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(Becigneul, Ganea)
- \(g_t = f'(p_t)^\top\)
- \(x_{t+1} = x_t - \alpha g_t\)
An \(n\)-manifold: "a 'nice' top. space \((M, \Theta)\) whose every \(p\in M\) has open neighbourhood homeomorphic to (a piece of) \(\mathbb{R}^n\)"
\[\phi: U\xrightarrow{\sim} \phi(U)\subset \mathbb{R}^n\]
"A manifold covered with a maximal atlas of smoothly-compatible charts", \((M, \Theta, \mathcal{A})\)
\[(f\circ \gamma)'(0).\]
\[X_{\gamma, p}f = (f\circ \gamma)'(0)\]
"Directional derivative of \(f\) along \(\gamma\) at \(p\)"
\[X_{\gamma, p}f = (f\circ \gamma)'(0).\]
\[T_pM = \left\{ X_{\gamma,p}\left| \gamma:I\to M~\text{smooth}, ~\gamma(0)=p \right. \right\}\]
curve segments via chartsA chart \((U,x)\), \(p\in U\), corresponds to a basis in \(T_p M\), represented by coordinate curves.
TODO: move to after geodesics
\[\exp_pX = \gamma(1)~\text{if defined}.\]
Given objective \(f:M\to\mathbb{R}\) and \(p_t\in M\), how to pick \(X_t\)?
Given smooth \(F:M\to N,\)
"the push-forward of \(F\) at \(p\)" is the linear map \(F_{*,p}:T_p{M}\to T_{F(p)}N\)
\[(\underbrace{F_{*,p}X}_{T_{F(p)}N})\underbrace{g}_{C^\infty(N)} = \underbrace{X}_{T_p M}(\underbrace{g\circ F}_{C^\infty(M)}).\]
Given objective \(f:M\to \mathbb{R}\) its derivative at point \(p\in M\) is a linear functional \[T_p{M}\to T\mathbb{R}\cong \mathbb{R}.\]
."A smooth manifold endowed with smooth assignment of (almost) 'inner products' \(g:T_p M\times T_p M\to \mathbb{R}\) at each \(p\)". Notes for details
linear functionals via inner product\[X_t = \operatorname{grad}f(p_t) \equiv f_{*,p_t}^\sharp,\] \[x_{t+1} = \exp_{p_t}(-\alpha X_t).\]
Ready to implement?
Immersion: smooth \(\phi: M\to\mathbb{R}^n\) such that \(\phi_{*,p}\) is injective for every \(p\in M\). Embedding: a diffeomorphic immersion.
[Strong] Whitney: can embed in \(\mathbb{R}^{2\operatorname{dim}M}.\)
\(M\subset\mathbb{R}^d\) be a level-set (\(\{p\left|F(p)=0\right.\}\)).
\(\left.\operatorname{id}\right|_{M}\) is an embedding and
can represent points and tangents via (redundant) \(\mathbb{R}^d\) coordinates,
can induce metric from Euclidean using \(F\),
need projection op.
(prevalent approach, including geoopt)
(examples using local charts?)
(Becigneul, Ganea)\[R_p:T_p M \to M,\] \[R_p(0) = p,\] \[(R_p)_{*,0} "=" \operatorname{id}_{T_p M}.\]
A Bundle is a triple \((E, \pi, M)\), \[E\xrightarrow{\pi} M\]
Tangent bundle: \[E = TM = \cup_p T_p M\] \[\pi = X_{\gamma, p} \mapsto p.\]
A smooth section of a bundle \(E\xrightarrow\pi M\) is a smooth map \[\sigma:M\to E\] such that \[\pi\circ\sigma = \operatorname{id}_M\]
Set of all smooth sections is denoted \(\Gamma(E)\).
A vector field is a smooth section of the tangent bundle \(TM\).
Another example: metric \(g\) is a smooth section of the bundle of bilinear forms (\((0,2)\)-tensor field)
"Covariant derivative of vf \(Y\) along \(X\)": \[\nabla_XY\] Wishlist:
In chart:
…
(Christoffel symbols)
…
Extendible if there is \(\tilde{X}\in\Gamma(TM)\) s.t. \(X_\lambda = \tilde{X}_{\gamma(\lambda)}.\)
If \(X:I \to TM\) is a vf along \(\gamma\) extendible to \(\tilde{X}\) and \[\nabla_X\tilde{X} = 0\] then \(X\) is called autoparallely-transported
(Becigneul, Ganea)(here should go some qualitative derivation vector transports for levelsets)
later addition
\(f: \mathbb{R} \to \mathbb{R},\) find \(x\) s.t. \(f(x)=0\).
later addition
\(f: \mathbb{R}\to \mathbb{R},\) find \(x\) s.t. \(f'(x)=0\).
later addition
\(f: \mathbb{R}^n\to\mathbb{R}^m\), find \(x\) s.t. \(f(x)=0\).
later addition
\(f: \mathbb{R}^n\to\mathbb{R}\), find \(x\) s.t. \(f'(x)=0\).
Solve \[\nabla_{X_t}\operatorname{grad}f(p_t) = -\operatorname{grad}f(p_t)\] for \(X_t\in T_{p_t}M\).
\[p_{t+1} = \exp_{p_t}X_t.\]
"Coordinates" on manifolds?
\[M = M_1\times \cdots\times M_n,\]
Product topology, product metric, …
RAdam: adaptivity term for each product component!
(Becigneul, Ganea)
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