From 3cbbab674c6a84d2554983f29a187358380e2d61 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Fri, 24 Nov 2023 12:40:14 +0100 Subject: [PATCH 01/27] Started adding markdown for general manifolds. --- docs/src/manifolds/manifolds.md | 19 +++++++++++++++++++ 1 file changed, 19 insertions(+) create mode 100644 docs/src/manifolds/manifolds.md diff --git a/docs/src/manifolds/manifolds.md b/docs/src/manifolds/manifolds.md new file mode 100644 index 000000000..e6c9f3553 --- /dev/null +++ b/docs/src/manifolds/manifolds.md @@ -0,0 +1,19 @@ +# (Matrix) Manifolds + +Manifolds are topological spaces that locally look like vector spaces. In the following we restrict ourselves to finite-dimensional manifolds. +__Definition__: A **finite-dimensional smooth manifold** of dimension $n$ is a second-countable Hausdorff space $\mathcal{M}$ for which $\forall{}x\in\mathcal{M}$ we can find a neighborhood $U$ that contains $x$ and a corresponding homeomorphism $\varphi_U:U\simeq{}W\sub\mathbb{R}^n$ where $W$ is an open subset. The homeomorphisms $\varphi_U$ are referred to as *coordinate charts*. If two such coordinate charts overlap, i.e. if $U_1\cup{}U_2\neq\{\}$, then the map $\varphi_{U_2}^{-1}\circ\varphi_{U_1}$ is $C^\infty$. + +One example of a manifold that is also important for `GeometricMachineLearning.jl` is the Lie group[^1] of orthonormal matrices $SO(N)$. Before we can proof that $SO(N)$ is a manifold we first need another definition and a theorem: + +[^1]: Lie groups are manifolds that also have a *group structure*, i.e. there is an operation $\mathcal{M}\times\mathcal{M}\to\mathcal{M},(a,b)\mapsto{}ab$ s.t. $(ab)c = a(bc)$ and $\exists{}e\mathcal{M}$ s.t. $ae$ = $a$ $\forall{}a\in\mathcal{M}$. + +__Definition__: Consider a smooth mapping $g: \mathcal{M}\to\mathcal{N}$ from one manifold to another. A point $B\in\mathcal{M}$ is called a **regular value** of $\mathcal{M}$ if $\forall{}A\in{}g^{-1}\{B\}$ the map $T_Ag:T_A\mathcal{M}\to{}T_{g(A)}\mathcal{N}$ is surjective. + +__Theorem__: Consider a smooth map $g:\mathcal{M}\to\mathcal{N}$ from one manifold into another. Then the preimage of a regular point $B$ of $\mathcal{N}$ is a submanifold of $\mathcal{M}$. Furthermore the codimension of $g^{-1}\{B\}$ is equal to the dimension of $\mathcal{N}$. This is known as the **preimage theorem**. +__Proof__: + + +__Theorem__: The group $SO(N)$ is a Lie group (i.e. has manifold structure). +__Proof__: The vector space $\mathbb{R}^{N\times{}N}$ clearly has manifold structure. The group $SO(N)$ is equivalent to one of the level sets of the mapping: $f:\mathbb{R}^{N\times{}N}\to\mathcal{S}(N), A\mapsto{}A^TA$, i.e. it is the component of $f^{-1}\{\mathbb{I}\}$ that contains $\mathbb{I}$. We still need to proof that $T_Af$ is surjective, i.e. $\forall{}B\in\mathcal{S}(N), A\in\mathbb{R}^{N\times{}N}$ $\exists{}C\in\mathbb{R}^{N\timesN}$ s.t. $C^TA + A^TC = B$. + +With the definition above we can generalize the notion of an ordinary differential equation on a vector space to an ordinary differential equation on a manifold. \ No newline at end of file From d1ad5712fa444c1a983310f5fb5988837abaf7d4 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Fri, 24 Nov 2023 16:02:08 +0100 Subject: [PATCH 02/27] Completed proof that SO(N) is a manifold. --- docs/src/manifolds/manifolds.md | 19 +++++++++++-------- 1 file changed, 11 insertions(+), 8 deletions(-) diff --git a/docs/src/manifolds/manifolds.md b/docs/src/manifolds/manifolds.md index e6c9f3553..816e8dd2e 100644 --- a/docs/src/manifolds/manifolds.md +++ b/docs/src/manifolds/manifolds.md @@ -1,19 +1,22 @@ # (Matrix) Manifolds Manifolds are topological spaces that locally look like vector spaces. In the following we restrict ourselves to finite-dimensional manifolds. -__Definition__: A **finite-dimensional smooth manifold** of dimension $n$ is a second-countable Hausdorff space $\mathcal{M}$ for which $\forall{}x\in\mathcal{M}$ we can find a neighborhood $U$ that contains $x$ and a corresponding homeomorphism $\varphi_U:U\simeq{}W\sub\mathbb{R}^n$ where $W$ is an open subset. The homeomorphisms $\varphi_U$ are referred to as *coordinate charts*. If two such coordinate charts overlap, i.e. if $U_1\cup{}U_2\neq\{\}$, then the map $\varphi_{U_2}^{-1}\circ\varphi_{U_1}$ is $C^\infty$. +__Definition__: A **finite-dimensional smooth manifold** of dimension ``n`` is a [second-countable Hausdorff space](basic_topology.md) ``\mathcal{M}`` for which ``\forall{}x\in\mathcal{M}`` we can find a neighborhood ``U`` that contains ``x`` and a corresponding homeomorphism ``\varphi_U:U\cong{}W\sub\mathbb{R}^n`` where ``W`` is an open subset. The homeomorphisms ``\varphi_U`` are referred to as *coordinate charts*. If two such coordinate charts overlap, i.e. if ``U_1\cap{}U_2\neq\{\}``, then the map ``\varphi_{U_2}^{-1}\circ\varphi_{U_1}`` is ``C^\infty``. -One example of a manifold that is also important for `GeometricMachineLearning.jl` is the Lie group[^1] of orthonormal matrices $SO(N)$. Before we can proof that $SO(N)$ is a manifold we first need another definition and a theorem: +One example of a manifold that is also important for `GeometricMachineLearning.jl` is the Lie group[^1] of orthonormal matrices ``SO(N)``. Before we can proof that ``SO(N)`` is a manifold we first need another definition and a theorem: -[^1]: Lie groups are manifolds that also have a *group structure*, i.e. there is an operation $\mathcal{M}\times\mathcal{M}\to\mathcal{M},(a,b)\mapsto{}ab$ s.t. $(ab)c = a(bc)$ and $\exists{}e\mathcal{M}$ s.t. $ae$ = $a$ $\forall{}a\in\mathcal{M}$. +[^1]: Lie groups are manifolds that also have a *group structure*, i.e. there is an operation ``\mathcal{M}\times\mathcal{M}\to\mathcal{M},(a,b)\mapsto{}ab`` s.t. ``(ab)c = a(bc)`` and ``\exists{}e\mathcal{M}`` s.t. ``ae`` = ``a`` ``\forall{}a\in\mathcal{M}``. -__Definition__: Consider a smooth mapping $g: \mathcal{M}\to\mathcal{N}$ from one manifold to another. A point $B\in\mathcal{M}$ is called a **regular value** of $\mathcal{M}$ if $\forall{}A\in{}g^{-1}\{B\}$ the map $T_Ag:T_A\mathcal{M}\to{}T_{g(A)}\mathcal{N}$ is surjective. +__Definition__: Consider a smooth mapping ``g: \mathcal{M}\to\mathcal{N}`` from one manifold to another. A point ``B\in\mathcal{M}`` is called a **regular value** of ``\mathcal{M}`` if ``\forall{}A\in{}g^{-1}\{B\}`` the map ``T_Ag:T_A\mathcal{M}\to{}T_{g(A)}\mathcal{N}`` is surjective. + +__Theorem__: Consider a smooth map ``g:\mathcal{M}\to\mathcal{N}`` from one manifold to another. Then the preimage of a regular point ``B`` of ``\mathcal{N}`` is a submanifold of ``\mathcal{M}``. Furthermore the codimension of ``g^{-1}\{B\}`` is equal to the dimension of ``\mathcal{N}`` and the tangent space ``T_A(g^{-1}\{B\})`` is equal to the kernel of ``T_Ag``. This is known as the **preimage theorem**. -__Theorem__: Consider a smooth map $g:\mathcal{M}\to\mathcal{N}$ from one manifold into another. Then the preimage of a regular point $B$ of $\mathcal{N}$ is a submanifold of $\mathcal{M}$. Furthermore the codimension of $g^{-1}\{B\}$ is equal to the dimension of $\mathcal{N}$. This is known as the **preimage theorem**. __Proof__: -__Theorem__: The group $SO(N)$ is a Lie group (i.e. has manifold structure). -__Proof__: The vector space $\mathbb{R}^{N\times{}N}$ clearly has manifold structure. The group $SO(N)$ is equivalent to one of the level sets of the mapping: $f:\mathbb{R}^{N\times{}N}\to\mathcal{S}(N), A\mapsto{}A^TA$, i.e. it is the component of $f^{-1}\{\mathbb{I}\}$ that contains $\mathbb{I}$. We still need to proof that $T_Af$ is surjective, i.e. $\forall{}B\in\mathcal{S}(N), A\in\mathbb{R}^{N\times{}N}$ $\exists{}C\in\mathbb{R}^{N\timesN}$ s.t. $C^TA + A^TC = B$. +__Theorem__: The group ``SO(N)`` is a Lie group (i.e. has manifold structure). +__Proof__: The vector space ``\mathbb{R}^{N\times{}N}`` clearly has manifold structure. The group ``SO(N)`` is equivalent to one of the level sets of the mapping: ``f:\mathbb{R}^{N\times{}N}\to\mathcal{S}(N), A\mapsto{}A^TA``, i.e. it is the component of ``f^{-1}\{\mathbb{I}\}`` that contains ``\mathbb{I}``. We still need to proof that ``\mathbb{I}`` is a regular point of ``f``, i.e. that for ``A\in{}SO(N)`` the mapping ``T_Af`` is surjective. This means that ``\forall{}B\in\mathcal{S}(N), A\in\mathbb{R}^{N\times{}N}`` ``\exists{}C\in\mathbb{R}^{N\times{}N}`` s.t. ``C^TA + A^TC = B``. The element ``C=\frac{1}{2}AB\in\mathcal{R}^{N\times{}N}`` satisfies this property. + +With the definition above we can generalize the notion of an ordinary differential equation (ODE) on a vector space to an ordinary differential equation on a manifold: -With the definition above we can generalize the notion of an ordinary differential equation on a vector space to an ordinary differential equation on a manifold. \ No newline at end of file +__Definition__: An **ODE on a manifold** is a mapping that assigns to each element of the manifold ``A\in\mathcal{M}`` an element of the corresponding tangent space ``T_A\mathcal{M}``. \ No newline at end of file From 3c2ff828f505887027665b21c7c54222f186e2d6 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Fri, 24 Nov 2023 16:02:42 +0100 Subject: [PATCH 03/27] Added file describing basic topological concepts needed for manifolds. --- docs/src/manifolds/basic_topology.md | 29 ++++++++++++++++++++++++++++ 1 file changed, 29 insertions(+) create mode 100644 docs/src/manifolds/basic_topology.md diff --git a/docs/src/manifolds/basic_topology.md b/docs/src/manifolds/basic_topology.md new file mode 100644 index 000000000..4cd5912d4 --- /dev/null +++ b/docs/src/manifolds/basic_topology.md @@ -0,0 +1,29 @@ +# Basic Notions of Topology + +On this page we discuss basic notions of topology that are necessary to define [manifolds](manifolds.md). Here we largely omit proofs and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. + +[^1]: Some authors (see e.g. [lang2012fundamentals](@cite)) do not require these properties. But since they constitute very weak restrictions and are always satisfied by the manifolds relevant for our purposes. Thus we require them here. + +__Definition__: A **topological space** is a set ``\mathcal{M}`` for which we define a collection of subsets of ``\mathcal{M}``, which we denote by ``\mathcal{T}`` and call the *open subsets*. ``\mathcal{T}`` further has to satisfy the following three conditions: +1. The empty set and ``\mathcal{M}`` belong to ``\mathcal{T}``. +2. Any union of an arbitrary number of elements of ``\mathcal{T}`` again belongs to ``\mathcal{T}``. +3. Any intersection of a finite number of elements of ``\mathcal{T}`` again belongs to ``\mathcal{T}``. + +Based on this definition of a topological space we can now define what it means to be *Hausdorff*: +__Definition__: A topological space ``\mathcal{M}`` is said to be **Hausdorff** if for any two points ``x,y\in\mathcal{M}`` we