Fix apply_diff_group for some actions

NEWS.md

@@ -5,6 +5,16 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
+## [0.9.3] - 2023-10-xx
+
+### Added
+
+- Functions `inv_diff` and `inv_diff!` that correspond to differentials of group inversion.
+
+### Fixed
+
+- Fixed issue with incorrect implementation of `apply_diff_group` in `GroupOperationAction` with left backward and right forward action [#669](https://github.com/JuliaManifolds/Manifolds.jl/issues/669).
+
 ## [0.9.2] - 2023-10-27
 
 ### Added

src/Manifolds.jl

@@ -989,6 +989,8 @@ export adjoint_action,
     identity_element!,
     inv,
     inv!,
+    inv_diff,
+    inv_diff!,
     inverse_apply,
     inverse_apply!,
     inverse_apply_diff,

src/groups/addition_operation.jl

@@ -55,6 +55,20 @@ function inv!(
     return q
 end
 
+"""
+    inv_diff(::AdditionGroupTrait, G::AbstractDecoratorManifold, p, X)
+
+Compute the value of differential of additive matrix inversion ``p ↦ -p`` at ``X``, i.e. ``-X``.
+"""
+function inv_diff(::AdditionGroupTrait, G::AbstractDecoratorManifold, p, X)
+    return -X
+end
+function inv_diff!(::AdditionGroupTrait, G::AbstractDecoratorManifold, Y, p, X)
+    Y .= X
+    Y .*= -1
+    return Y
+end
+
 function is_identity(::AdditionGroupTrait, G::AbstractDecoratorManifold, q; kwargs...)
     return isapprox(G, q, zero(q); kwargs...)
 end

src/groups/group.jl

@@ -529,6 +529,22 @@ function inv!(
     return e
 end
 
+@doc raw"""
+    inv_diff(G::AbstractDecoratorManifold, p, X)
+
+Compute the value of differential of inverse ``p^{-1} ∈ \mathcal{G}`` of an element
+``p ∈ \mathcal{G}`` at tangent vector `X` at `p`. The result is a tangent vector at ``p^{-1}``.
+"""
+inv_diff(G::AbstractDecoratorManifold, p)
+
+@trait_function inv_diff(G::AbstractDecoratorManifold, p, X)
+function inv_diff(::TraitList{<:IsGroupManifold}, G::AbstractDecoratorManifold, p, X)
+    Y = allocate_result(G, inv_diff, X, p)
+    return inv_diff!(G, Y, p, X)
+end
+
+@trait_function inv_diff!(G::AbstractDecoratorManifold, Y, p, X)
+
 function Base.copyto!(
     ::TraitList{IsGroupManifold{O}},
     ::AbstractDecoratorManifold,
@@ -902,7 +918,7 @@ function inverse_translate_diff(
     conv::ActionDirectionAndSide,
 )
     BG = base_group(G)
-    return translate_diff(BG, inv(BG, p), q, X, conv)
+    return translate_diff(BG, inv(BG, p), q, inv_diff(BG, p, X), conv)
 end
 
 @trait_function inverse_translate_diff!(
@@ -923,7 +939,7 @@ function inverse_translate_diff!(
     conv::ActionDirectionAndSide,
 )
     BG = base_group(G)
-    return translate_diff!(BG, Y, inv(BG, p), q, X, conv)
+    return translate_diff!(BG, Y, inv(BG, p), q, inv_diff(BG, p, X), conv)
 end
 
 @doc raw"""

src/groups/group_operation_action.jl

@@ -97,11 +97,34 @@ end
 
 function apply_diff_group(A::GroupOperationAction, a, X, p)
     G = base_group(A)
-    return translate_diff(G, p, a, X, reverse_direction_and_side(A))
+    if direction_and_side(A) === LeftForwardAction() ||
+       direction_and_side(A) === RightBackwardAction()
+        return translate_diff(G, p, a, X, reverse_direction_and_side(A))
+    else
+        return translate_diff(
+            G,
+            p,
+            a,
+            inv_diff(G, a, X),
+            (direction(A), switch_side(action_side(A))),
+        )
+    end
 end
 function apply_diff_group!(A::GroupOperationAction, Y, a, X, p)
     G = base_group(A)
-    return translate_diff!(G, Y, p, a, X, reverse_direction_and_side(A))
+    if direction_and_side(A) === LeftForwardAction() ||
+       direction_and_side(A) === RightBackwardAction()
+        return translate_diff!(G, Y, p, a, X, reverse_direction_and_side(A))
+    else
+        return translate_diff!(
+            G,
+            Y,
+            p,
+            a,
+            inv_diff(G, a, X),
+            (direction(A), switch_side(action_side(A))),
+        )
+    end
 end
 
 function inverse_apply_diff(A::GroupOperationAction, a, p, X)

src/groups/multiplication_operation.jl

@@ -124,6 +124,24 @@ function inv!(
     return q
 end
 
+"""
+    inv_diff(::MultiplicationGroupTrait, G::AbstractDecoratorManifold, p, X)
+
+Compute the value of differential of matrix inversion ``p ↦ p^{-1}`` at ``X``.
+The formula reads ``-p^{-1}Xp^{-1}``.
+"""
+function inv_diff(::MultiplicationGroupTrait, G::AbstractDecoratorManifold, p, X)
+    p_inv = inv(p)
+    return -p_inv * X * p_inv
+end
+function inv_diff!(::MultiplicationGroupTrait, G::AbstractDecoratorManifold, Y, p, X)
+    p_inv = inv(p)
+    Z = X * p_inv
+    mul!(Y, p_inv, Z)
+    Y .*= -1
+    return Y
+end
+
 compose(::MultiplicationGroupTrait, G::AbstractDecoratorManifold, p, q) = p * q
 
 function compose!(::MultiplicationGroupTrait, G::AbstractDecoratorManifold, x, p, q)