Use GeodesicInterpolationWithinRadius for Circle mean

Project.toml

@@ -1,7 +1,7 @@
 name = "Manifolds"
 uuid = "1cead3c2-87b3-11e9-0ccd-23c62b72b94e"
 authors = ["Seth Axen <[email protected]>", "Mateusz Baran <[email protected]>", "Ronny Bergmann <[email protected]>", "Antoine Levitt <[email protected]>"]
-version = "0.8.57"
+version = "0.8.58"
 
 [deps]
 Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"

src/manifolds/Circle.jl

@@ -317,61 +317,22 @@ i.e. $\dim(𝕊^1) = 1$.
 """
 manifold_dimension(::Circle) = 1
 
-@doc raw"""
-    mean(M::Circle{ℝ}, x::AbstractVector[, w::AbstractWeights])
-
-Compute the Riemannian [`mean`](@ref mean(M::AbstractManifold, args...)) of `x` of points on
-the [`Circle`](@ref) $𝕊^1$, reprsented by real numbers, i.e. the angular mean
-````math
-\operatorname{atan}\Bigl( \sum_{i=1}^n w_i\sin(x_i),  \sum_{i=1}^n w_i\sin(x_i) \Bigr).
-````
 """
-mean(::Circle{ℝ}, ::Any)
-function Statistics.mean(::Circle{ℝ}, x::AbstractVector{<:Real}; kwargs...)
-    return atan(1 / length(x) * sum(sin, x), 1 / length(x) * sum(cos, x))
-end
-function Statistics.mean(
-    ::Circle{ℝ},
-    x::AbstractVector{<:Real},
-    w::AbstractVector;
-    kwargs...,
-)
-    return atan(sum(w .* sin.(x)), sum(w .* cos.(x)))
-end
-@doc raw"""
-    mean(M::Circle{ℂ}, x::AbstractVector[, w::AbstractWeights])
-
-Compute the Riemannian [`mean`](@ref mean(M::AbstractManifold, args...)) of `x` of points on
-the [`Circle`](@ref) $𝕊^1$, reprsented by complex numbers, i.e. embedded in the complex plane.
-Comuting the sum
-````math
-s = \sum_{i=1}^n x_i
-````
-the mean is the angle of the complex number $s$, so represented in the complex plane as
-$\frac{s}{\lvert s \rvert}$, whenever $s \neq 0$.
+    mean(
+        M::Circle,
+        x::AbstractVector,
+        [w::AbstractWeights,]
+        method = GeodesicInterpolationWithinRadius(π/2);
+        kwargs...,
+    )
 
-If the sum $s=0$, the mean is not unique. For example for opposite points or equally spaced
-angles.
+Compute the Riemannian [`mean`](@ref mean(M::AbstractManifold, args...)) of points in vector
+`x` using [`GeodesicInterpolationWithinRadius`](@ref).
 """
-mean(::Circle{ℂ}, ::Any)
-function Statistics.mean(M::Circle{ℂ}, x::AbstractVector{<:Complex}; kwargs...)
-    s = sum(x)
-    abs(s) == 0 && return error(
-        "The mean for $(x) on $(M) is not defined/unique, since the sum of the complex numbers is zero",
-    )
-    return s / abs(s)
-end
-function Statistics.mean(
-    M::Circle{ℂ},
-    x::AbstractVector{<:Complex},
-    w::AbstractVector;
-    kwargs...,
-)
-    s = sum(w .* x)
-    abs(s) == 0 && error(
-        "The mean for $(x) on $(M) is not defined/unique, since the sum of the complex numbers is zero",
-    )
-    return s /= abs(s)
+mean(::Circle, ::Any...)
+
+function default_estimation_method(::Circle, ::typeof(mean))
+    return GeodesicInterpolationWithinRadius(π / 2)
 end
 
 mid_point(M::Circle{ℝ}, p1, p2) = exp(M, p1, 0.5 * log(M, p1, p2))

src/statistics.jl

@@ -71,8 +71,8 @@ points are in an open geodesic ball about the mean with corresponding radius
     + [`Euclidean`](@ref)
     + [`SymmetricPositiveDefinite`](@ref) [^Ho2013]
 * Other manifolds:
-    + [`Sphere`](@ref): $\frac{π}{2}$ [^Salehian2015]
-    + [`Grassmann`](@ref): $\frac{π}{4}$ [^Chakraborty2015]
+    + [`Sphere`](@ref)/[`Circle`](@ref): $\frac{π}{2}$ [^Salehian2015]
+    + [`Grassmann`](@ref)/[`ProjectiveSpace`](@ref): $\frac{π}{4}$ [^Chakraborty2015]
     + [`Stiefel`](@ref)/[`Rotations`](@ref): $\frac{π}{2 \sqrt 2}$ [^Chakraborty2019]
 
 For online variance computation, the algorithm additionally uses an analogous