can find two open sets ``U_x,U_y\in\mathcal{T}`` s.t. ``x\in{}U_x, y\in{}U_y`` and ``U_x\cap{}U_y=\{\}``. + +Lastly we define the notion of *second countability*: +__Definition__: A topological space ``\mathcal{M}`` is said to be **second-countable** if we can find a countable subcollection of ``\mathcal{T}`` called ``\mathcal{U}`` s.t. ``\forall{}U\in\mathcal{T}`` and ``x\in{}U`` we can find an element ``V\in\mathcal{U}`` for which ``x\in{}V\sub{}U``. + + +## References +- General Topology book. +- Lang + +@book{lang2012fundamentals, + title={Fundamentals of differential geometry}, + author={Lang, Serge}, + volume={191}, + year={2012}, + publisher={Springer Science \& Business Media} +} From a32fa22e0c10493c39bf248167f9bee49c95c8af Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Fri, 24 Nov 2023 16:05:50 +0100 Subject: [PATCH 04/27] Started basic description of NN optimizers. --- docs/src/optimizers/general_optimization.md | 12 ++++++++++++ 1 file changed, 12 insertions(+) create mode 100644 docs/src/optimizers/general_optimization.md diff --git a/docs/src/optimizers/general_optimization.md b/docs/src/optimizers/general_optimization.md new file mode 100644 index 000000000..325129190 --- /dev/null +++ b/docs/src/optimizers/general_optimization.md @@ -0,0 +1,12 @@ +# Optimization for Neural Networks + +Optimization for neural networks is (almost always) some variation on gradient descent. The most basic form of gradient descent is a discretization of the *gradient flow equation*: +```math +\dot{\theta} = -\nabla_\theta{}L, +``` +by means of a Euler time-stepping scheme: +```math +\theta^{t+1} = \theta^{t} - h\nabla_{\theta^{t}}L, +``` +where $\eta$ (the time step of the Euler scheme) is referred to as the *learning rate* +This equation can easily be generalized to [manifolds](manifolds.md) by replacing the *Euclidean gradient* $\nabla_{\theta^{t}L}$ by a *Riemannian gradient* $-h\mathrm{grad}_{\theta^{t}}L$ and addition by $-h\nabla_{\theta^{t}}L$ with a [retraction](retraction.md) by $-h\mathrm{grad}_{\theta^{t}}L$. \ No newline at end of file From 9f62150f26d53dfa585db87f4ad9de3272df178e Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Sun, 26 Nov 2023 18:02:28 +0100 Subject: [PATCH 05/27] Added files on the existence and uniqueness theorem, the inverse function theorem and the sumersion theorem. The inverse function theorem file is essentially done whereas the existence and uniqueness theorem as well as the submersion theorem have not really been started. Also expaned on the basic notions of topology section and added my favourite reference. --- docs/src/manifolds/basic_topology.md | 52 +++++++++++++++++-- .../existence_and_uniqueness_theorem.md | 24 +++++++++ .../src/manifolds/inverse_function_theorem.md | 29 +++++++++++ docs/src/manifolds/submersion_theorem.md | 3 ++ 4 files changed, 104 insertions(+), 4 deletions(-) create mode 100644 docs/src/manifolds/existence_and_uniqueness_theorem.md create mode 100644 docs/src/manifolds/inverse_function_theorem.md create mode 100644 docs/src/manifolds/submersion_theorem.md diff --git a/docs/src/manifolds/basic_topology.md b/docs/src/manifolds/basic_topology.md index 4cd5912d4..53bbc9355 100644 --- a/docs/src/manifolds/basic_topology.md +++ b/docs/src/manifolds/basic_topology.md @@ -1,8 +1,8 @@ -# Basic Notions of Topology +# Basic Concepts of General Topology -On this page we discuss basic notions of topology that are necessary to define [manifolds](manifolds.md). Here we largely omit proofs and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. +On this page we discuss basic notions of topology that are necessary to define and work [manifolds](manifolds.md). Here we largely omit concrete examples and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. For a wide range of examples and a detailed discussion of the theory see e.g. [lipschutz1965general](@cite). The here-presented theory is also (rudimentary) covered in most differential geometry books such as [lang2012](@cite) and [bishop2012tensor](@cite). -[^1]: Some authors (see e.g. [lang2012fundamentals](@cite)) do not require these properties. But since they constitute very weak restrictions and are always satisfied by the manifolds relevant for our purposes. Thus we require them here. +[^1]: Some authors (see e.g. [lang2012fundamentals](@cite)) do not require these properties. But since they constitute very weak restrictions and are always satisfied by the manifolds relevant for our purposes we require them here. __Definition__: A **topological space** is a set ``\mathcal{M}`` for which we define a collection of subsets of ``\mathcal{M}``, which we denote by ``\mathcal{T}`` and call the *open subsets*. ``\mathcal{T}`` further has to satisfy the following three conditions: 1. The empty set and ``\mathcal{M}`` belong to ``\mathcal{T}``. @@ -12,9 +12,45 @@ __Definition__: A **topological space** is a set ``\mathcal{M}`` for which we de Based on this definition of a topological space we can now define what it means to be *Hausdorff*: __Definition__: A topological space ``\mathcal{M}`` is said to be **Hausdorff** if for any two points ``x,y\in\mathcal{M}`` we can find two open sets ``U_x,U_y\in\mathcal{T}`` s.t. ``x\in{}U_x, y\in{}U_y`` and ``U_x\cap{}U_y=\{\}``. -Lastly we define the notion of *second countability*: +We now give the second definition that we need for defining manifolds, that of *second countability*: __Definition__: A topological space ``\mathcal{M}`` is said to be **second-countable** if we can find a countable subcollection of ``\mathcal{T}`` called ``\mathcal{U}`` s.t. ``\forall{}U\in\mathcal{T}`` and ``x\in{}U`` we can find an element ``V\in\mathcal{U}`` for which ``x\in{}V\sub{}U``. +We now give a few definitions and results that are needed for the [inverse function theorem](inverse_function_theorem.md) which is essential for practical applications of manifold theory. + +__Definition__: A mapping ``f`` between topological spaces ``\mathcal{M}`` and ``\mathcal{N}`` is called **continuous** if the preimage of every open set is again an open set, i.e. if ``f^{-1}\{U\}\in\mathcal{T}`` for ``U`` open in ``\mathcal{N}`` and ``\mathcal{T}`` the topology on ``\mathcal{M}``. + +__Definition__: A **closed set** of a topological space ``\mathcal{M}`` is one whose complement is an open set, i.e. ``F`` is closed if ``F^c\in\mathcal{T}``, where the superscript ``{}^c`` indicates the complement. For closed sets we thus have the following three properties: +1. The empty set and ``\mathcal{M}`` are closed sets. +2. Any union of a finite number of closed sets is again closed. +3. Any intersection of an arbitrary number of closed sets is again closed. + +__Theorem__: The definition of continuity is equivalent to the following, second definition: ``f:\mathcal{M}\to\mathcal{N}`` is continuous if ``f^{-1}\{F\}\sub\mathcal{M}`` is a closed set for each closed set ``F\sub\mathcal{N}``. + +__Proof__: First assume that ``f`` is continuous according to the first definition and not to the second. Then ``f^{-1}{F}`` is not closed but ``f^{-1}{F^c}`` is open. But ``f^{-1}\{F^c\} = \{x\in\mathcal{M}:f(x)\nin\mathcal{N}\} = (f^{-1}\{F\})^c`` cannot be open, else ``f^{-1}\{F\}`` would be closed. The implication of the first definition under assumption of the second can be shown analogously. + +__Theorem__: The property of a set ``F`` being closed is equivalent to the following statement: If a point ``y`` is such that for every open set ``U`` containing it we have ``U\cap{}F\neq\{\}`` then this point is contained in ``F``. + +__Proof__: We first proof that if a set is closed then the statement holds. Consider a closed set ``F`` and a point ``y\nin{}F`` s.t. every open set containing ``y`` has nonempty intersection with ``F``. But the complement ``F^c`` also is such a set, which is a clear contradiction. Now assume the above statement for a set ``F`` and further assume ``F`` is not closed. Its complement ``F^c`` is thus not open. Now consider the *interior* of this set: ``\mathrm{int}(F^c):=\cup\{U:U\sub{}F^c\}``, i.e. the biggest open set contained within ``F^c``. Hence there must be a point ``y`` which is in ``F^c`` but is not in its interior, else ``F^c`` would be equal to its interior, i.e. would be open. We further must be able to find an open set ``U`` that contains ``y`` but is also contained in ``F^c``, else ``y`` would be an element of ``F``. A contradiction. + +__Definition__: An **open cover** of a topological space ``\mathcal{M}`` is a (not necessarily countable) collection of open sets ``\{U_i\}_{i\mathcal{I}}`` s.t. their union contains ``\mathcal{M}``. A **finite open cover** is a collection of a finite number of open sets that cover ``\mathcal{M}``. We say that an open cover is **reducible** to a finite cover if we can find a finite number of elements in the open cover whose union still contains ``\mathcal{M}``. + +__Definition__: A topological space ``\mathcal{M}`` is called **compact** if every open cover is reducible to a finite cover. + +__Theorem__: Consider a continuous function ``f:\mathcal{M}\to\mathcal{N}`` and a compact set ``K\in\mathcal{M}``. Then ``f(K)`` is also compact. + +__Proof__: Consider an open cover of ``f(K)``: ``\{U_i\}_{i\in\mathcal{I}}``. Then ``\{f^{-1}\{U_i\}\}_{i\in\mathcal{I}}`` is an open cover of ``K`` and hence reducible to a finite cover ``\{f^{-1}\{U_i\}\}_{i\in\{i_1,\ldots,i_n}}``. But then ``\{{U_i\}_{i\in\{i_1,\ldots,i_n}}`` also covers ``f(K)``. + +__Theorem__: A closed subset of a compact space is compact: + +__Proof__: Call the closed set ``F`` and consider an open cover of this set: ``\{U\}_{i\in\mathcal{I}}``. Then this open cover combined with ``F^c`` is an open cover for the entire compact space, hence reducible to a finite cover. + +__Theorem__: A compact subset of a Hausdorff space is closed: + +__Proof__: Consider a compact subset ``K``. If ``K`` is not closed, then there has to be a point ``y\nin{}K`` s.t. every open set containing ``y`` intersects ``K``. Because the surrounding space is Hausdorff we can now find the following two collections of open sets: ``\{(U_z, U_{z,y}: U_z\cap{}U_{z,y}=\{\})\}_{z\in{}K}``. The open cover ``\{U_z}_{z\in{}K}`` is then reducible to a finite cover ``\{U_z}_{z\in{z_1, \ldots, z_n}\}``. The intersection ``\cap_{z\in{z_1, \ldots, z_n}}U_{z,y}`` is then an open set that contains ``y`` but has no intersection with ``K``. A contraction. + +__Theorem__: If ``\mathcal{M}`` is compact and ``\mathcal{N}`` is Hausdorff, then the inverse of a continuous function ``f:\mathcal{M}\to\mathcal{N}`` is again continuous, i.e. ``f(V)`` is an open set in ``\mathcal{N}`` for ``V\in\mathcal{T}``. + +__Proof__: We can equivalently show that every closed set is mapped to a closed set. First consider the set ``K\in\mathcal{M}``. Its image is again compact and hence closed because ``\mathcal{N}`` is Hausdorff. ## References - General Topology book. @@ -27,3 +63,11 @@ __Definition__: A topological space ``\mathcal{M}`` is said to be **second-count year={2012}, publisher={Springer Science \& Business Media} } + +@book{lipschutz1965general, + title={General Topology}, + author={Seymour Lipschutz}, + year={1965}, + publisher={McGraw-Hill Book Company}, + location={New York City, New York} +} \ No newline at end of file diff --git a/docs/src/manifolds/existence_and_uniqueness_theorem.md b/docs/src/manifolds/existence_and_uniqueness_theorem.md new file mode 100644 index 000000000..4aa89da45 --- /dev/null +++ b/docs/src/manifolds/existence_and_uniqueness_theorem.md @@ -0,0 +1,24 @@ +# The Existence-And-Uniqueness Theorem + +In order to proof the existence-and-uniqueness theorem we first need another theorem, the **Banach fixed-point theorem** for which we also need another definition. + +__Definition__: A **contraction mapping** is a map ``T:\mathbb{R}^N\to\mathbb{R}^N`` for which there exists ``q\in[0,1)`` s.t. ``\forall{}x,y\in\mathbb{R}^N,\,||T(x)-T(y)||\leq{}q||x-y||``. + +__Theorem (Banach fixed-point theorem)__: Every **contraction mapping** ``T`` admits a unique fixed point ``x^*`` (i.e. a point ``x^*`` s.t. ``F(x^*)=x^*``) and this point can be found by taking an arbitrary point ``x_0\in\mathbb{R}^N`` and taking the limit ``\lim_{n\to\infty}T^n(x_0)``. + +__Proof (Banach fixed-point theorem)__: Take an arbitrary point ``x_0\in\mathbb{R}^N`` and consider the sequence ``(x_n)_{n\in\mathbb{N}}`` with ``x_n:=T^n(x_0)``. Then it holds that (for ``m>n``): +```math +\begin{aligned} +|x_m - x_n| & \leq |x_m - x_{m-1}| + |x_{m-1} - x_{m-2}| + \cdots + |x_{m-(m-n+1)}-x_{n}| \\ + & = |x_{n+(m-n)} - x_{n+(m-n-1)}| + \cdots + |x_{n+1} - x_n| \\ + & \leq \sum_{i=0}^{m-n-1}q^i|x_{n+1} - x_n| \\ + & \leq \sum_{i=0}^{m-n-1}q^iq^n|x_1 - x_0| \\ + & = q^n|x_1 -x_0|\sum_{i=1}^{m-n-1}q^i, +\end{aligned} +``` +where we have used the triangle inequality in the first line. If we now let ``m`` on the right-hand side first go to infinity then we get +```math +|x_m-x_n| & \leq q^n|x_1 -x_0|\sum_{i=1}^{\infty}q^i + & =q^n|x_1 -x_0| \frac{1}{1-q}, +``` +proofing that the sequence is Cauchy. Because ``\mathbb{R}^N`` is a complete metric space we get that ``(x_n)_{n\in\mathbb{N}}`` is a convergent sequence. We call the limit of this sequence ``x^*``. This completes the proof of the Banach fixed-point theorem. diff --git a/docs/src/manifolds/inverse_function_theorem.md b/docs/src/manifolds/inverse_function_theorem.md new file mode 100644 index 000000000..4063c494e --- /dev/null +++ b/docs/src/manifolds/inverse_function_theorem.md @@ -0,0 +1,29 @@ +# The Inverse Function Theorem + +The **inverse function theorem** gives a sufficient condition on a vector-valued function to be invertible in a neighborhood of a specific point. This theorem is critical in developing a theory of [manifolds](manifolds.md) and serves as a basis for the [submersion theorem](submersion_theorem.md). Here we first state the theorem and then give a proof. + +__Theorem (Inverse function theorem)__: Consider a vector-valued differentiable function ``F:\mathbb{R}^N\to\mathbb{R}^N`` and assume its Jacobian is non-degenerate at a point ``x\in\mathbb{R}^N``. Then there exists a neighborhood ``U`` that contains ``F(x)`` and on which ``F`` is invertible, i.e. ``\exists{}H:U\to\mathbb{R}^N`` s.t. ``\forall{}y\inU,\,F\circ{}H(y) = y`` and the inverse is differentiable. + +__Proof__: Consider a mapping ``F:\mathbb{R}^N\to\mathbb{R}^N`` and assume its Jacobian has full rank at point ``x``, i.e. ``\det{}F'(x)\neq0``. Now consider a ball around ``x`` whose radius ``r`` we do not yet fix and two points ``y`` and ``z`` in that ball: ``y,z\in{}B(x,r)``. We further introduce the function ``G(y):=F(x)-F'(x)y``. By the *mean value theorem* we have ``|G(z) - G(y)|\leq|z-y|\sup_{0 Date: Mon, 27 Nov 2023 12:46:54 +0100 Subject: [PATCH 06/27] Added the right reference for the Bishop book at the bottom. --- docs/src/manifolds/basic_topology.md | 11 ++++++++++- 1 file changed, 10 insertions(+), 1 deletion(-) diff --git a/docs/src/manifolds/basic_topology.md b/docs/src/manifolds/basic_topology.md index 53bbc9355..14894909e 100644 --- a/docs/src/manifolds/basic_topology.md +++ b/docs/src/manifolds/basic_topology.md @@ -1,6 +1,6 @@ # Basic Concepts of General Topology -On this page we discuss basic notions of topology that are necessary to define and work [manifolds](manifolds.md). Here we largely omit concrete examples and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. For a wide range of examples and a detailed discussion of the theory see e.g. [lipschutz1965general](@cite). The here-presented theory is also (rudimentary) covered in most differential geometry books such as [lang2012](@cite) and [bishop2012tensor](@cite). +On this page we discuss basic notions of topology that are necessary to define and work [manifolds](manifolds.md). Here we largely omit concrete examples and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. For a wide range of examples and a detailed discussion of the theory see e.g. [lipschutz1965general](@cite). The here-presented theory is also (rudimentary) covered in most differential geometry books such as [lang2012](@cite) and [bishop1980tensor](@cite). [^1]: Some authors (see e.g. [lang2012fundamentals](@cite)) do not require these properties. But since they constitute very weak restrictions and are always satisfied by the manifolds relevant for our purposes we require them here. @@ -55,6 +55,7 @@ __Proof__: We can equivalently show that every closed set is mapped to a closed ## References - General Topology book. - Lang +- Bishop @book{lang2012fundamentals, title={Fundamentals of differential geometry}, @@ -70,4 +71,12 @@ __Proof__: We can equivalently show that every closed set is mapped to a closed year={1965}, publisher={McGraw-Hill Book Company}, location={New York City, New York} +} + +@book{bishop1980tensor, + title={Tensor Analysis on Manifolds}, + author={Richard L. Bishop, Samuel I. Goldberg}, + year={1980}, + publisher={Dover Publications}, + location={Mineola, New York} } \ No newline at end of file From 249e4b8b9ce94075668bba63eb83b56789edd6f0 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Fri, 24 Nov 2023 12:40:14 +0100 Subject: [PATCH 07/27] Started adding markdown for general manifolds. --- docs/src/manifolds/manifolds.md | 19 +++++++++++++++++++ 1 file changed, 19 insertions(+) create mode 100644 docs/src/manifolds/manifolds.md diff --git a/docs/src/manifolds/manifolds.md b/docs/src/manifolds/manifolds.md new file mode 100644 index 000000000..e6c9f3553 --- /dev/null +++ b/docs/src/manifolds/manifolds.md @@ -0,0 +1,19 @@ +# (Matrix) Manifolds + +Manifolds are topological spaces that locally look like vector spaces. In the following we restrict ourselves to finite-dimensional manifolds. +__Definition__: A **finite-dimensional smooth manifold** of dimension $n$ is a second-countable Hausdorff space $\mathcal{M}$ for which $\forall{}x\in\mathcal{M}$ we can find a neighborhood $U$ that contains $x$ and a corresponding homeomorphism $\varphi_U:U\simeq{}W\sub\mathbb{R}^n$ where $W$ is an open subset. The homeomorphisms $\varphi_U$ are referred to as *coordinate charts*. If two such coordinate charts overlap, i.e. if $U_1\cup{}U_2\neq\{\}$, then the map $\varphi_{U_2}^{-1}\circ\varphi_{U_1}$ is $C^\infty$. + +One example of a manifold that is also important for `GeometricMachineLearning.jl` is the Lie group[^1] of orthonormal matrices $SO(N)$. Before we can proof that $SO(N)$ is a manifold we first need another definition and a theorem: + +[^1]: Lie groups are manifolds that also have a *group structure*, i.e. there is an operation $\mathcal{M}\times\mathcal{M}\to\mathcal{M},(a,b)\mapsto{}ab$ s.t. $(ab)c = a(bc)$ and $\exists{}e\mathcal{M}$ s.t. $ae$ = $a$ $\forall{}a\in\mathcal{M}$. + +__Definition__: Consider a smooth mapping $g: \mathcal{M}\to\mathcal{N}$ from one manifold to another. A point $B\in\mathcal{M}$ is called a **regular value** of $\mathcal{M}$ if $\forall{}A\in{}g^{-1}\{B\}$ the map $T_Ag:T_A\mathcal{M}\to{}T_{g(A)}\mathcal{N}$ is surjective. + +__Theorem__: Consider a smooth map $g:\mathcal{M}\to\mathcal{N}$ from one manifold into another. Then the preimage of a regular point $B$ of $\mathcal{N}$ is a submanifold of $\mathcal{M}$. Furthermore the codimension of $g^{-1}\{B\}$ is equal to the dimension of $\mathcal{N}$. This is known as the **preimage theorem**. +__Proof__: + + +__Theorem__: The group $SO(N)$ is a Lie group (i.e. has manifold structure). +__Proof__: The vector space $\mathbb{R}^{N\times{}N}$ clearly has manifold structure. The group $SO(N)$ is equivalent to one of the level sets of the mapping: $f:\mathbb{R}^{N\times{}N}\to\mathcal{S}(N), A\mapsto{}A^TA$, i.e. it is the component of $f^{-1}\{\mathbb{I}\}$ that contains $\mathbb{I}$. We still need to proof that $T_Af$ is surjective, i.e. $\forall{}B\in\mathcal{S}(N), A\in\mathbb{R}^{N\times{}N}$ $\exists{}C\in\mathbb{R}^{N\timesN}$ s.t. $C^TA + A^TC = B$. + +With the definition above we can generalize the notion of an ordinary differential equation on a vector space to an ordinary differential equation on a manifold. \ No newline at end of file From d77882d773baf986889ddc731720b9e2e83e86c6 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Fri, 24 Nov 2023 16:02:08 +0100 Subject: [PATCH 08/27] Completed proof that SO(N) is a manifold. --- docs/src/manifolds/manifolds.md | 19 +++++++++++-------- 1 file changed, 11 insertions(+), 8 deletions(-) diff --git a/docs/src/manifolds/manifolds.md b/docs/src/manifolds/manifolds.md index e6c9f3553..816e8dd2e 100644 --- a/docs/src/manifolds/manifolds.md +++ b/docs/src/manifolds/manifolds.md @@ -1,19 +1,22 @@ # (Matrix) Manifolds Manifolds are topological spaces that locally look like vector spaces. In the following we restrict ourselves to finite-dimensional manifolds. -__Definition__: A **finite-dimensional smooth manifold** of dimension $n$ is a second-countable Hausdorff space $\mathcal{M}$ for which $\forall{}x\in\mathcal{M}$ we can find a neighborhood $U$ that contains $x$ and a corresponding homeomorphism $\varphi_U:U\simeq{}W\sub\mathbb{R}^n$ where $W$ is an open subset. The homeomorphisms $\varphi_U$ are referred to as *coordinate charts*. If two such coordinate charts overlap, i.e. if $U_1\cup{}U_2\neq\{\}$, then the map $\varphi_{U_2}^{-1}\circ\varphi_{U_1}$ is $C^\infty$. +__Definition__: A **finite-dimensional smooth manifold** of dimension ``n`` is a [second-countable Hausdorff space](basic_topology.md) ``\mathcal{M}`` for which ``\forall{}x\in\mathcal{M}`` we can find a neighborhood ``U`` that contains ``x`` and a corresponding homeomorphism ``\varphi_U:U\cong{}W\sub\mathbb{R}^n`` where ``W`` is an open subset. The homeomorphisms ``\varphi_U`` are