351 lines
11 KiB
Plaintext
351 lines
11 KiB
Plaintext
---
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title: "plots of a dependence"
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author: "Niclas"
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format: html
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editor: visual
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execute:
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echo: true
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working-directory: ../
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---
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```{r loading libraries}
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#| cache: true
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#| echo: false
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#| collapse: true
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# load local files
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source(here::here("R", "singular_values.R"))
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source(here::here("R", "graphon_distribution.R"))
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source(here::here("R","singular_value_plot.R"))
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# load libaries for data handling
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library(ggplot2)
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library(dplyr)
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library(latex2exp)
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```
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## Setup
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We consider the matrix $QQ^\top$ and look at the smallest eigenvalue, i.e. the
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smallest non-zero singular value of $Q$.
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The matrix $Q$ is given by
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$$
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Q_{ik} = \int_{\frac{k}{K}}^{\frac{k+1}{K}} p_a(u| X_i) \, du
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$$
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with
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$$
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p_a(u|X) = \frac{f_v(F_a^{-1}(u) - a^\top X)}{f_a(F_a^{-1}(u))}
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$$
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In this document we plot different the smallest eigenvalue in dependence of the
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parameter $a$ with different "ratios" of the parameters $n$ and
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$$
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k = \lfloor n^\alpha \rfloor
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$$
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with $\alpha = 0.1, 0.2, \dots 0.5$. The data matrix $X$ is a random matrix with
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i.i.d. distributed entries. We consider $x_{ij} \sim U[0,1]$ and $x_{ij} \sim Exp(\lambda)$.
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## Exponential distribution
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```{r k = n^alpha data generation, rate = 1}
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#| cache: true
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#| echo: false
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#| collapse: true
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ns <- seq(100, 1000, 100)
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as <- seq(0, 20, 2)
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alphas <- seq(0.1, 0.5, 0.1)
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set.seed(100)
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results <- data.frame(dim_n = integer(),
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dim_k = integer(),
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param_a = double(),
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param_alpha = double(),
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ssv = double())
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for (a in as) {
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for (i in 1:length(ns)) {
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for (j in 1:length(alphas)) {
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n <- ns[i]
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K <- floor(n^alphas[j])
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if (!K > 0) next # skip if K is equal to zero
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# use the default seed 1L
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Q <- compute_matrix(seed=1L,
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a= a,
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n = n,
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K = K,
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sample_X_fn = function(n) {matrix(rexp(n), ncol = 1L)},
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fv = function(x) {dnorm(x, mean=0, sd=1)},
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Fv = function(x) {pnorm(x, mean=0, sd=1)},
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guard = 1e-12)
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ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
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current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
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results <- rbind(results, current_res)
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}
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}
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}
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```
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```{r k = n^alpha plotting, rate = 1}
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results |>
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filter(param_a %in% c(0, 10, 20)) |>
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mutate(param_a = as.factor(param_a),
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param_alpha = as.factor(param_alpha)) |>
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group_by(param_a, param_alpha) |>
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ggplot(aes(dim_n, ssv, col=param_a, shape=param_alpha, interaction(param_a, param_alpha))) +
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geom_point(size=1.5) +
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geom_line() +
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geom_function(fun = function(x) {sqrt(x)}, colour="black") +
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#scale_y_log10() +
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theme_bw() +
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labs(x=latex2exp::TeX("$n$"),
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y=latex2exp::TeX("Smallest singular value of $Q$"),
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title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
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subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim Exp(1)$")),
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colour=latex2exp::TeX("$a$"),
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shape=latex2exp::TeX("$\\alpha$"))
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```
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```{r k = n^alpha data generation, rate = 3}
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#| cache: true
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#| echo: false
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#| collapse: true
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ns <- seq(100, 1000, 100)
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as <- seq(0, 20, 2)
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alphas <- seq(0.1, 0.5, 0.1)
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set.seed(100)
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results <- data.frame(dim_n = integer(),
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dim_k = integer(),
