---
title: "plots of a dependence"
author: "Niclas"
format: html
editor: visual
execute:
  echo: true
  working-directory: ../
---

```{r loading libraries}
#| cache: true
#| echo: false
#| collapse: true
# load local files
source(here::here("R", "singular_values.R"))
source(here::here("R", "graphon_distribution.R"))
source(here::here("R","singular_value_plot.R"))

# load libaries for data handling
library(ggplot2)
library(dplyr)
library(latex2exp)

```

## Setup

We consider the matrix $QQ^\top$ and look at the smallest eigenvalue, i.e. the smallest non-zero singular value of $Q$.

The matrix $Q$ is given by $$Q_{ik} = \int_{\frac{k}{K}}^{\frac{k+1}{K}} p_a(u| X_i) \, du$$ with $$p_a(u|X) = \frac{f_v(F_a^{-1}(u) - a^\top X)}{f_a(F_a^{-1}(u))}$$ In this document we plot different the smallest eigenvalue in dependence of the parameter $a$ with different "ratios" of the parameters $n$ and $$k = \lfloor n^\alpha \rfloor$$

with $\alpha = 0.1, 0.2, \dots 0.5$. The data matrix $X$ is a random matrix with i.i.d. distributed entries. We consider $x_{ij} \sim U[0,1]$ and $x_{ij} \sim Exp(\lambda)$.

## Exponential distribution

```{r k = n^alpha data generation, rate = 1}
#| cache: true
#| echo: false
#| collapse: true
ns <- seq(100, 10000, 100)
as <- seq(0, 20, 2)
alphas <-  seq(0.1, 0.5, 0.1)

set.seed(100)
results01 <- data.frame(dim_n = integer(),
                      dim_k = integer(),
                      param_a = double(),
                      param_alpha = double(),
                      ssv = double())
for (a in as) {
  for (i in 1:length(ns)) {
    for (j in 1:length(alphas)) {
      n <- ns[i]
      K <- floor(n^alphas[j])
      if (!K > 0) next # skip if K is equal to zero
      # use the default seed 1L
      Q <- compute_matrix(seed=1L,
                     a= a,
                     n = n,
                     K = K,
                     sample_X_fn = function(n) {matrix(rexp(n), ncol = 1L)},
                     fv = function(x) {dnorm(x, mean=0, sd=1)},
                     Fv = function(x) {pnorm(x, mean=0, sd=1)},
                     guard = 1e-12)
      
      ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
      
      current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
      results01 <- rbind(results01, current_res)
    }
  }
}
```

```{r k = n^alpha plotting, rate = 1}
# plot the results
results01 |>
  filter(param_a %in% c(0, 10, 20)) |>
  mutate(param_a = as.factor(param_a),
         param_alpha = as.factor(param_alpha)) |>
  group_by(param_a, param_alpha) |>
  ggplot(aes(dim_n, ssv, col=param_a, shape=param_alpha, interaction(param_a, param_alpha))) +
  geom_point(size=1.5) +
  geom_line() +
  geom_function(fun = function(x) {sqrt(x)}, colour="black") +
  #scale_y_log10() +
  theme_bw() +
  labs(x=latex2exp::TeX("$n$"),
       y=latex2exp::TeX("Smallest singular value of $Q$"),
       title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
       subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim Exp(1)$")),
       colour=latex2exp::TeX("$a$"),
       shape=latex2exp::TeX("$\\alpha$"))
```

```{r k = n^alpha data generation, rate = 3}
#| cache: true
#| echo: false
#| collapse: true
ns <- seq(100, 10000, 100)
as <- seq(0, 20, 2)
alphas <-  seq(0.1, 0.5, 0.1)

set.seed(100)
results02 <- data.frame(dim_n = integer(),
                      dim_k = integer(),
                      param_a = double(),
                      param_alpha = double(),
                      ssv = double())
for (a in as) {
  for (i in 1:length(ns)) {
    for (j in 1:length(alphas)) {
      n <- ns[i]
      K <- floor(n^alphas[j])
      if (!K > 0) next # skip if K is equal to zero
      # use the default seed 1L
      Q <- compute_matrix(seed=1L,
                     a= a,
                     n = n,
                     K = K,
                     sample_X_fn = function(n) {matrix(rexp(n, rate=3), ncol = 1L)},
                     fv = function(x) {dnorm(x, mean=0, sd=1)},
                     Fv = function(x) {pnorm(x, mean=0, sd=1)},
                     guard = 1e-12)
      
      ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
      
      current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
      results02 <- rbind(results02, current_res)
    }
  }
}
```

```{r k = n^alpha plotting, rate = 3}
results02 |>
  filter(param_a %in% c(0, 10, 20)) |>
  mutate(param_a = as.factor(param_a),
         param_alpha = as.factor(param_alpha)) |>
  group_by(param_a, param_alpha) |>
  ggplot(aes(dim_n, ssv, col=param_a, shape=param_alpha, interaction(param_a, param_alpha))) +
  geom_point(size=1.5) +
  geom_line() +
  geom_function(fun = function(x) {sqrt(x)}, colour="black") +
  #scale_y_log10() +
  theme_bw() +
  labs(x=latex2exp::TeX("$n$"),
       y=latex2exp::TeX("Smallest singular value of $Q$"),
       title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
       subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim Exp(3))$")),
       colour=latex2exp::TeX("$a$"),
       shape=latex2exp::TeX("$\\alpha$"))
```