referred to as *coordinate charts*. If two such coordinate charts overlap, i.e. if ``U_1\cap{}U_2\neq\{\}``, then the map ``\varphi_{U_2}^{-1}\circ\varphi_{U_1}`` is ``C^\infty``. -One example of a manifold that is also important for `GeometricMachineLearning.jl` is the Lie group[^1] of orthonormal matrices $SO(N)$. Before we can proof that $SO(N)$ is a manifold we first need another definition and a theorem: +One example of a manifold that is also important for `GeometricMachineLearning.jl` is the Lie group[^1] of orthonormal matrices ``SO(N)``. Before we can proof that ``SO(N)`` is a manifold we first need another definition and a theorem: -[^1]: Lie groups are manifolds that also have a *group structure*, i.e. there is an operation $\mathcal{M}\times\mathcal{M}\to\mathcal{M},(a,b)\mapsto{}ab$ s.t. $(ab)c = a(bc)$ and $\exists{}e\mathcal{M}$ s.t. $ae$ = $a$ $\forall{}a\in\mathcal{M}$. +[^1]: Lie groups are manifolds that also have a *group structure*, i.e. there is an operation ``\mathcal{M}\times\mathcal{M}\to\mathcal{M},(a,b)\mapsto{}ab`` s.t. ``(ab)c = a(bc)`` and ``\exists{}e\mathcal{M}`` s.t. ``ae`` = ``a`` ``\forall{}a\in\mathcal{M}``. -__Definition__: Consider a smooth mapping $g: \mathcal{M}\to\mathcal{N}$ from one manifold to another. A point $B\in\mathcal{M}$ is called a **regular value** of $\mathcal{M}$ if $\forall{}A\in{}g^{-1}\{B\}$ the map $T_Ag:T_A\mathcal{M}\to{}T_{g(A)}\mathcal{N}$ is surjective. +__Definition__: Consider a smooth mapping ``g: \mathcal{M}\to\mathcal{N}`` from one manifold to another. A point ``B\in\mathcal{M}`` is called a **regular value** of ``\mathcal{M}`` if ``\forall{}A\in{}g^{-1}\{B\}`` the map ``T_Ag:T_A\mathcal{M}\to{}T_{g(A)}\mathcal{N}`` is surjective. + +__Theorem__: Consider a smooth map ``g:\mathcal{M}\to\mathcal{N}`` from one manifold to another. Then the preimage of a regular point ``B`` of ``\mathcal{N}`` is a submanifold of ``\mathcal{M}``. Furthermore the codimension of ``g^{-1}\{B\}`` is equal to the dimension of ``\mathcal{N}`` and the tangent space ``T_A(g^{-1}\{B\})`` is equal to the kernel of ``T_Ag``. This is known as the **preimage theorem**. -__Theorem__: Consider a smooth map $g:\mathcal{M}\to\mathcal{N}$ from one manifold into another. Then the preimage of a regular point $B$ of $\mathcal{N}$ is a submanifold of $\mathcal{M}$. Furthermore the codimension of $g^{-1}\{B\}$ is equal to the dimension of $\mathcal{N}$. This is known as the **preimage theorem**. __Proof__: -__Theorem__: The group $SO(N)$ is a Lie group (i.e. has manifold structure). -__Proof__: The vector space $\mathbb{R}^{N\times{}N}$ clearly has manifold structure. The group $SO(N)$ is equivalent to one of the level sets of the mapping: $f:\mathbb{R}^{N\times{}N}\to\mathcal{S}(N), A\mapsto{}A^TA$, i.e. it is the component of $f^{-1}\{\mathbb{I}\}$ that contains $\mathbb{I}$. We still need to proof that $T_Af$ is surjective, i.e. $\forall{}B\in\mathcal{S}(N), A\in\mathbb{R}^{N\times{}N}$ $\exists{}C\in\mathbb{R}^{N\timesN}$ s.t. $C^TA + A^TC = B$. +__Theorem__: The group ``SO(N)`` is a Lie group (i.e. has manifold structure). +__Proof__: The vector space ``\mathbb{R}^{N\times{}N}`` clearly has manifold structure. The group ``SO(N)`` is equivalent to one of the level sets of the mapping: ``f:\mathbb{R}^{N\times{}N}\to\mathcal{S}(N), A\mapsto{}A^TA``, i.e. it is the component of ``f^{-1}\{\mathbb{I}\}`` that contains ``\mathbb{I}``. We still need to proof that ``\mathbb{I}`` is a regular point of ``f``, i.e. that for ``A\in{}SO(N)`` the mapping ``T_Af`` is surjective. This means that ``\forall{}B\in\mathcal{S}(N), A\in\mathbb{R}^{N\times{}N}`` ``\exists{}C\in\mathbb{R}^{N\times{}N}`` s.t. ``C^TA + A^TC = B``. The element ``C=\frac{1}{2}AB\in\mathcal{R}^{N\times{}N}`` satisfies this property. + +With the definition above we can generalize the notion of an ordinary differential equation (ODE) on a vector space to an ordinary differential equation on a manifold: -With the definition above we can generalize the notion of an ordinary differential equation on a vector space to an ordinary differential equation on a manifold. \ No newline at end of file +__Definition__: An **ODE on a manifold** is a mapping that assigns to each element of the manifold ``A\in\mathcal{M}`` an element of the corresponding tangent space ``T_A\mathcal{M}``. \ No newline at end of file From 3d4741202ffaffe791f13241a9ae33992143f9dd Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Fri, 24 Nov 2023 16:02:42 +0100 Subject: [PATCH 09/27] Added file describing basic topological concepts needed for manifolds. --- docs/src/manifolds/basic_topology.md | 29 ++++++++++++++++++++++++++++ 1 file changed, 29 insertions(+) create mode 100644 docs/src/manifolds/basic_topology.md diff --git a/docs/src/manifolds/basic_topology.md b/docs/src/manifolds/basic_topology.md new file mode 100644 index 000000000..4cd5912d4 --- /dev/null +++ b/docs/src/manifolds/basic_topology.md @@ -0,0 +1,29 @@ +# Basic Notions of Topology + +On this page we discuss basic notions of topology that are necessary to define [manifolds](manifolds.md). Here we largely omit proofs and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. + +[^1]: Some authors (see e.g. [lang2012fundamentals](@cite)) do not require these properties. But since they constitute very weak restrictions and are always satisfied by the manifolds relevant for our purposes. Thus we require them here. + +__Definition__: A **topological space** is a set ``\mathcal{M}`` for which we define a collection of subsets of ``\mathcal{M}``, which we denote by ``\mathcal{T}`` and call the *open subsets*. ``\mathcal{T}`` further has to satisfy the following three conditions: +1. The empty set and ``\mathcal{M}`` belong to ``\mathcal{T}``. +2. Any union of an arbitrary number of elements of ``\mathcal{T}`` again belongs to ``\mathcal{T}``. +3. Any intersection of a finite number of elements of ``\mathcal{T}`` again belongs to ``\mathcal{T}``. + +Based on this definition of a topological space we can now define what it means to be *Hausdorff*: +__Definition__: A topological space ``\mathcal{M}`` is said to be **Hausdorff** if for any two points ``x,y\in\mathcal{M}`` we can find two open sets ``U_x,U_y\in\mathcal{T}`` s.t. ``x\in{}U_x, y\in{}U_y`` and ``U_x\cap{}U_y=\{\}``. + +Lastly we define the notion of *second countability*: +__Definition__: A topological space ``\mathcal{M}`` is said to be **second-countable** if we can find a countable subcollection of ``\mathcal{T}`` called ``\mathcal{U}`` s.t. ``\forall{}U\in\mathcal{T}`` and ``x\in{}U`` we can find an element ``V\in\mathcal{U}`` for which ``x\in{}V\sub{}U``. + + +## References +- General Topology book. +- Lang + +@book{lang2012fundamentals, + title={Fundamentals of differential geometry}, + author={Lang, Serge}, + volume={191}, + year={2012}, + publisher={Springer Science \& Business Media} +} From 8d6bc4af0de538cc6b2303612c6638235452011a Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Fri, 24 Nov 2023 16:05:50 +0100 Subject: [PATCH 10/27] Started basic description of NN optimizers. --- docs/src/optimizers/general_optimization.md | 12 ++++++++++++ 1 file changed, 12 insertions(+) create mode 100644 docs/src/optimizers/general_optimization.md diff --git a/docs/src/optimizers/general_optimization.md b/docs/src/optimizers/general_optimization.md new file mode 100644 index 000000000..325129190 --- /dev/null +++ b/docs/src/optimizers/general_optimization.md @@ -0,0 +1,12 @@ +# Optimization for Neural Networks + +Optimization for neural networks is (almost always) some variation on gradient descent. The most basic form of gradient descent is a discretization of the *gradient flow equation*: +```math +\dot{\theta} = -\nabla_\theta{}L, +``` +by means of a Euler time-stepping scheme: +```math +\theta^{t+1} = \theta^{t} - h\nabla_{\theta^{t}}L, +``` +where $\eta$ (the time step of the Euler scheme) is referred to as the *learning rate* +This equation can easily be generalized to [manifolds](manifolds.md) by replacing the *Euclidean gradient* $\nabla_{\theta^{t}L}$ by a *Riemannian gradient* $-h\mathrm{grad}_{\theta^{t}}L$ and addition by $-h\nabla_{\theta^{t}}L$ with a [retraction](retraction.md) by $-h\mathrm{grad}_{\theta^{t}}L$. \ No newline at end of file From 9e9bd258fb5b2d9949a30bdbad51f76edaf017e7 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Sun, 26 Nov 2023 18:02:28 +0100 Subject: [PATCH 11/27] Added files on the existence and uniqueness theorem, the inverse function theorem and the sumersion theorem. The inverse function theorem file is essentially done whereas the existence and uniqueness theorem as well as the submersion theorem have not really been started. Also expaned on the basic notions of topology section and added my favourite reference. --- docs/src/manifolds/basic_topology.md | 52 +++++++++++++++++-- .../existence_and_uniqueness_theorem.md | 24 +++++++++ .../src/manifolds/inverse_function_theorem.md | 29 +++++++++++ docs/src/manifolds/submersion_theorem.md | 3 ++ 4 files changed, 104 insertions(+), 4 deletions(-) create mode 100644 docs/src/manifolds/existence_and_uniqueness_theorem.md create mode 100644 docs/src/manifolds/inverse_function_theorem.md create mode 100644 docs/src/manifolds/submersion_theorem.md diff --git a/docs/src/manifolds/basic_topology.md b/docs/src/manifolds/basic_topology.md index 4cd5912d4..53bbc9355 100644 --- a/docs/src/manifolds/basic_topology.md +++ b/docs/src/manifolds/basic_topology.md @@ -1,8 +1,8 @@ -# Basic Notions of Topology +# Basic Concepts of General Topology -On this page we discuss basic notions of topology that are necessary to define [manifolds](manifolds.md). Here we largely omit proofs and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. +On this page we discuss basic notions of topology that are necessary to define and work [manifolds](manifolds.md). Here we largely omit concrete examples and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. For a wide range of examples and a detailed discussion of the theory see e.g. [lipschutz1965general](@cite). The here-presented theory is also (rudimentary) covered in most differential geometry books such as [lang2012](@cite) and [bishop2012tensor](@cite). -[^1]: Some authors (see e.g. [lang2012fundamentals](@cite)) do not require these properties. But since they constitute very weak restrictions and are always satisfied by the manifolds relevant for our purposes. Thus we require them here. +[^1]: Some authors (see e.g. [lang2012fundamentals](@cite)) do not require these properties. But since they constitute very weak restrictions and are always satisfied by the manifolds relevant for our purposes we require them here. __Definition__: A **topological space** is a set ``\mathcal{M}`` for which we define a collection of subsets of ``\mathcal{M}``, which we denote by ``\mathcal{T}`` and