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Introduce r-norms on manifolds with components #206

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Oct 19, 2024
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61e6443
On manifolds, that consist of components, allow to (outer) r-norms.
kellertuer Oct 18, 2024
f229843
Resolve an ambiguity.
kellertuer Oct 18, 2024
77d5eb0
NEWS.md and Project.TOML.
kellertuer Oct 18, 2024
3f34424
adapt `norm(M, p, X)` as well to have an `r=2.0` for product and Defa…
kellertuer Oct 18, 2024
204dceb
Fix a final test for today.
kellertuer Oct 18, 2024
b64d5e5
runs formatter.
kellertuer Oct 18, 2024
d29bfba
Finish code features with the norm on pwer manifold to also have an r.
kellertuer Oct 18, 2024
ffafbeb
Finish norms.
kellertuer Oct 18, 2024
cab6856
Refactor power to be loadable more than once, extend code coverage.
kellertuer Oct 19, 2024
6765f14
Apply suggestions from code review
kellertuer Oct 19, 2024
151fbaf
further checks.
kellertuer Oct 19, 2024
afac395
Merge branch 'kellertuer/product_norms' of github.com:JuliaManifolds/…
kellertuer Oct 19, 2024
5258cf0
Maybe run formatter?
kellertuer Oct 19, 2024
e1b6059
add and test also -inf case.
kellertuer Oct 19, 2024
4947ba3
Fix a typo in code.
kellertuer Oct 19, 2024
7950513
Inverse Product Retractions.
kellertuer Oct 19, 2024
47ad4d3
Apply suggestions from code review
kellertuer Oct 19, 2024
aed8ed4
Rephrase docs.
kellertuer Oct 19, 2024
470e4a7
fix syntax errors that appeared in accepting changes.
kellertuer Oct 19, 2024
5637e7b
runs formatter.
kellertuer Oct 19, 2024
b6c48f1
Add yet another corner case.
kellertuer Oct 19, 2024
2489652
Maybe now?
kellertuer Oct 19, 2024
53a7ddf
improve style.
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008cc54
Apply suggestions from code review
kellertuer Oct 19, 2024
e7b3190
reactivate a few tests.
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577a5ad
Fix a few tests.
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9 changes: 9 additions & 0 deletions NEWS.md
Original file line number Diff line number Diff line change
Expand Up @@ -5,6 +5,15 @@ 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).

## [0.15.18] 18/10/2024

### Added

* `distance(M, p, q, r)` to compute `r`-norms on manifolds that have components.
* `distance(M, p, q, m, r)` to compute (approximate) `r`-norms on manifolds that have components
using an `AbstractInverseRetractionMethod m` within every (inner) distance call.
* `norm(M, p, X, r)` to compute `r`-norms on manifolds that have components.

## [0.15.17] 04/10/2024

### Changed
Expand Down
2 changes: 1 addition & 1 deletion Project.toml
Original file line number Diff line number Diff line change
@@ -1,7 +1,7 @@
name = "ManifoldsBase"
uuid = "3362f125-f0bb-47a3-aa74-596ffd7ef2fb"
authors = ["Seth Axen <[email protected]>", "Mateusz Baran <[email protected]>", "Ronny Bergmann <[email protected]>", "Antoine Levitt <[email protected]>"]
version = "0.15.17"
version = "0.15.18"

[deps]
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Expand Down
6 changes: 4 additions & 2 deletions src/DefaultManifold.jl
Original file line number Diff line number Diff line change
Expand Up @@ -57,7 +57,7 @@ function check_approx(M::DefaultManifold, p, X, Y; kwargs...)
return ApproximatelyError(v, s)
end

distance(::DefaultManifold, p, q) = norm(p - q)
distance(::DefaultManifold, p, q, r::Real = 2) = norm(p - q, r)

embed!(::DefaultManifold, q, p) = copyto!(q, p)