For $a = 0$ the smallest singular value is very close to zero.

```{r k = n^alpha data generation, rate = 5}
#| cache: true
#| echo: false
#| collapse: true
ns <- seq(100, 10000, 100)
as <- seq(0, 20, 2)
alphas <-  seq(0.1, 0.5, 0.1)

set.seed(100)
results03 <- data.frame(dim_n = integer(),
                      dim_k = integer(),
                      param_a = double(),
                      param_alpha = double(),
                      ssv = double())
for (a in as) {
  for (i in 1:length(ns)) {
    for (j in 1:length(alphas)) {
      n <- ns[i]
      K <- floor(n^alphas[j])
      if (!K > 0) next # skip if K is equal to zero
      # use the default seed 1L
      Q <- compute_matrix(seed=1L,
                     a= a,
                     n = n,
                     K = K,
                     sample_X_fn = function(n) {matrix(rexp(n, rate=5), ncol = 1L)},
                     fv = function(x) {dnorm(x, mean=0, sd=1)},
                     Fv = function(x) {pnorm(x, mean=0, sd=1)},
                     guard = 1e-12)
      
      ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
      
      current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
      results03 <- rbind(results03, current_res)
    }
  }
}
```

```{r k = n^alpha plotting, rate = 5}
results03 |>
  filter(param_a %in% c(0, 10, 20)) |>
  mutate(param_a = as.factor(param_a),
         param_alpha = as.factor(param_alpha)) |>
  group_by(param_a, param_alpha) |>
  ggplot(aes(dim_n, ssv, col=param_a, shape=param_alpha, interaction(param_a, param_alpha))) +
  geom_point(size=1.5) +
  geom_line() +
  geom_function(fun = function(x) {sqrt(x)}, colour="black") +
  #scale_y_log10() +
  theme_bw() +
  labs(x=latex2exp::TeX("$n$"),
       y=latex2exp::TeX("Smallest singular value of $Q$"),
       title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
       subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim Exp(5))$")),
       colour=latex2exp::TeX("$a$"),
       shape=latex2exp::TeX("$\\alpha$"))
```

Why is here a perfect match for $\alpha = 0.1$ and $a = 20$ to the square function? The difference is of the order of $10^{-11}$!

## Uniform distribution

```{r k = n^alpha data generation, U[0,1]}
#| cache: true
#| echo: false
#| collapse: true
ns <- seq(100, 10000, 100)
as <- seq(0, 20, 2)
alphas <-  seq(0.1, 0.5, 0.1)

set.seed(100)
results04 <- data.frame(dim_n = integer(),
                      dim_k = integer(),
                      param_a = double(),
                      param_alpha = double(),
                      ssv = double())
for (a in as) {
  for (i in 1:length(ns)) {
    for (j in 1:length(alphas)) {
      n <- ns[i]
      K <- floor(n^alphas[j])
      if (!K > 0) next # skip if K is equal to zero
      # use the default seed 1L
      Q <- compute_matrix(seed=1L,
                     a= a,
                     n = n,
                     K = K,
                     sample_X_fn = function(n) {matrix(runif(n), ncol = 1L)},
                     fv = function(x) {dnorm(x, mean=0, sd=1)},
                     Fv = function(x) {pnorm(x, mean=0, sd=1)},
                     guard = 1e-12)
      
      ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
      
      current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
      results04 <- rbind(results04, current_res)
    }
  }
}
```

```{r k = n^alpha plotting, U[0,1]}
results04 |>
  filter(param_a %in% c(0, 10, 20)) |>
  mutate(param_a = as.factor(param_a),
         param_alpha = as.factor(param_alpha)) |>
  group_by(param_a, param_alpha) |>
  ggplot(aes(dim_n, ssv, col=param_a, shape=param_alpha, interaction(param_a, param_alpha))) +
  geom_point(size=1.5) +
  geom_line() +
  geom_function(fun = function(x) {sqrt(x)}, colour="black") +
  #scale_y_log10() +
  theme_bw() +
  labs(x=latex2exp::TeX("$n$"),
       y=latex2exp::TeX("Smallest singular value of $Q$"),
       title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
       subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim U[0,1] $")),
       colour=latex2exp::TeX("$a$"),
       shape=latex2exp::TeX("$\\alpha$"))
```

