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add plots for nonnormal X

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Niclas
2026-03-31 15:32:06 +02:00
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---
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, 1000, 100)
as <- seq(0, 20, 2)
alphas <- seq(0.1, 0.5, 0.1)
set.seed(100)
results <- 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)
results <- rbind(results, current_res)
}
}
}
```
```{r k = n^alpha plotting, rate = 1}
results |>
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, 1000, 100)
as <- seq(0, 20, 2)
alphas <- seq(0.1, 0.5, 0.1)
set.seed(100)
results <- 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)
results <- rbind(results, current_res)
}
}
}
```
```{r k = n^alpha plotting, rate = 3}
results |>
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, 1000, 100)
as <- seq(0, 20, 2)
alphas <- seq(0.1, 0.5, 0.1)
set.seed(100)
results <- 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)
results <- rbind(results, current_res)
}
}
}
```
```{r k = n^alpha plotting, rate = 5}
results |>
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, 1000, 100)
as <- seq(0, 20, 2)
alphas <- seq(0.1, 0.5, 0.1)
set.seed(100)
results <- 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)
results <- rbind(results, current_res)
}
}
}
```
```{r k = n^alpha plotting, U[0,1]}
results |>
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, 1000, 100)
as <- seq(0, 20, 2)
alphas <- seq(0.1, 0.5, 0.1)
set.seed(100)
results <- 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)
results <- rbind(results, current_res)
}
}
}
```
```{r k = n^alpha plotting, U[0,2]}
results |>
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$"))
```