
To build the package the `devtools` package is used. This also provides `roxygen2` which is used for documentation and automatic creation of the `NAMESPACE` file.
```R
setwd("./CVE_R")  # Set path to the package root.
library(devtools) # Load required `devtools` package.
document()        # Create `.Rd` files and write `NAMESPACE`.
```
Next the package needs to be build, therefore (if pure `R` package, aka. `C/C++`, `Fortran`, ... code) just do the following.
```bash
R CMD build CVE_C; R CMD INSTALL CVE_0.2.tar.gz
```
Then we are ready for using the package.
As well as building the `NAMESPACE` and `*.Rd` files using `devtools` (`roxygen2`) the following resembles an entire build pipeline including checks.
```bash
R -q -e 'library(devtools); setwd("CVE_C"); pkgbuild::compile_dll(); document(); pkgbuild::clean_dll()'
R CMD build CVE_C; R CMD check CVE_0.2.tar.gz;
R CMD INSTALL CVE_0.2.tar.gz
```
## Build and install from within `R`.
An alternative approach is the following.
```R
## Installing CVE (C implementation)
(setwd('~/Projects/CVE/CVE_C'))
# equiv to Rcpp::compileAttributes().
library(devtools)
pkgbuild::compile_dll() # required for packages with C/C++ code
document()              # See bug: https://github.com/stan-dev/rstantools/issues/52
pkgbuild::clean_dll()
(path <- build(vignettes = FALSE))
install.packages(path, repos = NULL, type = "source")
library(CVE)
```
**Note: I only recommend this approach during development.**

# Package Structure

## Demos
A demo is an `.R` file that lives in `demo/`. Demos are like examples but tend to
be longer. Instead of focusing on a single function, they show how to weave
together multiple functions to solve a problem.

You list and access demos with `demo()`:
   * Show all available demos: `demo()`.
   * Show all demos in a package: `demo(package = "CVE")`.
   * Run a specific demo: `demo("runtime_test", package = "CVE")`.
   * Find a demo: `system.file("demo", "runtime_test.R", package = "CVE")`.

Each demo must be listed in `demo/00Index` in the following form:
`demo-name Demo description`.
The demo name is the name of the file without the extension,
e.g. `demo/runtime_test.R` becomes `runtime_test`.

By default the demo ask for human input for each plot: "Hit to see next plot".
This behavior can be overridden by adding `devAskNewPage(ask = FALSE)` to
the demo file. You can add pauses by adding:
`readline("press any key to continue")`.

**Note**: Demos are not automatically tested by `R CMD check`. This means that they
can easily break without your knowledge.


# General Notes for Source Code analysis
## Search in multiple files.
Using the Linux `grep` program with the parameters `-rnw` and specifying a include files filter like the following example.
```bash
grep --include=*\.{c,h,R} -rnw '.' -e "sweep"
```
searches in all `C` source and header files as well as `R` source files for the term _sweep_.

## Recursive directory compare with colored structure (more or less).
```bash
diff -r CVE_R/ CVE_C/ | grep -E "^([<>]|[^<>].*)"
```

## Parsing `bash` script parameters.
```bash
usage="$0 [-v|--verbose] [-n|--dry-run] [(-s|--stack-size) <size>] [-h|--help] [-- [p1, [p2, ...]]]"
verbose=false
help=false
dry_run=false
stack_size=0

while [ $# -gt 0 ]; do
    case "$1" in
        -v | --verbose )    verbose=true;      shift ;;
        -n | --dry-run )    dry_run=true;      shift ;;
        -s | --stack-size ) stack_size="$2";   shift; shift ;;
        -h | --help )       echo $usage;       exit ;; # On help print usage and exit.
        -- ) shift; break ;;            # Break param "parsing".
         * ) echo $usage >&2; exit 1 ;; # Print usage and exit with failure.
    esac
done

echo verbose=$verbose
echo dry_run=$dry_run
echo stack_size=$stack_size
```