Here we have the same effect for $\alpha = 0.1$ and $a = 20$.

```{r k = n^alpha data generation, U[0,2]}
#| cache: true
#| echo: false
#| collapse: true
ns <- seq(100, 10000, 100)
as <- seq(0, 20, 2)
alphas <-  seq(0.1, 0.5, 0.1)

set.seed(100)
results05 <- data.frame(dim_n = integer(),
                      dim_k = integer(),
                      param_a = double(),
                      param_alpha = double(),
                      ssv = double())
for (a in as) {
  for (i in 1:length(ns)) {
    for (j in 1:length(alphas)) {
      n <- ns[i]
      K <- floor(n^alphas[j])
      if (!K > 0) next # skip if K is equal to zero
      # use the default seed 1L
      Q <- compute_matrix(seed=1L,
                     a= a,
                     n = n,
                     K = K,
                     sample_X_fn = function(n) {matrix(runif(n, min = 0, max=2), ncol = 1L)},
                     fv = function(x) {dnorm(x, mean=0, sd=1)},
                     Fv = function(x) {pnorm(x, mean=0, sd=1)},
                     guard = 1e-12)
      
      ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
      
      current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
      results05 <- rbind(results05, current_res)
    }
  }
}
```

```{r k = n^alpha plotting, U[0,2]}
results05 |>
  filter(param_a %in% c(0, 10, 20)) |>
  mutate(param_a = as.factor(param_a),
         param_alpha = as.factor(param_alpha)) |>
  group_by(param_a, param_alpha) |>
  ggplot(aes(dim_n, ssv, col=param_a, shape=param_alpha, interaction(param_a, param_alpha))) +
  geom_point(size=1.5) +
  geom_line() +
  geom_function(fun = function(x) {sqrt(x)}, colour="black") +
  #scale_y_log10() +
  theme_bw() +
  labs(x=latex2exp::TeX("$n$"),
       y=latex2exp::TeX("Smallest singular value of $Q$"),
       title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
       subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim U[0,2] $")),
       colour=latex2exp::TeX("$a$"),
       shape=latex2exp::TeX("$\\alpha$"))
```

```{r k = n^alpha data generation, N(0,1)}
#| cache: true
#| echo: false
#| collapse: true
ns <- seq(100, 10000, 100)
as <- seq(0, 20, 2)
alphas <-  seq(0.1, 0.5, 0.1)

set.seed(100)
results06 <- data.frame(dim_n = integer(),
                      dim_k = integer(),
                      param_a = double(),
                      param_alpha = double(),
                      ssv = double())
for (a in as) {
  for (i in 1:length(ns)) {
    for (j in 1:length(alphas)) {
      n <- ns[i]
      # HERE WE USE THE CEILING AND NOT FLOOR!
      K <- ceiling(n^alphas[j])
      if (!K > 0) next # skip if K is equal to zero
      # use the default seed 1L
      Q <- compute_matrix(seed=1L,
                     a= a,
                     n = n,
                     K = K,
                     sample_X_fn = function(n) {matrix(rnorm(n), ncol = 1L)},
                     fv = function(x) {dnorm(x, mean=0, sd=1)},
                     Fv = function(x) {pnorm(x, mean=0, sd=1)},
                     guard = 1e-12)
      
      ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
      
      current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
      results06 <- rbind(results06, current_res)
    }
  }
}
```

```{r k = n^alpha plotting, U[0,2]}
results06 |>
  filter(param_a %in% c(0, 10, 20)) |>
  mutate(param_a = as.factor(param_a),
         param_alpha = as.factor(param_alpha)) |>
  group_by(param_a, param_alpha) |>
  ggplot(aes(dim_n, ssv * dim_k, col=param_a, shape=param_alpha, interaction(param_a, param_alpha))) +
  geom_point(size=1.5) +
  geom_line() +
  geom_function(fun = function(x) {x^(0.5)}, colour="black") +
  #scale_y_log10() +
  theme_bw() +
  labs(x=latex2exp::TeX("$n$"),
       y=latex2exp::TeX("Smallest singular value of $Q$"),
       title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
       subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim N(0,1) $, use ceil function instead of floor for rounding.")),
       colour=latex2exp::TeX("$a$"),
       shape=latex2exp::TeX("$\\alpha$"))
```

```{r k = n^alpha plotting, U[0,2]}
results06 |>
  filter(param_a %in% c(0, 10, 20)) |>
  mutate(param_a = as.factor(param_a),
         param_alpha = as.factor(param_alpha)) |>
  group_by(param_a, param_alpha) |>
  ggplot(aes(dim_n, ssv / sqrt(dim_n) * dim_k, col=param_a, shape=param_alpha, interaction(param_a, param_alpha))) +
  geom_point(size=1.5) +
  geom_line() +
 # geom_function(fun = function(x) {x^(0.5)}, colour="black") +
  #scale_y_log10() +
  theme_bw() +
  labs(x=latex2exp::TeX("$n$"),
       y=latex2exp::TeX("Smallest singular value of $Q$ / sqrt(n)"),
       title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
       subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim N(0,1) $")),
       colour=latex2exp::TeX("$a$"),
       shape=latex2exp::TeX("$\\alpha$"))
```

```{r}
results <- list(results01, results02, results03, results04, results05, results06)
save(results, file="results.RData")
```