call the *open subsets*. ``\mathcal{T}`` further has to satisfy the following three conditions: 1. The empty set and ``\mathcal{M}`` belong to ``\mathcal{T}``. @@ -12,9 +12,45 @@ __Definition__: A **topological space** is a set ``\mathcal{M}`` for which we de Based on this definition of a topological space we can now define what it means to be *Hausdorff*: __Definition__: A topological space ``\mathcal{M}`` is said to be **Hausdorff** if for any two points ``x,y\in\mathcal{M}`` we can find two open sets ``U_x,U_y\in\mathcal{T}`` s.t. ``x\in{}U_x, y\in{}U_y`` and ``U_x\cap{}U_y=\{\}``. -Lastly we define the notion of *second countability*: +We now give the second definition that we need for defining manifolds, that of *second countability*: __Definition__: A topological space ``\mathcal{M}`` is said to be **second-countable** if we can find a countable subcollection of ``\mathcal{T}`` called ``\mathcal{U}`` s.t. ``\forall{}U\in\mathcal{T}`` and ``x\in{}U`` we can find an element ``V\in\mathcal{U}`` for which ``x\in{}V\sub{}U``. +We now give a few definitions and results that are needed for the [inverse function theorem](inverse_function_theorem.md) which is essential for practical applications of manifold theory. + +__Definition__: A mapping ``f`` between topological spaces ``\mathcal{M}`` and ``\mathcal{N}`` is called **continuous** if the preimage of every open set is again an open set, i.e. if ``f^{-1}\{U\}\in\mathcal{T}`` for ``U`` open in ``\mathcal{N}`` and ``\mathcal{T}`` the topology on ``\mathcal{M}``. + +__Definition__: A **closed set** of a topological space ``\mathcal{M}`` is one whose complement is an open set, i.e. ``F`` is closed if ``F^c\in\mathcal{T}``, where the superscript ``{}^c`` indicates the complement. For closed sets we thus have the following three properties: +1. The empty set and ``\mathcal{M}`` are closed sets. +2. Any union of a finite number of closed sets is again closed. +3. Any intersection of an arbitrary number of closed sets is again closed. + +__Theorem__: The definition of continuity is equivalent to the following, second definition: ``f:\mathcal{M}\to\mathcal{N}`` is continuous if ``f^{-1}\{F\}\sub\mathcal{M}`` is a closed set for each closed set ``F\sub\mathcal{N}``. + +__Proof__: First assume that ``f`` is continuous according to the first definition and not to the second. Then ``f^{-1}{F}`` is not closed but ``f^{-1}{F^c}`` is open. But ``f^{-1}\{F^c\} = \{x\in\mathcal{M}:f(x)\nin\mathcal{N}\} = (f^{-1}\{F\})^c`` cannot be open, else ``f^{-1}\{F\}`` would be closed. The implication of the first definition under assumption of the second can be shown analogously. + +__Theorem__: The property of a set ``F`` being closed is equivalent to the following statement: If a point ``y`` is such that for every open set ``U`` containing it we have ``U\cap{}F\neq\{\}`` then this point is contained in ``F``. + +__Proof__: We first proof that if a set is closed then the statement holds. Consider a closed set ``F`` and a point ``y\nin{}F`` s.t. every open set containing ``y`` has nonempty intersection with ``F``. But the complement ``F^c`` also is such a set, which is a clear contradiction. Now assume the above statement for a set ``F`` and further assume ``F`` is not closed. Its complement ``F^c`` is thus not open. Now consider the *interior* of this set: ``\mathrm{int}(F^c):=\cup\{U:U\sub{}F^c\}``, i.e. the biggest open set contained within ``F^c``. Hence there must be a point ``y`` which is in ``F^c`` but is not in its interior, else ``F^c`` would be equal to its interior, i.e. would be open. We further must be able to find an open set ``U`` that contains ``y`` but is also contained in ``F^c``, else ``y`` would be an element of ``F``. A contradiction. + +__Definition__: An **open cover** of a topological space ``\mathcal{M}`` is a (not necessarily countable) collection of open sets ``\{U_i\}_{i\mathcal{I}}`` s.t. their union contains ``\mathcal{M}``. A **finite open cover** is a collection of a finite number of open sets that cover ``\mathcal{M}``. We say that an open cover is **reducible** to a finite cover if we can find a finite number of elements in the open cover whose union still contains ``\mathcal{M}``. + +__Definition__: A topological space ``\mathcal{M}`` is called **compact** if every open cover is reducible to a finite cover. + +__Theorem__: Consider a continuous function ``f:\mathcal{M}\to\mathcal{N}`` and a compact set ``K\in\mathcal{M}``. Then ``f(K)`` is also compact. + +__Proof__: Consider an open cover of ``f(K)``: ``\{U_i\}_{i\in\mathcal{I}}``. Then ``\{f^{-1}\{U_i\}\}_{i\in\mathcal{I}}`` is an open cover of ``K`` and hence reducible to a finite cover ``\{f^{-1}\{U_i\}\}_{i\in\{i_1,\ldots,i_n}}``. But then ``\{{U_i\}_{i\in\{i_1,\ldots,i_n}}`` also covers ``f(K)``. + +__Theorem__: A closed subset of a compact space is compact: + +__Proof__: Call the closed set ``F`` and consider an open cover of this set: ``\{U\}_{i\in\mathcal{I}}``. Then this open cover combined with ``F^c`` is an open cover for the entire compact space, hence reducible to a finite cover. + +__Theorem__: A compact subset of a Hausdorff space is closed: + +__Proof__: Consider a compact subset ``K``. If ``K`` is not closed, then there has to be a point ``y\nin{}K`` s.t. every open set containing ``y`` intersects ``K``. Because the surrounding space is Hausdorff we can now find the following two collections of open sets: ``\{(U_z, U_{z,y}: U_z\cap{}U_{z,y}=\{\})\}_{z\in{}K}``. The open cover ``\{U_z}_{z\in{}K}`` is then reducible to a finite cover ``\{U_z}_{z\in{z_1, \ldots, z_n}\}``. The intersection ``\cap_{z\in{z_1, \ldots, z_n}}U_{z,y}`` is then an open set that contains ``y`` but has no intersection with ``K``. A contraction. + +__Theorem__: If ``\mathcal{M}`` is compact and ``\mathcal{N}`` is Hausdorff, then the inverse of a continuous function ``f:\mathcal{M}\to\mathcal{N}`` is again continuous, i.e. ``f(V)`` is an open set in ``\mathcal{N}`` for ``V\in\mathcal{T}``. + +__Proof__: We can equivalently show that every closed set is mapped to a closed set. First consider the set ``K\in\mathcal{M}``. Its image is again compact and hence closed because ``\mathcal{N}`` is Hausdorff. ## References - General Topology book. @@ -27,3 +63,11 @@ __Definition__: A topological space ``\mathcal{M}`` is said to be **second-count year={2012}, publisher={Springer Science \& Business Media} } + +@book{lipschutz1965general, + title={General Topology}, + author={Seymour Lipschutz}, + year={1965}, + publisher={McGraw-Hill Book Company}, + location={New York City, New York} +} \ No newline at end of file diff --git a/docs/src/manifolds/existence_and_uniqueness_theorem.md b/docs/src/manifolds/existence_and_uniqueness_theorem.md new file mode 100644 index 000000000..4aa89da45 --- /dev/null +++ b/docs/src/manifolds/existence_and_uniqueness_theorem.md @@ -0,0 +1,24 @@ +# The Existence-And-Uniqueness Theorem + +In order to proof the existence-and-uniqueness theorem we first need another theorem, the **Banach fixed-point theorem** for which we also need another definition. + +__Definition__: A **contraction mapping** is a map ``T:\mathbb{R}^N\to\mathbb{R}^N`` for which there exists ``q\in[0,1)`` s.t. ``\forall{}x,y\in\mathbb{R}^N,\,||T(x)-T(y)||\leq{}q||x-y||``. + +__Theorem (Banach fixed-point theorem)__: Every **contraction mapping** ``T`` admits a unique fixed point ``x^*`` (i.e. a point ``x^*`` s.t. ``F(x^*)=x^*``) and this point can be found by taking an arbitrary point ``x_0\in\mathbb{R}^N`` and taking the limit ``\lim_{n\to\infty}T^n(x_0)``. + +__Proof (Banach fixed-point theorem)__: Take an arbitrary point ``x_0\in\mathbb{R}^N`` and consider the sequence ``(x_n)_{n\in\mathbb{N}}`` with ``x_n:=T^n(x_0)``. Then it holds that (for ``m>n``): +```math +\begin{aligned} +|x_m - x_n| & \leq |x_m - x_{m-1}| + |x_{m-1} - x_{m-2}| + \cdots + |x_{m-(m-n+1)}-x_{n}| \\ + & = |x_{n+(m-n)} - x_{n+(m-n-1)}| + \cdots + |x_{n+1} - x_n| \\ + & \leq \sum_{i=0}^{m-n-1}q^i|x_{n+1} - x_n| \\ + & \leq \sum_{i=0}^{m-n-1}q^iq^n|x_1 - x_0| \\ + & = q^n|x_1 -x_0|\sum_{i=1}^{m-n-1}q^i, +\end{aligned} +``` +where we have used the triangle inequality in the first line. If we now let ``m`` on the right-hand side first go to infinity then we get +```math +|x_m-x_n| & \leq q^n|x_1 -x_0|\sum_{i=1}^{\infty}q^i + & =q^n|x_1 -x_0| \frac{1}{1-q}, +``` +proofing that the sequence is Cauchy. Because ``\mathbb{R}^N`` is a complete metric space we get that ``(x_n)_{n\in\mathbb{N}}`` is a convergent sequence. We call the limit of this sequence ``x^*``. This completes the proof of the Banach fixed-point theorem. diff --git a/docs/src/manifolds/inverse_function_theorem.md b/docs/src/manifolds/inverse_function_theorem.md new file mode 100644 index 000000000..4063c494e --- /dev/null +++ b/docs/src/manifolds/inverse_function_theorem.md @@ -0,0 +1,29 @@ +# The Inverse Function Theorem + +The **inverse function theorem** gives a sufficient condition on a vector-valued function to be invertible in a neighborhood of a specific point. This theorem is critical in developing a theory of [manifolds](manifolds.md) and serves as a basis for the [submersion theorem](submersion_theorem.md). Here we first state the theorem and then give a proof. + +__Theorem (Inverse function theorem)__: Consider a vector-valued differentiable function ``F:\mathbb{R}^N\to\mathbb{R}^N`` and assume its Jacobian is non-degenerate at a point ``x\in\mathbb{R}^N``. Then there exists a neighborhood ``U`` that contains ``F(x)`` and on which ``F`` is invertible, i.e. ``\exists{}H:U\to\mathbb{R}^N`` s.t. ``\forall{}y\inU,\,F\circ{}H(y) = y`` and the inverse is differentiable. + +__Proof__: Consider a mapping ``F:\mathbb{R}^N\to\mathbb{R}^N`` and assume its Jacobian has full rank at point ``x``, i.e. ``\det{}F'(x)\neq0``. Now consider a ball around ``x`` whose radius ``r`` we do not yet fix and two points ``y`` and ``z`` in that ball: ``y,z\in{}B(x,r)``. We further introduce the function ``G(y):=F(x)-F'(x)y``. By the *mean value theorem* we have ``|G(z) - G(y)|\leq|z-y|\sup_{0 Date: Mon, 27 Nov 2023 12:46:54 +0100 Subject: [PATCH 12/27] Added the right reference for the Bishop book at the bottom. --- docs/src/manifolds/basic_topology.md | 11 ++++++++++- 1 file changed, 10 insertions(+), 1 deletion(-) diff --git a/docs/src/manifolds/basic_topology.md b/docs/src/manifolds/basic_topology.md index 53bbc9355..14894909e 100644 --- a/docs/src/manifolds/basic_topology.md +++ b/docs/src/manifolds/basic_topology.md @@ -1,6 +1,6 @@ # Basic Concepts of General Topology -On this page we discuss basic notions of topology that are necessary to define and work [manifolds](manifolds.md). Here we largely omit concrete examples and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. For a wide range of examples and a detailed discussion of the theory see