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param_a = double(),
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param_alpha = double(),
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ssv = double())
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for (a in as) {
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for (i in 1:length(ns)) {
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for (j in 1:length(alphas)) {
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n <- ns[i]
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K <- floor(n^alphas[j])
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if (!K > 0) next # skip if K is equal to zero
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# use the default seed 1L
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Q <- compute_matrix(seed=1L,
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a= a,
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n = n,
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K = K,
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sample_X_fn = function(n) {matrix(rexp(n, rate=3), ncol = 1L)},
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fv = function(x) {dnorm(x, mean=0, sd=1)},
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Fv = function(x) {pnorm(x, mean=0, sd=1)},
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guard = 1e-12)
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ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
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current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
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results <- rbind(results, current_res)
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}
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}
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}
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```
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```{r k = n^alpha plotting, rate = 3}
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results |>
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filter(param_a %in% c(0, 10, 20)) |>
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mutate(param_a = as.factor(param_a),
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param_alpha = as.factor(param_alpha)) |>
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group_by(param_a, param_alpha) |>
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ggplot(aes(dim_n, ssv, col=param_a, shape=param_alpha, interaction(param_a, param_alpha))) +
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geom_point(size=1.5) +
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geom_line() +
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geom_function(fun = function(x) {sqrt(x)}, colour="black") +
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#scale_y_log10() +
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theme_bw() +
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labs(x=latex2exp::TeX("$n$"),
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y=latex2exp::TeX("Smallest singular value of $Q$"),
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title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
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subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim Exp(3))$")),
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colour=latex2exp::TeX("$a$"),
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shape=latex2exp::TeX("$\\alpha$"))
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```
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For $a = 0$ the smallest singular value is very close to zero.
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```{r k = n^alpha data generation, rate = 5}
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#| cache: true
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#| echo: false
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#| collapse: true
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ns <- seq(100, 1000, 100)
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as <- seq(0, 20, 2)
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alphas <- seq(0.1, 0.5, 0.1)
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set.seed(100)
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results <- data.frame(dim_n = integer(),
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dim_k = integer(),
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param_a = double(),
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param_alpha = double(),
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ssv = double())
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for (a in as) {
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for (i in 1:length(ns)) {
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for (j in 1:length(alphas)) {
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n <- ns[i]
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K <- floor(n^alphas[j])
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if (!K > 0) next # skip if K is equal to zero
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# use the default seed 1L
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Q <- compute_matrix(seed=1L,
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a= a,
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n = n,
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K = K,
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sample_X_fn = function(n) {matrix(rexp(n, rate=5), ncol = 1L)},
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fv = function(x) {dnorm(x, mean=0, sd=1)},
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Fv = function(x) {pnorm(x, mean=0, sd=1)},
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guard = 1e-12)
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ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
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current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
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results <- rbind(results, current_res)
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}
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}
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}
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```
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```{r k = n^alpha plotting, rate = 5}
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results |>
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filter(param_a %in% c(0, 10, 20)) |>
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mutate(param_a = as.factor(param_a),
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param_alpha = as.factor(param_alpha)) |>
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group_by(param_a, param_alpha) |>
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ggplot(aes(dim_n, ssv, col=param_a, shape=param_alpha, interaction(param_a, param_alpha))) +
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geom_point(size=1.5) +
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geom_line() +
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geom_function(fun = function(x) {sqrt(x)}, colour="black") +
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#scale_y_log10() +
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theme_bw() +
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labs(x=latex2exp::TeX("$n$"),
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y=latex2exp::TeX("Smallest singular value of $Q$"),
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title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
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subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim Exp(5))$")),
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colour=latex2exp::TeX("$a$"),
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shape=latex2exp::TeX("$\\alpha$"))
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```
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Why is here a perfect match for $\alpha = 0.1$ and $a = 20$ to the square function?
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The difference is of the order of $10^{-11}$!