Expand Down Expand Up @@ -123,6 +123,8 @@ function get_vector_orthonormal!(M::DefaultManifold{ℂ}, Y, p, c, ::RealNumbers
return copyto!(Y, reshape(c[1:n] + c[(n + 1):(2n)] * 1im, representation_size(M)))
end

has_components(::DefaultManifold) = true

injectivity_radius(::DefaultManifold) = Inf

@inline inner(::DefaultManifold, p, X, Y) = dot(X, Y)
Expand All @@ -138,7 +140,7 @@ end

number_system(::DefaultManifold{𝔽}) where {𝔽} = 𝔽

norm(::DefaultManifold, p, X) = norm(X)
norm(::DefaultManifold, p, X, r::Real = 2) = norm(X, r)

project!(::DefaultManifold, q, p) = copyto!(q, p)
project!(::DefaultManifold, Y, p, X) = copyto!(Y, X)
Expand Down
10 changes: 10 additions & 0 deletions src/ManifoldsBase.jl
Original file line number Diff line number Diff line change
Expand Up @@ -594,6 +594,15 @@ function embed_project!(M::AbstractManifold, Y, p, X)
return project!(M, Y, p, embed(M, p, X))
end

"""
has_components(M::AbstractManifold)

Return whether the [`AbstractManifold`](@ref)`(M)` consists of components,
like the [`PowerManifold`](@ref) or the [`ProductManifold`](@ref), that one can iterate over.
By default, this function returns `false`.
"""
has_components(M::AbstractManifold) = false

@doc raw"""
injectivity_radius(M::AbstractManifold)

Expand Down Expand Up @@ -1354,6 +1363,7 @@ export ×,
get_vector,
get_vector!,
get_vectors,
has_components,
hat,
hat!,
injectivity_radius,
Expand Down
182 changes: 165 additions & 17 deletions src/PowerManifold.jl
Original file line number Diff line number Diff line change
Expand Up @@ -530,22 +530,132 @@ function default_vector_transport_method(M::PowerManifold, t::Type)
return default_vector_transport_method(M.manifold, eltype(t))
end

@doc raw"""
distance(M::AbstractPowerManifold, p, q)
_doc_distance_pow = """
distance(M::AbstractPowerManifold, p, q, r::Real=2)
distance(M::AbstractPowerManifold, p, q, m::AbstractInverseRetractionMethod=LogarithmicInverseRetraction(), r::Real=2)

Compute the distance between `q` and `p` on an [`AbstractPowerManifold`](@ref).

First, the componentwise distances are computed using the Riemannian distance function
on `M.manifold`. These can be approximated using the
`norm` of an [`AbstractInverseRetractionMethod`](@ref) `m`.
This yields an array of distance values.

Compute the distance between `q` and `p` on an [`AbstractPowerManifold`](@ref),
i.e. from the element wise distances the Forbenius norm is computed.
Second, we compute the `r`-norm on this array of distances.
This is also the only place, there the `r` is used.
"""

function distance(M::AbstractPowerManifold, p, q)
sum_squares = zero(number_eltype(p))
return _distance_r(M, p, q, 2)
end

@doc "$(_doc_distance_pow)"
function distance(M::AbstractPowerManifold, p, q, r::Real)
(isinf(r) && r > 0) && return _distance_max(M, p, q)
(isinf(r) && r < 0) && return _distance_min(M, p, q)
(r == 1) && return _distance_1(M, p, q)
return _distance_r(M, p, q, r)
end
function distance(
M::AbstractPowerManifold,
p,
q,
::LogarithmicInverseRetraction,
r::Real = 2,
)
return distance(M, p, q, r)
end

