added plot with alpha dependence

scripts/plots_a_dependence.qmd

@@ -241,4 +241,73 @@ title(main="Vectors for a rescaled to norm one.")
 
  It seems, that the direction of the parameter $a$ has a small influence in the 
  smallest singular value of the matrix $Q$, but not the norm of it??
+ 
+ 
+ ## Michael's Plot
+ Here we plot for each $n$ the smallest singular value of $Q$. We choose the
+ parameter $K$ as
+ $$
+ K = \lfloor n^\alpha \rfloor
+ $$
+ with $\alpha \in \{0.1, 0.2, 0.3, 0.4, 0.5\}$.
+ 
+ 
+```{r Plot for different alphas}
+#| cache: true
+#| echo: false
+#| collapse: true
+ns <- seq(100, 1000, 100)
+as <- c(1.0, 2, 5, 10, 20)
+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(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)
+      results <- rbind(results, current_res)
+    }
+  }
+}
+```
+
+```{r hyperparameter n / k^alpha = const plotting}
+results |>
+  mutate(param_alpha = as.factor(param_alpha),
+         param_a = as.factor(param_a)) |>
+  group_by(param_a, param_alpha) |>
+  filter(dim_k == max(dim_k)) |>
+  ggplot(aes(dim_n, ssv, col=param_a, shape=param_alpha)) +
+  geom_point(size=1.5) +
+  geom_line() +
+  #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 $\\alpha$."),
+       subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$")),
+       colour=latex2exp::TeX("$n$"),
+       shape=latex2exp::TeX("$\\alpha$"))
+```
+