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A Julia Notebook on a diffferential
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### A Pluto.jl notebook ### | |
# v0.18.0 | |
using Markdown | |
using InteractiveUtils | |
# ╔═╡ b8662ee7-3af9-4703-978f-6ce42408b581 | |
using Manifolds, Manopt, Plots, Random | |
# ╔═╡ a474d350-9689-11ec-2178-65e3d33c9f05 | |
md""" | |
# A small note on a certain differential | |
""" | |
# ╔═╡ d3f21250-521d-4438-a774-3fcd09fd315f | |
Random.seed!(42) | |
# ╔═╡ 610a9702-524e-4294-957c-fc601f7061d1 | |
md""" | |
Let ``\mathcal M`` be a manifold, ``p,q \in \mathcal M`` and ``t \in \mathbb R`` be given. | |
We consider the function $f \colon T_p\mathcal M \to \mathbb R$ | |
```math | |
f(X) = \frac{1}{2}d_{\mathcal M}^2(q, \exp_p(tX)),\qquad X \in T_p\mathcal M | |
``` | |
Hence the differential is a map from ``T_p\mathcal M`` to ``\mathbb R`` since both domain and codomain are their own tangent spaces. | |
""" | |
# ╔═╡ 35cdd1a3-ba0f-47e3-83b3-2258a480a957 | |
md""" | |
If we use this within minimisation, we can even specify the minmiser ``X^* = \frac{1}{t}\log_pq`` aince then the distance is zero. | |
""" | |
# ╔═╡ 996217a9-83db-47f9-b6d5-f28eedbf6e67 | |
md""" | |
We decompose ``f`` into the functions | |
```math | |
\begin{align} | |
g(s) &= \frac{1}{2}d_{\mathcal M}(q, s), \qquad &D_s g(s)[X] &= \langle -\log_sq, X \rangle_s, \quad X \in T_s\mathcal M | |
\\ | |
h(Y) &= \exp_p(Y), \qquad &D_Y h(Y)[Z] &= D_Y \exp_p(Y)[Z] | |
\\ | |
i(X) &= tX, \qquad &D_X i(X)[Y] &= tY. | |
\end{align} | |
``` | |
And we use the Chain rule on manifolds, i.e. for ``f, g`` the concatenation ``f(g(p)) = (f\circ g)(p)`` then the chain rule reads ``D (f\circ g)(p)[X] = D f(g(p))\bigl[ Dg(p)[X] \bigr]`` (cf. AMS08 p. 195). | |
""" | |
# ╔═╡ 9c17c5ee-7a4d-4bc4-a86e-d606f76fcd40 | |
md""" | |
For our case this reads ``f(X) = g(h(i(X)))`` so that the differential reads | |
```math | |
Df(X)[Y] = D\, g\bigl( h(i(X)) \bigr)\Bigl[ D h(i(X)) \bigl[ Di(X)[Y] \bigr] \Bigr] | |
``` | |
where | |
* ``Di(X)[Y] = tY`` | |
* ``h(i(X)) = \exp_p(tX)`` | |
* hence ``D h(i(X)) \bigl[ Di(X)[Y] \bigr] = D_exp_p(tX)[tY]`` | |
so that we finally have with ``s=\exp_ptX`` that | |
```math | |
\begin{split} | |
Df(X)[Y] &= D \frac{1}{2}d^2_{\mathcal M}\bigl(q, \exp_p (tX)\bigr) [Y] = D \frac{1}{2}d^2_{\mathcal M}(q,exp_p tX)\bigl[D_exp_p(tX)[tY]\bigr] | |
\\ | |
&= \langle -\log_{\exp_p (tX)} q, D \exp_p(tX)[tY] \rangle_{\exp_p(tX)} | |
\end{split} | |
``` | |
""" | |
# ╔═╡ dc9bbe53-abcf-4e02-922b-632234ecff74 | |
md"Let's check this in code" | |