e.g. [lipschutz1965general](@cite). The here-presented theory is also (rudimentary) covered in most differential geometry books such as [lang2012](@cite) and [bishop2012tensor](@cite). +On this page we discuss basic notions of topology that are necessary to define and work [manifolds](manifolds.md). Here we largely omit concrete examples and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. For a wide range of examples and a detailed discussion of the theory see e.g. [lipschutz1965general](@cite). The here-presented theory is also (rudimentary) covered in most differential geometry books such as [lang2012](@cite) and [bishop1980tensor](@cite). [^1]: Some authors (see e.g. [lang2012fundamentals](@cite)) do not require these properties. But since they constitute very weak restrictions and are always satisfied by the manifolds relevant for our purposes we require them here. @@ -55,6 +55,7 @@ __Proof__: We can equivalently show that every closed set is mapped to a closed ## References - General Topology book. - Lang +- Bishop @book{lang2012fundamentals, title={Fundamentals of differential geometry}, @@ -70,4 +71,12 @@ __Proof__: We can equivalently show that every closed set is mapped to a closed year={1965}, publisher={McGraw-Hill Book Company}, location={New York City, New York} +} + +@book{bishop1980tensor, + title={Tensor Analysis on Manifolds}, + author={Richard L. Bishop, Samuel I. Goldberg}, + year={1980}, + publisher={Dover Publications}, + location={Mineola, New York} } \ No newline at end of file From a9a642999331b693a2f7fbdcf164160da5712a87 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 4 Dec 2023 18:35:50 +0100 Subject: [PATCH 13/27] Added the basic topological and analysis files as well as the bsfg optimizer. --- docs/make.jl | 11 ++++++++++- 1 file changed, 10 insertions(+), 1 deletion(-) diff --git a/docs/make.jl b/docs/make.jl index e71674e4e..555e6f329 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -27,14 +27,22 @@ makedocs(; "SympNet" => "architectures/sympnet.md", ], "Manifolds" => [ + "Concepts from General Topology" => "manifolds/basic_topology.md", + "General Theory on Manifolds" => "manifolds/manifolds.md", + "The Inverse Function Theorem" => "manifolds/inverse_function_theorem.md", + "The Submersion Theorem" => "manifolds/submersion_theorem.md", "Homogeneous Spaces" => "manifolds/homogeneous_spaces.md", "Stiefel" => "manifolds/stiefel_manifold.md", "Grassmann" => "manifolds/grassmann_manifold.md", + "Differential Equations and the EAU theorem" => "manifolds/existence_and_uniqueness_theorem.md", ], "Arrays" => [ "Global Tangent Space" => "arrays/stiefel_lie_alg_horizontal.md", ], - "Optimizer Framework" => "Optimizer.md", + "Optimizer Framework" => [ + "Optimizers" => "Optimizer.md", + "General Optimization" => "optimizers/general_optimization.md", + ], "Optimizer Functions" => [ "Horizontal Lift" => "optimizers/manifold_related/horizontal_lift.md", "Global Sections" => "optimizers/manifold_related/global_sections.md", @@ -42,6 +50,7 @@ makedocs(; "Geodesic Retraction" => "optimizers/manifold_related/geodesic.md", "Cayley Retraction" => "optimizers/manifold_related/cayley.md", "Adam Optimizer" => "optimizers/adam_optimizer.md", + "BFGS Optimizer" => "optimizers/bfgs_optimizer.md", ], "Special Neural Network Layers" => [ "Attention" => "layers/attention_layer.md", From fca4061f0b7152e995d513216828be50f31d7c41 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 4 Dec 2023 18:36:13 +0100 Subject: [PATCH 14/27] Added new references. --- docs/src/GeometricMachineLearning.bib | 113 ++++++++++++++++++++++++++ 1 file changed, 113 insertions(+) diff --git a/docs/src/GeometricMachineLearning.bib b/docs/src/GeometricMachineLearning.bib index 9f711ef09..67cabf9f1 100644 --- a/docs/src/GeometricMachineLearning.bib +++ b/docs/src/GeometricMachineLearning.bib @@ -38,3 +38,116 @@ @book{leimkuhler2004simulating year={2004}, publisher={Cambridge university press} } + +@book{lang2012fundamentals, + title={Fundamentals of differential geometry}, + author={Lang, Serge}, + volume={191}, + year={2012}, + publisher={Springer Science \& Business Media} +} + +@book{lipschutz1965general, + title={General Topology}, + author={Seymour Lipschutz}, + year={1965}, + publisher={McGraw-Hill Book Company}, + location={New York City, New York} +} + +@book{bishop1980tensor, + title={Tensor Analysis on Manifolds}, + author={Richard L. Bishop, Samuel I. Goldberg}, + year={1980}, + publisher={Dover Publications}, + location={Mineola, New York} +} + +@book{wright2006numerical, + title={Numerical optimization}, + author={Stephen J. Wright, Jorge Nocedal}, + year={2006}, + publisher={Springer Science+Business Media}, + location={New York, NY} +} + +@article{fresca2021comprehensive, + title={A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs}, + author={Fresca, Stefania and Dede’, Luca and Manzoni, Andrea}, + journal={Journal of Scientific Computing}, + volume={87}, + pages={1--36}, + year={2021}, + publisher={Springer} +} + +@article{buchfink2023symplectic, + title={Symplectic model reduction of Hamiltonian systems on nonlinear manifolds and approximation with weakly symplectic autoencoder}, + author={Buchfink, Patrick and Glas, Silke and Haasdonk, Bernard}, + journal={SIAM Journal on Scientific Computing}, + volume={45}, + number={2}, + pages={A289--A311}, + year={2023}, + publisher={SIAM} +} + +@article{peng2016symplectic, + title={Symplectic model reduction of Hamiltonian systems}, + author={Peng, Liqian and Mohseni, Kamran}, + journal={SIAM Journal on Scientific Computing}, + volume={38}, + number={1}, + pages={A1--A27}, + year={2016}, + publisher={SIAM} +} + +@article{luong2015effective, + title={Effective approaches to attention-based neural machine translation}, + author={Luong, Minh-Thang and Pham, Hieu and Manning, Christopher D}, + journal={arXiv preprint arXiv:1508.04025}, + year={2015} +} + +@article{bahdanau2014neural, + title={Neural machine translation by jointly learning to align and translate}, + author={Bahdanau, Dzmitry and Cho, Kyunghyun and Bengio, Yoshua}, + journal={arXiv preprint arXiv:1409.0473}, + year={2014} +} + +@article{greif2019decay, + title={Decay of the Kolmogorov N-width for wave problems}, + author={Greif, Constantin and Urban, Karsten}, + journal={Applied Mathematics Letters}, + volume={96}, + pages={216--222}, + year={2019}, + publisher={Elsevier} +} + +@article{blickhan2023registration, + title={A registration method for reduced basis problems using linear optimal transport}, + author={Blickhan, Tobias}, + journal={arXiv preprint arXiv:2304.14884}, + year={2023} +} + +@article{lee2020model, + title={Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders}, + author={Lee, Kookjin and Carlberg, Kevin T}, + journal={Journal of Computational Physics}, + volume={404}, + pages={108973}, + year={2020}, + publisher={Elsevier} +} + +@article{vaswani2017attention, + title={Attention is all you need}, + author={Vaswani, Ashish and Shazeer, Noam and Parmar, Niki and Uszkoreit, Jakob and Jones, Llion and Gomez, Aidan N and Kaiser, Lukasz and Polosukhin, Illia}, + journal={Advances in neural information processing systems}, + volume={30}, + year={2017} +} From 4f1cba97d4afacfd6ad8cc85a569e56aacff4f42 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 4 Dec 2023 18:37:16 +0100 Subject: [PATCH 15/27] Fixed references in files. They are now all called with @bibliography. --- docs/src/Optimizer.md | 10 ++++- docs/src/layers/attention_layer.md | 10 ++++- docs/src/layers/multihead_attention_layer.md | 8 +++- docs/src/manifolds/basic_topology.md | 38 +++++-------------- .../src/manifolds/inverse_function_theorem.md | 13 +++---- docs/src/optimizers/bfgs_optimizer.md | 14 +++---- docs/src/optimizers/general_optimization.md | 2 +- .../src/reduced_order_modeling/autoencoder.md | 6 ++- .../kolmogorov_n_width.md | 12 ++++-- .../symplectic_autoencoder.md | 10 ++++- 10 files changed, 69 insertions(+), 54 deletions(-) diff --git a/docs/src/Optimizer.md b/docs/src/Optimizer.md index f7cbe5fae..2b6005a28 100644 --- a/docs/src/Optimizer.md +++ b/docs/src/Optimizer.md @@ -4,6 +4,8 @@ In order to generalize neural network optimizers to [homogeneous spaces](manifol Starting from an element of the tangent space $T_Y\mathcal{M}$[^1], we need to perform two mappings to arrive at $\mathfrak{g}^\mathrm{hor}$, which we refer to by $\Omega$ and a red horizontal arrow: +[^1]: In practice this is obtained by first using an AD routine on a loss function $L$, and then computing the Riemannian gradient based on this. See the section of the [Stiefel manifold](manifolds/stiefel_manifold.md) for an example of this. + ![](tikz/general_optimization_with_boundary.png) Here the mapping $\Omega$ is a [horizontal lift](optimizers/manifold_related/horizontal_lift.md) from the tangent space onto the **horizontal component of the Lie algebra at $Y$**. @@ -13,6 +15,10 @@ The red line maps the horizontal component at $Y$, i.e. $\mathfrak{g}^{\mathrm{h The $\mathrm{cache}$ stores information about previous optimization steps and is dependent on the optimizer. The elements of the $\mathrm{cache}$ are also in $\mathfrak{g}^\mathrm{hor}$. Based on this the optimer ([Adam](optimizers/adam_optimizer.md) in this case) computes a final velocity, which is the input of a [retraction](optimizers/manifold_related/retractions.md). Because this *update* is done for $\mathfrak{g}^{\mathrm{hor}}\equiv{}T_Y\mathcal{M}$, we still need to perform a mapping, called `apply_section` here, that then finally updates the network parameters. The two red lines are described in [global sections](optimizers/manifold_related/global_sections.md). ## References -- Brantner B. Generalizing Adam To Manifolds For Efficiently Training Transformers[J]. arXiv preprint arXiv:2305.16901, 2023. -[^1]: In practice this is obtained by first using an AD routine on a loss function $L$, and then computing the Riemannian gradient based on this. See the section of the [Stiefel manifold](manifolds/stiefel_manifold.md) for an example of this. \ No newline at end of file +```@bibliography +Pages = [] +Canonical = false + +brantner2023generalizing +``` \ No newline at end of file diff --git a/docs/src/layers/attention_layer.md b/docs/src/layers/attention_layer.md index f72adc979..fce2da1a5 100644 --- a/docs/src/layers/attention_layer.md +++ b/docs/src/layers/attention_layer.md @@ -84,5 +84,11 @@ Attention was used before, but always in connection with **recurrent neural netw ## References -- Luong M T, Pham H, Manning C D. Effective approaches to attention-based neural machine translation[J]. arXiv preprint arXiv:1508.04025, 2015. -- Bahdanau D, Cho K, Bengio Y. Neural machine translation by jointly learning to align and translate[J]. arXiv preprint arXiv:1409.0473, 2014. \ No newline at end of file + +```@bibliography +Pages = [] +Canonical = false + +bahdanau2014neural +luong2015effective +``` \ No newline at end of file diff --git a/docs/src/layers/multihead_attention_layer.md b/docs/src/layers/multihead_attention_layer.md index d117ba4ea..c97193316 100644 --- a/docs/src/layers/multihead_attention_layer.md +++ b/docs/src/layers/multihead_attention_layer.md @@ -51,4 +51,10 @@ Because the main task of the $W_i^V$, $W_i^K$ and $W_i^Q$ matrices here is for t ## References -- Vaswani, Ashish, et al. "Attention is all you need." Advances in neural information processing systems 30 (2017). \ No newline at end of file + +```@bibliography +Pages = [] +Canonical = false + +vaswani2017attention +``` \ No newline at end of file diff --git a/docs/src/manifolds/basic_topology.md b/docs/src/manifolds/basic_topology.md index 14894909e..9537f97ce 100644 --- a/docs/src/manifolds/basic_topology.md +++ b/docs/src/manifolds/basic_topology.md @@ -1,6 +1,6 @@ # Basic Concepts of General Topology -On this page we discuss basic notions of topology that are necessary to define and work [manifolds](manifolds.md). Here we largely omit concrete examples and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. For a wide range of examples and a detailed discussion of the theory see e.g. [lipschutz1965general](@cite). The here-presented theory is also (rudimentary) covered in most differential geometry books such as [lang2012](@cite) and [bishop1980tensor](@cite). +On this page we discuss basic notions of topology that are necessary to define and work [manifolds](manifolds.md). Here we largely omit concrete examples and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. For a wide range of examples and a detailed discussion of the theory see e.g. [lipschutz1965general](@cite). The here-presented theory is also (rudimentary) covered in most differential geometry books such as [lang2012fundamentals](@cite) and [bishop1980tensor](@cite). [^1]: Some authors (see e.g. [lang2012fundamentals](@cite)) do not require these properties. But since they constitute very weak restrictions and are always satisfied by the manifolds relevant for our purposes we require them here. @@ -53,30 +53,12 @@ __Theorem__: If ``\mathcal{M}`` is compact and ``\mathcal{N}`` is Hausdorff, the __Proof__: We can equivalently show that every closed set is mapped to a closed set. First consider the set ``K\in\mathcal{M}``. Its image is again compact and hence closed because ``\mathcal{N}`` is Hausdorff. ## References -- General Topology book. -- Lang -- Bishop - -@book{lang2012fundamentals, - title={Fundamentals of differential geometry}, - author={Lang, Serge}, - volume={191}, - year={2012}, - publisher={Springer Science \& Business Media} -} - -@book{lipschutz1965general, - title={General Topology}, - author={Seymour Lipschutz}, - year={1965}, - publisher={McGraw-Hill Book Company}, - location={New York City, New York} -} - -@book{bishop1980tensor, - title={Tensor Analysis on Manifolds}, - author={Richard L. Bishop, Samuel I. Goldberg}, - year={1980}, - publisher={Dover Publications}, - location={Mineola, New York} -} \ No newline at end of file + +```@bibliography +Pages = [] +Canonical = false + +bishop1980tensor +lang2012fundamentals +lipschutz1965general +``` \ No newline at end of file diff --git a/docs/src/manifolds/inverse_function_theorem.md b/docs/src/manifolds/inverse_function_theorem.md index 4063c494e..cc13d3927 100644 --- a/docs/src/manifolds/inverse_function_theorem.md +++ b/docs/src/manifolds/inverse_function_theorem.md @@ -20,10 +20,9 @@ and this goes to zero as ``\eta`` goes to zero, because ``H`` is continuous and ## References -@book{lang2012fundamentals, - title={Fundamentals of differential geometry}, - author={Lang, Serge}, - volume={191}, - year={2012}, - publisher={Springer Science \& Business Media} -} \ No newline at end of file +```@bibliography +Pages = [] +Canonical = false + +lang2012fundamentals +``` \ No newline at end of file diff --git a/docs/src/optimizers/bfgs_optimizer.md b/docs/src/optimizers/bfgs_optimizer.md index b7bcd6e3a..2f319c822 100644 --- a/docs/src/optimizers/bfgs_optimizer.md +++ b/docs/src/optimizers/bfgs_optimizer.md @@ -128,10 +128,10 @@ H_{k} = H_{k-1} - \frac{H_{k-1}y_{k-1}y_{k-1}^TH_{k-1}}{y_{k-1}^TH_{k-1}y_{k-1}} TODO: Example where this works well! ## References -@book{wright2006numerical, - title={Numerical optimization}, - author={Stephen J. Wright, Jorge Nocedal}, - year={2006}, - publisher={Springer Science+Business Media}, - location={New York, NY} -} \ No newline at end of file + +```@biblioraphy +Pages = [] +Canonical = false + +wright2006numerical +``` \ No newline at end of file diff --git a/docs/src/optimizers/general_optimization.md b/docs/src/optimizers/general_optimization.md index 325129190..13f1f43d7 100644 --- a/docs/src/optimizers/general_optimization.md +++ b/docs/src/optimizers/general_optimization.md @@ -9,4 +9,4 @@ by means of a Euler time-stepping scheme: \theta^{t+1} = \theta^{t} - h\nabla_{\theta^{t}}L, ``` where $\eta$ (the time step of the Euler scheme) is referred to as the *learning rate* -This equation can easily be generalized to [manifolds](manifolds.md) by replacing the *Euclidean gradient* $\nabla_{\theta^{t}L}$ by a *Riemannian gradient* $-h\mathrm{grad}_{\theta^{t}}L$ and addition by $-h\nabla_{\theta^{t}}L$ with a [retraction](retraction.md) by $-h\mathrm{grad}_{\theta^{t}}L$. \ No newline at end of file +This equation can easily be generalized to [manifolds](../manifolds/manifolds.md) by replacing the *Euclidean gradient* $\nabla_{\theta^{t}L}$ by a *Riemannian gradient* $-h\mathrm{grad}_{\theta^{t}}L$ and addition by $-h\nabla_{\theta^{t}}L$ with a [retraction](../optimizers/manifold_related/retractions.md) by $-h\mathrm{grad}_{\theta^{t}}L$. \ No newline at end of file diff --git a/docs/src/reduced_order_modeling/autoencoder.md b/docs/src/reduced_order_modeling/autoencoder.md index 044cede0b..50da204dc 100644 --- a/docs/src/reduced_order_modeling/autoencoder.md +++ b/docs/src/reduced_order_modeling/autoencoder.md @@ -46,5 +46,9 @@ In reduced order modeling we aim to construct a mapping to a space that is close The third step can be done with various machine learning (ML) techniques. Traditionally the most popular of these has been **Proper orthogonal decomposition** (POD), but in recent years **autoencoders** have also become a popular alternative (see (Fresca et al, 2021)). ## References +```@bibliography +Pages = [] +Canonical = false -- Fresca, Stefania, Luca Dede, and Andrea Manzoni. "A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs." Journal of Scientific Computing 87 (2021): 1-36. \ No newline at end of file +fresca2021comprehensive +``` \ No newline at end of file diff --git a/docs/src/reduced_order_modeling/kolmogorov_n_width.md b/docs/src/reduced_order_modeling/kolmogorov_n_width.md index 65595990b..f97cb1898 100644 --- a/docs/src/reduced_order_modeling/kolmogorov_n_width.md +++ b/docs/src/reduced_order_modeling/kolmogorov_n_width.md @@ -12,6 +12,12 @@ In order to overcome this, techniques based on neural networks (see e.g. (Lee an ## References -- Greif, Constantin, and Karsten Urban. "Decay of the Kolmogorov N-width for wave problems." Applied Mathematics Letters 96 (2019): 216-222. -- Blickhan, Tobias. "A registration method for reduced basis problems using linear optimal transport." arXiv preprint arXiv:2304.14884 (2023). -- Lee, Kookjin, and Kevin T. Carlberg. "Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders." Journal of Computational Physics 404 (2020): 108973. + +```@bibliography +Pages = [] +Canonical = false + +blickhan2023registration +greif2019decay +lee2020model +``` \ No newline at end of file diff --git a/docs/src/reduced_order_modeling/symplectic_autoencoder.md b/docs/src/reduced_order_modeling/symplectic_autoencoder.md index 3d5d40a97..2f35fb92d 100644 --- a/docs/src/reduced_order_modeling/symplectic_autoencoder.md +++ b/docs/src/reduced_order_modeling/symplectic_autoencoder.md @@ -83,5 +83,11 @@ So we alternate between SympNet and PSD layers. Because all the PSD layers are b ## References -- Buchfink, Patrick, Silke Glas, and Bernard Haasdonk. "Symplectic model reduction of Hamiltonian systems on nonlinear manifolds and approximation with weakly symplectic autoencoder." SIAM Journal on Scientific Computing 45.2 (2023): A289-A311. -- Peng, Liqian, and Kamran Mohseni. "Symplectic model reduction of Hamiltonian systems." SIAM Journal on Scientific Computing 38.1 (2016): A1-A27. \ No newline at end of file + +```@bibliography +Pages = [] +Canonical = false + +buchfink2023symplectic +peng2016symplectic +``` \ No newline at end of file From 6b274ef47576b84497e4c2d633e71290ffe1b691 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 4 Dec 2023 18:37:27 +0100 Subject: [PATCH 16/27] Fixed typo (space after #). --- docs/src/references.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/src/references.md b/docs/src/references.md index a967e234b..0f5214267 100644 --- a/docs/src/references.md +++ b/docs/src/references.md @@ -1,4 +1,4 @@ -#References +# References ```@bibliography ``` \ No newline at end of file From 981b04b66805f1940c389226e1592109afabfc97 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 4 Dec 2023 18:38:42 +0100 Subject: [PATCH 17/27] Fixed typo. --- docs/src/optimizers/bfgs_optimizer.