# Analysis
## Logging (a `cve` run).
To log `loss`, `error` (estimated) the true error (error of current estimated `B` against the true `B`) or even the step size one can use the `logger` parameter. A `logger` is a function that gets the current `environment` of the CVE optimization methods (__do not alter this environment, only read from it__). This can be used to create logs like in the following example.
```R
library(CVE)

# Setup histories.
(epochs <- 50)
(attempts <- 10)
loss.history       <- matrix(NA, epochs + 1, attempts)
error.history      <- matrix(NA, epochs + 1, attempts)
tau.history        <- matrix(NA, epochs + 1, attempts)
true.error.history <- matrix(NA, epochs + 1, attempts)

# Create a dataset
ds <- dataset("M1")
X <- ds$X
Y <- ds$Y
B <- ds$B # the true `B`
(k <- ncol(ds$B))

# True projection matrix.
P <- B %*% solve(t(B) %*% B) %*% t(B)
# Define the logger for the `cve()` method.
logger <- function(env) {
    # Note the `<<-` assignement!
    loss.history[env$epoch + 1, env$attempt] <<- env$loss
    error.history[env$epoch + 1, env$attempt] <<- env$error
    tau.history[env$epoch + 1, env$attempt] <<- env$tau
    # Compute true error by comparing to the true `B`
    B.est <- null(env$V) # Function provided by CVE
    P.est <- B.est %*% solve(t(B.est) %*% B.est) %*% t(B.est)
    true.error <- norm(P - P.est, 'F') / sqrt(2 * k)
    true.error.history[env$epoch + 1, env$attempt] <<- true.error
}
# Performa SDR
dr <- cve(Y ~ X, k = k, logger = logger, epochs = epochs, attempts = attempts)
# Plot history's
par(mfrow = c(2, 2))
matplot(loss.history,       type = 'l', log = 'y', xlab = 'iter',
        main = 'loss', ylab = expression(L(V[iter])))
matplot(error.history,      type = 'l', log = 'y', xlab = 'iter',
        main = 'error', ylab = 'error')
matplot(tau.history,        type = 'l', log = 'y', xlab = 'iter',
        main = 'tau', ylab = 'tau')
matplot(true.error.history, type = 'l', log = 'y', xlab = 'iter',
        main = 'true error', ylab = 'true error')
```

## Reading log files.
The run-time tests (upcoming further tests) are creating log files saved in `tmp/`. These log files are `CSV` files (actually `TSV`) with a header storing the test results. Depending on the test the files may contain different data. As an example we use the run-time test logs which store in each line the `dataset`, the used `method` as well as the `error` (actual error of estimated `B` against real `B`) and the `time`. For reading and analyzing the data see the following example.
```R
# Load log as `data.frame`
log <- read.csv('tmp/test0.log', sep = '\t')
# Create a error boxplot grouped by dataset.
boxplot(error ~ dataset, log)

# Overview
for (ds.name in paste0('M', seq(5))) {
    ds <- subset(log, dataset == ds.name, select = c('method', 'dataset', 'time', 'error'))
    print(summary(ds))
}
```

## Environments and variable lookup.
In the following a view simple examples of how `R` searches for variables.
In addition we manipulate function closures to alter the search path in variable lookup and outer scope variable manipulation.
```R
droids <- "These aren't the droids you're looking for."

search <- function() {
    print(droids)
}

trooper.seeks <- function() {
    droids <- c("R2-D2", "C-3PO")
    search()
}

jedi.seeks <- function() {
    droids <- c("R2-D2", "C-3PO")
    environment(search) <- environment()
    search()
}

trooper.seeks()
# [1] "These aren't the droids you're looking for."
jedi.seeks()
# [1] "R2-D2", "C-3PO"
```

The next example illustrates how to write (without local copies) to variables outside the functions local environment.
```R
counting <- function() {
    count <<- count + 1 # Note the `<<-` assignment.
}

(function() {
    environment(counting) <- environment()
    count <- 0

    for (i in 1:10) {
        counting()
    }

    return(count)
})()

(function () {
    closure <- new.env()
    environment(counting) <- closure
    assign("count", 0, envir = closure)

    for (i in 1:10) {
        counting()
    }

    return(closure$count)
})()
```

Another example for the usage of `do.call` where the evaluation of parameters is illustrated (example taken (and altered) from `?do.call`).
```R
## examples of where objects will be found.
A <- "A.Global"
f <- function(x) print(paste("f.new", x))
env <- new.env()
assign("A", "A.new", envir = env)
assign("f", f, envir = env)
f <- function(x) print(paste("f.Global", x))
f(A)                                          # f.Global A.Global
do.call("f", list(A))                         # f.Global A.Global
do.call("f", list(A), envir = env)            # f.new A.Global
do.call(f,   list(A), envir = env)            # f.Global A.Global
do.call("f", list(quote(A)), envir = env)     # f.new A.new
do.call(f,   list(quote(A)), envir = env)     # f.Global A.new
do.call("f", list(as.name("A")), envir = env) # f.new A.new
do.call("f", list(as.name("A")), envir = env) # f.new A.new
```

# Performance benchmarks
In this section alternative implementations of simple algorithms are compared for there performance.

### Computing the trace of a matrix multiplication.
```R
library(microbenchmark)

A <- matrix(runif(120), 12, 10)

# Check correctnes and benckmark performance.
stopifnot(
    all.equal(
        sum(diag(t(A) %*% A)), sum(diag(crossprod(A, A)))
    ),
    all.equal(
        sum(diag(t(A) %*% A)), sum(A * A)
    )
)
microbenchmark(
    MM = sum(diag(t(A) %*% A)),
    cross = sum(diag(crossprod(A, A))),
    elem = sum(A * A)
)
# Unit: nanoseconds
#   expr  min     lq    mean median     uq   max neval
#     MM 4232 4570.0 5138.81   4737 4956.0 40308   100
#  cross 2523 2774.5 2974.93   2946 3114.5  5078   100
#   elem  582  762.5  973.02    834  964.0 12945   100
```

```R
n <- 200
M <- matrix(runif(n^2), n, n)

dnorm2 <- function(x) exp(-0.5 * x^2) / sqrt(2 * pi)

stopifnot(
    all.equal(dnorm(M), dnorm2(M))
)
microbenchmark(
    dnorm = dnorm(M),
    dnorm2 = dnorm2(M),
    exp = exp(-0.5 * M^2) # without scaling -> irrelevant for usage
)
# Unit: microseconds
#   expr     min      lq     mean   median       uq      max neval
#  dnorm 841.503 843.811 920.7828 855.7505 912.4720 2405.587   100
# dnorm2 543.510 580.319 629.5321 597.8540 607.3795 2603.763   100
#    exp 502.083 535.943 577.2884 548.3745 561.3280 2113.220   100
```