# ╔═╡ c9cc1f8c-f9b0-4417-8b46-ddaf70681c1f | |
M = Sphere(2) | |
# ╔═╡ 040f9163-cdb8-4b67-925f-6d4938bc4007 | |
p = random_point(M) | |
# ╔═╡ f8f54021-991b-4afa-b9fa-91ef2e9135aa | |
q = random_point(M) | |
# ╔═╡ ad5e006b-7173-49c5-b3f7-e3d95045403e | |
t = 0.3 | |
# ╔═╡ 86423703-b26c-4f25-89ae-139f5572e7c4 | |
Xstar = 1/t * log(M, p, q) | |
# ╔═╡ 5f7192e9-c56a-4d55-ab31-550b7bb358e6 | |
f(X) = 0.5 * distance(M, q, exp(M, p, t*X)).^2 | |
# ╔═╡ d0cfa119-3e93-468d-b039-d5928cadf2b4 | |
f(Xstar) | |
# ╔═╡ ffeb8c41-0fd8-47f2-8f76-a6b85a2a98b3 | |
Df(X, Y) = inner(M, | |
exp(M, p, t*X), | |
-log(M, exp(M, p, t*X), q), | |
differential_exp_argument(M, p, t*X, t*Y), | |
) | |
# ╔═╡ 7e88a79e-3d94-4ab4-a841-f3cc055b245f | |
md""" the following ready checks, that in ``X^*`` the differential is zero """ | |
# ╔═╡ 43b84248-8a24-4a54-810c-ca88344224ba | |
Df(Xstar, random_tangent(M, p)) | |
# ╔═╡ bd26abad-62e7-49c5-9694-3e80b4a8992f | |
# ╔═╡ 00000000-0000-0000-0000-000000000001 | |
PLUTO_PROJECT_TOML_CONTENTS = """ | |
[deps] | |
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Manopt = "0fc0a36d-df90-57f3-8f93-d78a9fc72bb5" | |
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" | |
Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" | |
[compat] | |
Manifolds = "~0.7.5" | |
Manopt = "~0.3.17" | |
Plots = "~1.25.11" | |
""" | |
# ╔═╡ 00000000-0000-0000-0000-000000000002 | |
PLUTO_MANIFEST_TOML_CONTENTS = """ | |
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version = "0.9.1+5" | |
""" | |
# ╔═╡ Cell order: | |
# ╟─a474d350-9689-11ec-2178-65e3d33c9f05 | |
# ╠═b8662ee7-3af9-4703-978f-6ce42408b581 | |
# ╟─d3f21250-521d-4438-a774-3fcd09fd315f | |
# ╟─610a9702-524e-4294-957c-fc601f7061d1 | |
# ╟─35cdd1a3-ba0f-47e3-83b3-2258a480a957 | |
# ╟─996217a9-83db-47f9-b6d5-f28eedbf6e67 | |
# ╟─9c17c5ee-7a4d-4bc4-a86e-d606f76fcd40 | |
# ╟─dc9bbe53-abcf-4e02-922b-632234ecff74 | |
# ╠═c9cc1f8c-f9b0-4417-8b46-ddaf70681c1f | |
# ╠═040f9163-cdb8-4b67-925f-6d4938bc4007 | |
# ╟─f8f54021-991b-4afa-b9fa-91ef2e9135aa | |
# ╠═ad5e006b-7173-49c5-b3f7-e3d95045403e | |
# ╠═86423703-b26c-4f25-89ae-139f5572e7c4 | |
# ╠═5f7192e9-c56a-4d55-ab31-550b7bb358e6 | |
# ╠═d0cfa119-3e93-468d-b039-d5928cadf2b4 | |
# ╠═ffeb8c41-0fd8-47f2-8f76-a6b85a2a98b3 | |
# ╟─7e88a79e-3d94-4ab4-a841-f3cc055b245f | |
# ╠═43b84248-8a24-4a54-810c-ca88344224ba | |
# ╠═bd26abad-62e7-49c5-9694-3e80b4a8992f | |
# ╟─00000000-0000-0000-0000-000000000001 | |
# ╟─00000000-0000-0000-0000-000000000002 |
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