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/src/optimizers/bfgs_optimizer.md b/docs/src/optimizers/bfgs_optimizer.md index 2f319c822..203021f3a 100644 --- a/docs/src/optimizers/bfgs_optimizer.md +++ b/docs/src/optimizers/bfgs_optimizer.md @@ -129,7 +129,7 @@ TODO: Example where this works well! ## References -```@biblioraphy +```@bibliography Pages = [] Canonical = false From 9ae14b8ae251831a4fbfed165a0fc92725971701 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Tue, 5 Dec 2023 10:21:23 +0100 Subject: [PATCH 18/27] Fixed yaml file in branch documentation_for_optimizer_yaml_test. --- .github/workflows/Documenter.yml | 30 ++++++++++++------------------ 1 file changed, 12 insertions(+), 18 deletions(-) diff --git a/.github/workflows/Documenter.yml b/.github/workflows/Documenter.yml index e3df5d1d8..c2a50b165 100644 --- a/.github/workflows/Documenter.yml +++ b/.github/workflows/Documenter.yml @@ -4,7 +4,6 @@ on: push: branches: - main - tags: '*' pull_request: jobs: @@ -13,31 +12,26 @@ jobs: runs-on: ubuntu-latest steps: - uses: actions/checkout@v3 + - run: | + sudo apt-get install imagemagick + sudo apt-get install poppler-utils + sudo apt-get install texlive-xetex + sudo apt-get install texlive-science + make all -C docs/src/tikz - uses: julia-actions/setup-julia@latest - - name: Install dependencies - run: | + - run: | julia --project=docs -e ' using Pkg Pkg.develop(PackageSpec(path=pwd())) Pkg.instantiate() Pkg.build() - Pkg.precompile()' - sudo apt-get install imagemagick - sudo apt-get install poppler-utils - sudo apt-get install texlive-xetex - sudo apt-get install texlive-science - -name: Generate tikz pictures - run: | - make all -C src/tikz - - name: Run doctests - run: | - julia --project=docs -e ' + Pkg.precompile() using Documenter: doctest using GeometricMachineLearning doctest(GeometricMachineLearning)' - - name: Build and deploy Documentation - run: julia --project make.jl - working-directory: docs + julia --project=docs docs/make.jl + - uses: julia-actions/julia-buildpkg@v1 + - uses: julia-actions/julia-docdeploy@v1 env: GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} - DOCUMENTER_KEY: ${{ secrets.DOCUMENTER_KEY }} + DOCUMENTER_KEY: ${{ secrets.DOCUMENTER_KEY }} \ No newline at end of file From 883706ab97a76c2ee1dd2d8dba0848930d87fd51 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Tue, 5 Dec 2023 10:22:17 +0100 Subject: [PATCH 19/27] Added package dependencies. --- docs/Project.toml | 17 +++++++++++++++++ 1 file changed, 17 insertions(+) diff --git a/docs/Project.toml b/docs/Project.toml index 1dec04c32..ed78630a5 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -1,4 +1,21 @@ [deps] +AbstractNeuralNetworks = "60874f82-5ada-4c70-bd1c-fa6be7711c8a" +BandedMatrices = "aae01518-5342-5314-be14-df237901396f" +ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4" +Distances = "b4f34e82-e78d-54a5-968a-f98e89d6e8f7" Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" +DocumenterCitations = "daee34ce-89f3-4625-b898-19384cb65244" +ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210" +GeometricBase = "9a0b12b7-583b-4f04-aa1f-d8551b6addc9" +GeometricEquations = "c85262ba-a08a-430a-b926-d29770767bf2" +GeometricIntegrators = "dcce2d33-59f6-5b8d-9047-0defad88ae06" GeometricMachineLearning = "194d25b2-d3f5-49f0-af24-c124f4aa80cc" +InteractiveUtils = "b77e0a4c-d291-57a0-90e8-8db25a27a240" +KernelAbstractions = "63c18a36-062a-441e-b654-da1e3ab1ce7c" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +NNlib = "872c559c-99b0-510c-b3b7-b6c96a88d5cd" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +ProgressMeter = "92933f4c-e287-5a05-a399-4b506db050ca" +Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" +TimerOutputs = "a759f4b9-e2f1-59dc-863e-4aeb61b1ea8f" +Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f" From ed48915a8db44a44e431ad6ed1dda208c56a0e66 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Tue, 5 Dec 2023 10:31:20 +0100 Subject: [PATCH 20/27] Resolved merge conflicts. --- docs/src/assets/logo.png | Bin 1583515 -> 0 bytes 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a/docs/src/manifolds/basic_topology.md +++ b/docs/src/manifolds/basic_topology.md @@ -57,7 +57,6 @@ __Theorem__: If ``\mathcal{M}`` is compact and ``\mathcal{N}`` is Hausdorff, the __Proof__: We can equivalently show that every closed set is mapped to a closed set. First consider the set ``K\in\mathcal{M}``. Its image is again compact and hence closed because ``\mathcal{N}`` is Hausdorff. ## References -<<<<<<< HEAD ```@bibliography Pages = [] @@ -67,32 +66,3 @@ bishop1980tensor lang2012fundamentals lipschutz1965general ``` -======= -- General Topology book. -- Lang -- Bishop - -@book{lang2012fundamentals, - title={Fundamentals of differential geometry}, - author={Lang, Serge}, - volume={191}, - year={2012}, - publisher={Springer Science \& Business Media} -} - -@book{lipschutz1965general, - title={General Topology}, - author={Seymour Lipschutz}, - year={1965}, - publisher={McGraw-Hill Book Company}, - location={New York City, New York} -} - -@book{bishop1980tensor, - title={Tensor Analysis on Manifolds}, - author={Richard L. Bishop, Samuel I. Goldberg}, - year={1980}, - publisher={Dover Publications}, - location={Mineola, New York} -} ->>>>>>> 9daa61fb1fc0177fc30f51b3df504196a03c688a diff --git a/docs/src/manifolds/inverse_function_theorem.md b/docs/src/manifolds/inverse_function_theorem.md index 5e8bad4d7..52648d9a1 100644 --- a/docs/src/manifolds/inverse_function_theorem.md +++ b/docs/src/manifolds/inverse_function_theorem.md @@ -20,19 +20,10 @@ and this goes to zero as ``\eta`` goes to zero, because ``H`` is continuous and ## References -<<<<<<< HEAD ```@bibliography Pages = [] Canonical = false lang2012fundamentals ``` -======= -@book{lang2012fundamentals, - title={Fundamentals of differential geometry}, - author={Lang, Serge}, - volume={191}, - year={2012}, - publisher={Springer Science \& Business Media} -} ->>>>>>> 9daa61fb1fc0177fc30f51b3df504196a03c688a + diff --git a/docs/src/optimizers/general_optimization.md b/docs/src/optimizers/general_optimization.md index c3e13866f..60c9da538 100644 --- a/docs/src/optimizers/general_optimization.md +++ b/docs/src/optimizers/general_optimization.md @@ -9,8 +9,6 @@ by means of a Euler time-stepping scheme: \theta^{t+1} = \theta^{t} - h\nabla_{\theta^{t}}L, ``` where $\eta$ (the time step of the Euler scheme) is referred to as the *learning rate* -<<<<<<< HEAD + This equation can easily be generalized to [manifolds](../manifolds/manifolds.md) by replacing the *Euclidean gradient* $\nabla_{\theta^{t}L}$ by a *Riemannian gradient* $-h\mathrm{grad}_{\theta^{t}}L$ and addition by $-h\nabla_{\theta^{t}}L$ with a [retraction](../optimizers/manifold_related/retractions.md) by $-h\mathrm{grad}_{\theta^{t}}L$. -======= -This equation can easily be generalized to [manifolds](manifolds.md) by replacing the *Euclidean gradient* $\nabla_{\theta^{t}L}$ by a *Riemannian gradient* $-h\mathrm{grad}_{\theta^{t}}L$ and addition by $-h\nabla_{\theta^{t}}L$ with a [retraction](retraction.md) by $-h\mathrm{grad}_{\theta^{t}}L$. ->>>>>>> 9daa61fb1fc0177fc30f51b3df504196a03c688a + From 5aa570bc42f3163f8e9f866216d06e6b6c880356 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Tue, 5 Dec 2023 10:39:28 +0100 Subject: [PATCH 21/27] Now making directory assets for the Documenter. --- .github/workflows/Documenter.yml | 1 + 1 file changed, 1 insertion(+) diff --git a/.github/workflows/Documenter.yml b/.github/workflows/Documenter.yml index c2a50b165..e44e57f1f 100644 --- a/.github/workflows/Documenter.yml +++ b/.github/workflows/Documenter.yml @@ -17,6 +17,7 @@ jobs: sudo apt-get install poppler-utils sudo apt-get install texlive-xetex sudo apt-get install texlive-science + mkdir docs/src/assets make all -C docs/src/tikz - uses: julia-actions/setup-julia@latest - run: | From 9445f19c71fd9d627b80084a7ab4001605b467cf Mon Sep 17 00:00:00 2001 From: benedict-96 <55493704+benedict-96@users.noreply.github.com> Date: Tue, 5 Dec 2023 10:46:58 +0100 Subject: [PATCH 22/27] Added two logos (normal and dark mode) via upload. --- logo.png | 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Subject: [PATCH 23/27] Updated path for logos. --- README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index 12eeb3b2b..65025009e 100644 --- a/README.md +++ b/README.md @@ -1,6 +1,6 @@

          - - + +

          [![Stable](https://img.shields.io/badge/docs-stable-blue.svg)](https://juliagni.github.io/GeometricMachineLearning.jl/stable) From d7fc42a5ba81cc9fd20f0d905c67a82ec9f73255 Mon Sep 17 00:00:00 2001 From: benedict-96 <55493704+benedict-96@users.noreply.github.com> Date: Tue, 5 Dec 2023 11:02:20 +0100 Subject: [PATCH 24/27] Update README.md Now using urls for the image files --- README.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/README.md b/README.md index 65025009e..d2ce1105a 100644 --- a/README.md +++ b/README.md @@ -1,6 +1,6 @@

          - - + +

          [![Stable](https://img.shields.io/badge/docs-stable-blue.svg)](https://juliagni.github.io/GeometricMachineLearning.jl/stable) @@ -55,4 +55,4 @@ plot(trajectory_to_plot) The optimization of the first layer is done on the Stiefel Manifold $St(n, N)$, and the optimizer used is the manifold version of Adam (see (Brantner, 2023)). ## References -- Brantner B. Generalizing Adam To Manifolds For Efficiently Training Transformers[J]. arXiv preprint arXiv:2305.16901, 2023. \ No newline at end of file +- Brantner B. Generalizing Adam To Manifolds For Efficiently Training Transformers[J]. arXiv preprint arXiv:2305.16901, 2023. From 6856585c6f47a1892b0f4f008141b27d6a199aca Mon Sep 17 00:00:00 2001 From: benedict-96 <55493704+benedict-96@users.noreply.github.com> Date: Tue, 5 Dec 2023 11:17:51 +0100 Subject: [PATCH 25/27] Now not using local pngs for logos in README.md --- README.md | 10 ++++++---- 1 file changed, 6 insertions(+), 4 deletions(-) diff --git a/README.md b/README.md index d2ce1105a..0f0d2d088 100644 --- a/README.md +++ b/README.md @@ -1,7 +1,9 @@ -

          - - -

          + + + + Shows a black logo in light color mode and a white one in dark color mode. + + [![Stable](https://img.shields.io/badge/docs-stable-blue.svg)](https://juliagni.github.io/GeometricMachineLearning.jl/stable) [![Latest](https://img.shields.io/badge/docs-latest-blue.svg)](https://juliagni.github.io/GeometricMachineLearning.jl/latest) From 7e000dd77d851d1ec6732a0f0c7faaf58b46b07b Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Tue, 5 Dec 2023 11:18:47 +0100 Subject: [PATCH 26/27] Removed pngs. 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27/27] fixed reference. --- docs/src/manifolds/basic_topology.md | 5 +---- 1 file changed, 1 insertion(+), 4 deletions(-) diff --git a/docs/src/manifolds/basic_topology.md b/docs/src/manifolds/basic_topology.md index e9df582a0..ada034814 100644 --- a/docs/src/manifolds/basic_topology.md +++ b/docs/src/manifolds/basic_topology.md @@ -1,10 +1,7 @@ # Basic Concepts of General Topology -<<<<<<< HEAD On this page we discuss basic notions of topology that are necessary to define and work [manifolds](manifolds.md). Here we largely omit concrete examples and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. For a wide range of examples and a detailed discussion of the theory see e.g. [lipschutz1965general](@cite). The here-presented theory is also (rudimentary) covered in most differential geometry books such as [lang2012fundamentals](@cite) and [bishop1980tensor](@cite). -======= -On this page we discuss basic notions of topology that are necessary to define and work [manifolds](manifolds.md). Here we largely omit concrete examples and only define concepts that are necessary for defining a manifold[^1], namely the properties of being *Hausdorff* and *second countable*. For a wide range of examples and a detailed discussion of the theory see e.g. [lipschutz1965general](@cite). The here-presented theory is also (rudimentary) covered in most differential geometry books such as [lang2012](@cite) and [bishop1980tensor](@cite). ->>>>>>> 9daa61fb1fc0177fc30f51b3df504196a03c688a + [^1]: Some authors (see e.g. [lang2012fundamentals](@cite)) do not require these properties. But since they constitute very weak restrictions and are always satisfied by the manifolds relevant for our purposes we require them here.

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