Add dimensions plot

.gitignore

@@ -52,4 +52,4 @@ rsconnect/
 
 /.quarto/
 **/*.quarto_ipynb
-./_freeze/
+_freeze/

_quarto.yml

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+project:
+  type: default
+
+execute:
+  freeze: auto
+  cache: false

scripts/plots_dimensions.qmd

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+---
+title: "Plots of n vs. k"
+author: "Niclas"
+format: html
+editor: visual
+execute:
+  echo: true
+  working-directory: ../
+---
+
+# Plots of the dimensions
+
+## 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
+$$
+p_a(u|X) = \frac{f_v(F_a^{-1}(u) - a^\top X)}{f_a(F_a^{-1}(u))}
+$$
+
+## Plots of n vs. k
+
+- The $v$'s are normally distributed with $v \sim \mathcal N(0,1)$
+- Plot $n = 100, 200, 300, 400$ and $k = 1, \dots, K$ with $K = \sqrt n$.
+
+```{r Load Libraries}
+# 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)
+```
+
+```{r Compute the data}
+#| cache: true
+#| echo: false
+#| collapse: true
+ns <- c(100, 200, 300, 400, 500)
+Ks <- floor(sqrt(ns))
+as <- c(0.5, 1.0, 1.5, 2.0)
+
+# set a global seed
+set.seed(42)
+results <- data.frame(dim_n = integer(),
+                      dim_k = integer(),
+                      param_a = double(),
+                      ssv = double())
+for (a in as) {
+  for (i in 1:length(ns)) {
+    n <- ns[i]
+    K <- Ks[i]
+    # use the default seed 1L
+    out <- smallest_sv_sequence(
+      a = a,
+      n = n,
+      maxK = K,
+      sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
+      guard=1e-12,
+      plot=FALSE,
+      fv = function(x) {dnorm(x, mean=0, sd=1)},
+      Fv = function(x) {pnorm(x, mean=0, sd=1)}
+      )
+    
+    current_res <- data.frame(dim_n = rep(n, K), dim_k = out$K, param_a = rep(a, K), ssv = out$sv)
+    results <- rbind(results, current_res)
+  }
+}
+
+```
+
+```{r plot the results}
+#| cache: true
+#| echo: false
+#| collapse: true
+#| fig-cap: "Simulation of the smallest singular values w.r.t. a, n and k"
+results |>
+  mutate(param_a =  as.factor(param_a),
+         dim_n = as.factor(dim_n)) |>
+  group_by(param_a, dim_n) |>
+  ggplot(aes(dim_k, ssv, col=dim_n, shape=param_a, interaction(dim_n, param_a))) +
+  geom_point(size=1.5) +
+  geom_line() +
+  scale_y_log10() +
+  theme_bw() +
+  labs(x=latex2exp::TeX("$k$"),
+       y=latex2exp::TeX("Smallest singular value of $Q$"),
+       title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $n$, $k$, and $a$."),
+       colour=latex2exp::TeX("$n$"),
+       shape=latex2exp::TeX("$a$"))
+```
+The data for $n = 100$ is covered by the data for $n = 200$.
+
+## Analysis of the convergence
+
+We assume that the smallest singular value $\sigma$ can be approximated by:
+$$
+\sigma = C \cdot n^\eta \cdot k^\kappa \cdot a^\alpha
+$$
+to estimate the coefficients we make a log-transform and perform a linear regression, i.e.
+$$
+\log(\sigma) = \log (C) + \eta \log(n) + \kappa\log(k) + \alpha \log(a).
+$$
+
+```{r estimate coeffs}
+model1 <- results |>
+  filter(ssv > 1e-15) |> # exclude to small values
+  lm(formula = log(ssv) ~ log(dim_n) + log(dim_k) + log(param_a))
+
+summary(model1)
+plot(model1)
+  
+```
+