### Using `crosspord()`
```R
p <- 12
q <- 10
V <- matrix(runif(p * q), p, q)

stopifnot(
    all.equal(V %*% t(V), tcrossprod(V)),
    all.equal(V %*% t(V), tcrossprod(V, V))
)
microbenchmark(
    V %*% t(V),
    tcrossprod(V),
    tcrossprod(V, V)
)
# Unit: microseconds
#              expr   min     lq    mean median     uq    max neval
#        V %*% t(V) 2.293 2.6335 2.94673 2.7375 2.9060 19.592   100
#     tcrossprod(V) 1.148 1.2475 1.86173 1.3440 1.4650 30.688   100
#  tcrossprod(V, V) 1.003 1.1575 1.28451 1.2400 1.3685  2.742   100
```

### Recycling vs. Sweep
```R
(n <- 200)
(p <- 12)
(q <- 10)
X_diff <- matrix(runif(n * (n - 1) / 2 * p), n * (n - 1) / 2, p)
V <- matrix(rnorm(p * q), p, q)
vecS <- runif(n * (n - 1) / 2)

stopifnot(
    all.equal((X_diff %*% V) * rep(vecS, q),
              sweep(X_diff %*% V, 1, vecS, `*`)),
    all.equal((X_diff %*% V) * rep(vecS, q),
              (X_diff %*% V) * vecS)
)
microbenchmark(
    rep = (X_diff %*% V) * rep(vecS, q),
    sweep = sweep(X_diff %*% V, 1, vecS, `*`, check.margin = FALSE),
    recycle = (X_diff %*% V) * vecS
)
# Unit: microseconds
#     expr      min        lq     mean    median       uq      max neval
#      rep  851.723  988.3655 1575.639 1203.6385 1440.578 18999.23   100
#    sweep 1313.177 1522.4010 2355.269 1879.2605 2065.399 18783.24   100
#  recycle  719.001  786.1265 1157.285  881.8825 1163.202 19091.79   100
```
### Scaled `crossprod` with matrix multiplication order.
```R
(n <- 200)
(p <- 12)
(q <- 10)
X_diff <- matrix(runif(n * (n - 1) / 2 * p), n * (n - 1) / 2, p)
V <- matrix(rnorm(p * q), p, q)
vecS <- runif(n * (n - 1) / 2)

ref <- crossprod(X_diff, X_diff * vecS) %*% V
stopifnot(
    all.equal(ref, crossprod(X_diff, (X_diff %*% V) * vecS)),
    all.equal(ref, crossprod(X_diff, (X_diff %*% V) * vecS))
)
microbenchmark(
    inner = crossprod(X_diff, X_diff * vecS) %*% V,
    outer = crossprod(X_diff, (X_diff %*% V) * vecS)
)
# Unit: microseconds
#   expr      min       lq     mean    median       uq       max neval
#  inner  789.065  867.939 1683.812  987.9375 1290.055 16800.265   100
#  outer 1141.479 1216.929 1404.702 1317.7315 1582.800  2531.766   100
```

### Fast dist matrix computation (aka. row sum of squares).
```R
library(microbenchmark)
library(CVE)

(n <- 200)
(N <- n * (n - 1) / 2)
(p <- 12)
M <- matrix(runif(N * p), N, p)

stopifnot(
    all.equal(rowSums(M^2), rowSums.c(M^2)),
    all.equal(rowSums(M^2), rowSquareSums.c(M))
)
microbenchmark(
    sums = rowSums(M^2),
    sums.c = rowSums.c(M^2),
    sqSums.c = rowSquareSums.c(M)
)
# Unit: microseconds
#      expr     min       lq      mean    median       uq      max neval
#      sums 666.311 1051.036 1612.3100 1139.0065 1547.657 13940.97   100
#    sums.c 342.647  672.453 1009.9109  740.6255 1224.715 13765.90   100
#  sqSums.c 115.325  142.128  175.6242  153.4645  169.678   759.87   100
```

## Using `Rprof()` for performance.
The standard method for profiling where an algorithm is spending its time is with `Rprof()`.
```R
path <- '../tmp/R.prof' # path to profiling file
Rprof(path)
cve.res <- cve.call(X, Y, k = k)
Rprof(NULL)
(prof <- summaryRprof(path)) # Summarise results
```
**Note: consider to run `gc()` before measuring**, aka cleaning up by explicitly calling the garbage collector.