244 lines
7.4 KiB
Plaintext
244 lines
7.4 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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# Plots of the dimensions
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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 $k$.
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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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## Hyperparameter with n vs. k = log(n)
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```{r n / log(n) hyperparameters data generation}
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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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Ks <- floor(log(ns))
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as <- seq(0, 20, 2)
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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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ssv = double())
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for (a in as) {
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for (i in 1:length(ns)) {
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n <- ns[i]
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K <- Ks[i]
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# use the default seed 1L
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out <- smallest_sv_sequence(
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a = a,
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n = n,
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maxK = K,
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sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
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guard=1e-12,
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plot=FALSE,
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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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)
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current_res <- data.frame(dim_n = rep(n, K), dim_k = out$K, param_a = rep(a, K), ssv = out$sv)
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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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```{r hyperparameter n vs log(n) plotting}
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results |>
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mutate(dim_n = as.factor(dim_n)) |>
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group_by(dim_n) |>
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filter(dim_k == max(dim_k) & param_a > 0) |>
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ggplot(aes(param_a, ssv, col=dim_n)) +
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geom_point(size=1.5) +
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geom_line() +
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#scale_y_log10() +
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theme_bw() +
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labs(x=latex2exp::TeX("$a$"),
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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 $k = \\lfloor\\log(n) \\rfloor$")),
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colour=latex2exp::TeX("$n$"),
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shape=latex2exp::TeX("$a$"))
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```
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Here we have relatively large values of the smallest singular value (ssv) of $Q$ since
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the values for $k$ are relatively small. We have already observed, that with
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larger $k$ the ssv shrinks rapidly towards zero. However the largest value for $k$
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is at most $k = `r floor(log(1000))`$ for $n = 1000$.
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We omitted the value $a = 0$, as this results in a ssv of $10^{-61}$.
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Note that this is only one sample and it could produce sighlty different results
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for other seeds.
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## Hyperparameter $n vs k = n^alpha$
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```{r n / log(n) hyperparameters data generation}
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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(rnorm(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 hyperparameter n / k^alpha = const plotting}
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results |>
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filter(dim_n %in% c(100, 500, 1000)) |>
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mutate(dim_n = as.factor(dim_n),
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param_alpha = as.factor(param_alpha)) |>
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group_by(dim_n, param_alpha) |>
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ggplot(aes(param_a, ssv, col=dim_n, shape=param_alpha)) +
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geom_point(size=1.5) +
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geom_line() +
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#scale_y_log10() +
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theme_bw() +
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labs(x=latex2exp::TeX("$a$"),
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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 $k = n^{\\alpha}$")),
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colour=latex2exp::TeX("$n$"),
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shape=latex2exp::TeX("$\\alpha$"))
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```
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Here we use $K = \lfloor n^\alpha\rfloor$ as hyperparameter. Why don't we see
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any change w.r.t. $a$ for $\alpha = 0.1$?
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## Two dimensional example
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Here consider $p = 2$ covariate variables for each node. In order to make the
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parameter $a$ invariant to the direction, we sample 5 different vectors for
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each $a$ and rescale them. We use $K = 5$ as hyperparameter.
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```{r data generation for two d example}
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#| cache: true
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#| echo: false
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#| collapse: true
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set.seed(10)
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ns <- seq(100, 1000, 100)
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as <- matrix(rnorm(8), ncol=4)
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as_norm <- seq(1, 20, 2)
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for (i in 1:ncol(as)){
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as[, i] <- as[, i] / sqrt(sum(as[, i]^2))
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}
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results <- data.frame(dim_n = integer(),
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dim_k = integer(),
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param_a = integer(),
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param_a_norm = double(),
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ssv = double())
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for (a_norm in as_norm) {
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for (i in 1:length(ns)) {
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for (j in 1:ncol(as)) {
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n <- ns[i]
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K <- 5 # floor(sqrt(n))
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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= as.vector(t(as[, j])),
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n = n,
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K = K,
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sample_X_fn = function(n) {matrix(rnorm(2 * n), ncol = 2L)},
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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 = j, param_a_norm=a_norm, 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 2d a parameter plotting}
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results |>
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filter(dim_n %in% c(100, 500, 1000)) |>
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mutate(dim_n = as.factor(dim_n),
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param_a = as.factor(param_a)) |>
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group_by(dim_n, param_a) |>
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ggplot(aes(param_a_norm, ssv, col=dim_n, shape=param_a, interaction(dim_n, param_a))) +
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geom_point(size=1.5) +
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geom_line() +
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#scale_y_log10() +
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theme_bw() +
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labs(x=latex2exp::TeX("$\\|a\\|$"),
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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 the norm of $a$."),
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subtitle = latex2exp::TeX(("Hyperparameter $k = 5$")),
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colour=latex2exp::TeX("$n$"),
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shape=latex2exp::TeX("$\\|\\alpha\\|$"))
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```
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```{r plot the vectors}
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plot(0, 0, xlim=c(-1.5, 1.5), ylim=c(-1.5, 1.5), type="n", xlab="x", ylab="y", asp=1)
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arrows(0, 0, as[1, ], as[2, ], col=1:4, lwd=2)
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title(main="Vectors for a rescaled to norm one.")
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```
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It seems, that the direction of the parameter $a$ has a small influence in the
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smallest singular value of the matrix $Q$, but not the norm of it??
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