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## Uniform distribution
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```{r k = n^alpha data generation, U[0,1]}
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#| cache: true
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#| echo: false
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#| collapse: true
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ns <- seq(100, 1000, 100)
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as <- seq(0, 20, 2)
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alphas <- seq(0.1, 0.5, 0.1)
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set.seed(100)
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results <- data.frame(dim_n = integer(),
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dim_k = integer(),
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param_a = double(),
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param_alpha = double(),
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ssv = double())
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for (a in as) {
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for (i in 1:length(ns)) {
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for (j in 1:length(alphas)) {
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n <- ns[i]
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K <- floor(n^alphas[j])
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if (!K > 0) next # skip if K is equal to zero
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# use the default seed 1L
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Q <- compute_matrix(seed=1L,
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a= a,
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n = n,
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K = K,
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sample_X_fn = function(n) {matrix(runif(n), ncol = 1L)},
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fv = function(x) {dnorm(x, mean=0, sd=1)},
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Fv = function(x) {pnorm(x, mean=0, sd=1)},
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guard = 1e-12)
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ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
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current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
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results <- rbind(results, current_res)
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}
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}
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}
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```
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```{r k = n^alpha plotting, U[0,1]}
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results |>
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filter(param_a %in% c(0, 10, 20)) |>
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mutate(param_a = as.factor(param_a),
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param_alpha = as.factor(param_alpha)) |>
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group_by(param_a, param_alpha) |>
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ggplot(aes(dim_n, ssv, col=param_a, shape=param_alpha, interaction(param_a, param_alpha))) +
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geom_point(size=1.5) +
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geom_line() +
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geom_function(fun = function(x) {sqrt(x)}, colour="black") +
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#scale_y_log10() +
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theme_bw() +
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labs(x=latex2exp::TeX("$n$"),
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y=latex2exp::TeX("Smallest singular value of $Q$"),
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title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
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subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim U[0,1] $")),
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colour=latex2exp::TeX("$a$"),
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shape=latex2exp::TeX("$\\alpha$"))
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```
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Here we have the same effect for $\alpha = 0.1$ and $a = 20$.
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```{r k = n^alpha data generation, U[0,2]}
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#| cache: true
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#| echo: false
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#| collapse: true
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ns <- seq(100, 1000, 100)
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as <- seq(0, 20, 2)
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alphas <- seq(0.1, 0.5, 0.1)
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set.seed(100)
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results <- data.frame(dim_n = integer(),
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dim_k = integer(),
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param_a = double(),
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param_alpha = double(),
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ssv = double())
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for (a in as) {
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for (i in 1:length(ns)) {
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for (j in 1:length(alphas)) {
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n <- ns[i]
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K <- floor(n^alphas[j])
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if (!K > 0) next # skip if K is equal to zero
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# use the default seed 1L
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Q <- compute_matrix(seed=1L,
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a= a,
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n = n,
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K = K,
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sample_X_fn = function(n) {matrix(runif(n, min = 0, max=2), ncol = 1L)},
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fv = function(x) {dnorm(x, mean=0, sd=1)},
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Fv = function(x) {pnorm(x, mean=0, sd=1)},
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guard = 1e-12)
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ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
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current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
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results <- rbind(results, current_res)
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}
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}
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}
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```
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```{r k = n^alpha plotting, U[0,2]}
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results |>
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filter(param_a %in% c(0, 10, 20)) |>
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mutate(param_a = as.factor(param_a),
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param_alpha = as.factor(param_alpha)) |>
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group_by(param_a, param_alpha) |>
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ggplot(aes(dim_n, ssv, col=param_a, shape=param_alpha, interaction(param_a, param_alpha))) +
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geom_point(size=1.5) +
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geom_line() +
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geom_function(fun = function(x) {sqrt(x)}, colour="black") +
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#scale_y_log10() +
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theme_bw() +
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labs(x=latex2exp::TeX("$n$"),
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y=latex2exp::TeX("Smallest singular value of $Q$"),
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title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
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subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$. Black line is $\\sqrt{n}$, and $X \\sim U[0,2] $")),
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colour=latex2exp::TeX("$a$"),
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shape=latex2exp::TeX("$\\alpha$"))
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```
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