A few edits to the notebooks (#19)

* Run formatter

* Fix some old typos

---------

Co-authored-by: Ronny Bergmann <git@ronnybergmann.net>

docs/src/examples/H2-Signal-TV.md

@@ -100,10 +100,6 @@ function artificial_H2_signal(
             )
         end
     end
-    #! In order to have length(data) ∝ pts, we need typeof(l) == Int and mod(pts, l) == 0.
-    if pts != length(data)
-        @warn "The length of the output signal will differ from the input number of points."
-    end
     return data, geodesics
 end
 function matrixify_Poincare_ball(input)

docs/src/examples/RCBM-Median.md

@@ -425,6 +425,8 @@ We can take a look at how the algorithms compare to each other in their performa
 
 ## The Median on the Sphere
 
+For the last experiment, note that a major difference here is that the sphere has constant positive sectional curvature equal to $1$. In this case, we lose the global convexity of the Riemannian distance and thus of the objective. Minimizers still exist, but they may, in general, be non-unique.
+
 ``` julia
 subexperiment_name = "Sn"
 k_max_sn = 1.0

docs/src/examples/Spectral-Procrustes.md

@@ -21,7 +21,7 @@ using ManifoldDiff, Manifolds, Manopt, ManoptExamples
 
 ## The Problem
 
-Given two matrices $A, B \in \mathbb R^{n \times d}$ we aim to solve the orthogonal Procrustes problem
+Given two matrices $A, B \in \mathbb R^{n \times d}$ we aim to solve the Procrustes problem
 
 ``` math
     {\arg\min}_{p \in \mathrm{SO}(d)}\ \Vert A - B \, p \Vert_2
@@ -31,22 +31,22 @@ Given two matrices $A, B \in \mathbb R^{n \times d}$ we aim to solve the orthogo
 where $\mathrm{SO}(d)$ is equipped with the standard bi-invariant metric, and where $\Vert \,\cdot\, \Vert_2$ denotes the spectral norm of a matrix, , its largest singular value.
 We aim to find the best matrix $p \in \mathbb R^{d \times d}$ such that $p^\top p = \mathrm{id}$ is the identity matrix, or in other words $p$ is the best rotation.
 Note that the spectral norm is convex in the Euclidean sense, but not geodesically convex on $\mathrm{SO}(d)$.
-If we define the objective as
+Let us define the objective as
 
 ``` math
     f (p)
     =
     \Vert A - B \, p \Vert_2
-    ,
+    .
 ```
 
-its subdifferential is given by
+To obtain subdifferential information, we use
 
 ``` math
-    \partial f(p) = \mathrm{proj}_p(-B^\top UV^\top)
+    \mathrm{proj}_p(-B^\top UV^\top)
 ```
 
-where $U$ and $V$ are some left and right singular vectors, respectively, corresponding to the largest singular value of $A - B \, p$, and $\mathrm{proj}_p$ is the projection onto
+as a substitute for $\partial f(p)$, where $U$ and $V$ are some left and right singular vectors, respectively, corresponding to the largest singular value of $A - B \, p$, and $\mathrm{proj}_p$ is the projection onto
 
 ``` math
     \mathcal T_p \mathrm{SO}(d)

examples/H2-Signal-TV.qmd

@@ -123,10 +123,6 @@ function artificial_H2_signal(
             )
         end
     end
-    #! In order to have length(data) ∝ pts, we need typeof(l) == Int and mod(pts, l) == 0.
-    if pts != length(data)
-        @warn "The length of the output signal will differ from the input number of points."
-    end
     return data, geodesics
 end
 function matrixify_Poincare_ball(input)

examples/RCBM-Median.qmd

@@ -430,6 +430,9 @@ benchmarking && pretty_table(A2_SPD, tf = tf_markdown, header=col_names_2)
 ```
 
 ## The Median on the Sphere
+
+For the last experiment, note that a major difference here is that the sphere has constant positive sectional curvature equal to $1$. In this case, we lose the global convexity of the Riemannian distance and thus of the objective. Minimizers still exist, but they may, in general, be non-unique.
+
 ```{julia}
 #| output: false
 subexperiment_name = "Sn"

examples/Spectral-Procrustes.qmd

@@ -44,7 +44,7 @@ using ManifoldDiff, Manifolds, Manopt, ManoptExamples
 
 ## The Problem
 
-Given two matrices $A, B \in \mathbb R^{n \times d}$ we aim to solve the orthogonal Procrustes problem
+Given two matrices $A, B \in \mathbb R^{n \times d}$ we aim to solve the Procrustes problem
 ```math
 	{\arg\min}_{p \in \mathrm{SO}(d)}\ \Vert A - B \, p \Vert_2
 	,
@@ -53,18 +53,18 @@ Given two matrices $A, B \in \mathbb R^{n \times d}$ we aim to solve the orthogo
 where $\mathrm{SO}(d)$ is equipped with the standard bi-invariant metric, and where $\Vert \,\cdot\, \Vert_2$ denotes the spectral norm of a matrix, \ie, its largest singular value.
 We aim to find the best matrix $p \in \mathbb R^{d \times d}$ such that $p^\top p = \mathrm{id}$ is the identity matrix, or in other words $p$ is the best rotation.
 Note that the spectral norm is convex in the Euclidean sense, but not geodesically convex on $\mathrm{SO}(d)$.
-If we define the objective as
+Let us define the objective as
 ```math
     f (p)
 	=
     \Vert A - B \, p \Vert_2
-    ,
+    .
 ```
-its subdifferential is given by
+To obtain subdifferential information, we use
 ```math
-    \partial f(p) = \mathrm{proj}_p(-B^\top UV^\top)
+    \mathrm{proj}_p(-B^\top UV^\top)
 ```
-where $U$ and $V$ are some left and right singular vectors, respectively, corresponding to the largest singular value of $A - B \, p$, and $\mathrm{proj}_p$ is the projection onto
+as a substitute for $\partial f(p)$, where $U$ and $V$ are some left and right singular vectors, respectively, corresponding to the largest singular value of $A - B \, p$, and $\mathrm{proj}_p$ is the projection onto
 ```math
 	\mathcal T_p \mathrm{SO}(d)
 	=