@doc "$(_doc_distance_pow)"
function distance(
M::AbstractPowerManifold,
p,
q,
m::AbstractInverseRetractionMethod,
r::Real = 2,
)
(isinf(r) && r > 0) && return _distance_max(M, p, q, m)
(isinf(r) && r < 0) && return _distance_min(M, p, q, m)
(r == 1) && return _distance_1(M, p, q, m)
return _distance_r(M, p, q, m, r)
end
#
#
# The three special cases
function _distance_r(
M::AbstractPowerManifold,
p,
q,
m::AbstractInverseRetractionMethod,
r::Real,
)
rep_size = representation_size(M.manifold)
values = [
distance(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, q, i), m) for
i in get_iterator(M)
]
return norm(values, r)
end
function _distance_r(M::AbstractPowerManifold, p, q, r::Real)
rep_size = representation_size(M.manifold)
values = [
distance(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, q, i)) for
i in get_iterator(M)
]
return norm(values, r)
end
function _distance_1(M::AbstractPowerManifold, p, q, m::AbstractInverseRetractionMethod)
s = zero(number_eltype(p))
rep_size = representation_size(M.manifold)
for i in get_iterator(M)
sum_squares +=
distance(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, q, i))^2
s += distance(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, q, i), m)
end
return sqrt(sum_squares)
return s
end
function _distance_1(M::AbstractPowerManifold, p, q)
s = zero(number_eltype(p))
rep_size = representation_size(M.manifold)
for i in get_iterator(M)
s += distance(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, q, i))
end
return s
end
function _distance_max(M::AbstractPowerManifold, p, q, m::AbstractInverseRetractionMethod)
d = float(zero(number_eltype(p)))
rep_size = representation_size(M.manifold)
for i in get_iterator(M)
v = distance(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, q, i), m)
d = (v > d) ? v : d
end
return d
end
function _distance_max(M::AbstractPowerManifold, p, q)
d = float(zero(number_eltype(p)))
rep_size = representation_size(M.manifold)
for i in get_iterator(M)
v = distance(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, q, i))
d = (v > d) ? v : d
end
return d
end
function _distance_min(M::AbstractPowerManifold, p, q, m::AbstractInverseRetractionMethod)
d = Inf
rep_size = representation_size(M.manifold)
for i in get_iterator(M)
v = distance(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, q, i), m)
d = (v < d) ? v : d
end
return d
end
function _distance_min(M::AbstractPowerManifold, p, q)
d = Inf
rep_size = representation_size(M.manifold)
for i in get_iterator(M)
v = distance(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, q, i))
d = (v < d) ? v : d
end
return d
end

@doc raw"""
exp(M::AbstractPowerManifold, p, X)

Expand Down Expand Up @@ -881,6 +991,13 @@ function Base.getindex(
return TangentSpace(M.manifold, p[M, I...])
end

"""
has_components(::AbstractPowerManifold)

Return `true, since points on an [`AbstractPowerManifold`](@ref) consist of components.
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"""
has_components(::AbstractPowerManifold) = true

@doc raw"""
injectivity_radius(M::AbstractPowerManifold[, p])

Expand Down Expand Up @@ -1092,20 +1209,51 @@ function mid_point!(M::PowerManifoldNestedReplacing, q, p1, p2)
end

@doc raw"""
norm(M::AbstractPowerManifold, p, X)
norm(M::AbstractPowerManifold, p, X, r::Real=2)

Compute the norm of `X` from the tangent space of `p` on an
[`AbstractPowerManifold`](@ref) `M`, i.e. from the element wise norms the
Frobenius norm is computed.
[`AbstractPowerManifold`](@ref) `M`, i.e. from the element wise norms `r`-norm is computed,
where the default `r=2.0` yields the Frobenius norm is computed.
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"""
function LinearAlgebra.norm(M::AbstractPowerManifold, p, X)
sum_squares = zero(number_eltype(X))
function LinearAlgebra.norm(M::AbstractPowerManifold, p, X, r::Real = 2)
(isinf(r) && r > 0) && return _norm_max(M, p, X)
(isinf(r) && r < 0) && return _norm_min(M, p, X)
(r == 1) && return _norm_1(M, p, X)
return _norm_r(M, p, X, r)
end
function _norm_r(M::AbstractPowerManifold, p, X, r::Real)
rep_size = representation_size(M.manifold)
values = [
norm(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, X, i)) for
i in get_iterator(M)
]
return norm(values, r)
end
function _norm_1(M::AbstractPowerManifold, p, X)
s = zero(number_eltype(p))
rep_size = representation_size(M.manifold)
for i in get_iterator(M)
s += norm(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, X, i))
end
return s
end
function _norm_max(M::AbstractPowerManifold, p, X)
d = 0.0
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rep_size = representation_size(M.manifold)
for i in get_iterator(M)
v = norm(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, X, i))
d = (v > d) ? v : d
end
return d
end
function _norm_min(M::AbstractPowerManifold, p, X)
d = Inf
rep_size = representation_size(M.manifold)
for i in get_iterator(M)
sum_squares +=
norm(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, X, i))^2
v = norm(M.manifold, _read(M, rep_size, p, i), _read(M, rep_size, X, i))
d = (v < d) ? v : d
end
return sqrt(sum_squares)
return d
end


Expand Down
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