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Max Snijders, max.snijders@gmail.com
December 7th, 2018
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Prof. Dr. Ulrich Bauer | Prof. Dr. Oliver Junge | Prof. Dr. Konstantin Mischaikow | Max Snijders |
Technical University of Munich | Technical University of Munich | Rutgers University | Technical University of Munich & Ludwig-Maximilians University Munich |
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![]() Example complex |
![]() Cycles |
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![]() Boundaries |
![]() Reps. for Homology |
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\[ H^1(X; \mathbb{Z}) \simeq \lbrack X, S^1 \rbrack \]
\[ \frac{\partial}{\partial t} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \sigma (y - x) \\ x (\rho - z) - y \\ x y - \beta z \end{pmatrix} \]
We use: \( \beta = \nicefrac{8}{3}, \sigma = 10, \rho = 28 \)
This work relies on a large body of pre-existing work, including:
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Paper | Persistent Homology |
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Comparison against competition | Real-world data |
Thesis refs.: |
Backup slides: |
Normal slides: |
Def. | For a given point metric space \( (X, d) \) we denote by \( \mathrm{Rips}_r(X,d) \) the Rips complex of radius \(r \in \mathbb{R}\) with vertex set \(X\) and as simplices all subsets of \(X\) whose constituent vertices are pairwise within distance \(r\). |
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Starting from a cocycle \( \eta \) which represents a cohomology class, which can be seen as a map on the complex' edges, we perform the following optimisation programme: \[ \begin{align} \bar{f} &= \mathrm{argmin}_{f \in C^0(\bar{X}, \Phi; \mathbb{R})} ||\eta + d_0 f||^2_2 \\ ||\eta||_2^2 &= \sum_{ab \in (\bar{X}, \Phi)} || \eta(ab) ||^2 \end{align} \]
Backup Index
![]() \( \eta_{\alpha}, || \eta_{\alpha} ||^2_2 = 1 \)
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![]() \( \eta_{\beta}, || \eta_{\beta} ||^2_2 = \nicefrac{1}{2} \)
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![]() \( \eta_{\gamma}, || \eta_{\gamma} ||^2_2 = \nicefrac{1}{7} \)
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![]() Coordinate \( \alpha \) s.t. \( \eta_{\gamma} = \eta_{\alpha} + d_0 \alpha\)
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Cohomology is defined by dualizing the chains and the boundary maps between them by applying the \( \mathrm{Hom}\left( \_, G \right) \) functor as follows:
Intuitively, whereas an \(n\)-chain is a sum of \(n\)-simplices, an \(n\)-cochain is a function taking values on each \(n\)-simplex.
Backup Index
Def. |
A simplicial complex \( (K, \Phi) \) is a totally ordered vertex set \( K \) together with a family of nonempty subsets \(\Psi\) of \( K \) called the simplices such that the following hold:
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Def. | We define the \(n\)-chains of a simplicial complex \( (K, \Phi) \), \(C_n(K,\Phi)\) to be the group generated by the \(n\)-simplices of \((K,\Phi)\) under formal summation. |
Def. | For a given \(n \in \mathbb{Z}\) we define the \(n\)th boundary operator \( \partial_n: C_n(K, \Phi) \rightarrow C_{n-1}(K,\Phi)\) on individual \(n\)-simplices: \[ \partial_n: \{ v_0, ..., v_N \} \mapsto \sum_{i=0}^n (-1)^i \{ v_0, ..., v_{i-1}, v_{i+1}, ..., v_N \} \] and extend it linearly w.r.t. the summation on simplices. |
Def. |
For a given simplicial complex \( K, \Phi \) and integer \(n \in \mathbb{Z} \) we denote two special subgroups of \(C_n(K,\Phi)\), namely the \(n\)-cycles \( Z_n(K, \Phi) = \mathrm{ker}\left( \partial_n: C_n(K, \Phi) \rightarrow C_{n-1}(K, \Phi)\right)\) and the \(n\)-boundaries \( B_n(K, \Phi) = \mathrm{im}\left( \partial_{n+1}: C_{n+1}(K, \Phi) \rightarrow C_{n}(K, \Phi)\right)\). We note that \( B_n(K, \Phi) \subseteq Z_n(K, \Phi) \) as \( \partial_{n} \circ \partial_{n+1} = 0 \) and so the following group is well-defined: \[ H_n(K, \Phi) = \nicefrac{Z_n(K, \Phi)}{B_n(K, \Phi)} \] we call this group the \(n\)th homology group for \((K, \Phi)\). |
We will only need the first two homology groups, \( H_0 \) and \( H_1 \).
Backup Index
Def. | A lifted coordinate function \( \theta: \{ t_1, ..., t_N \} \rightarrow \mathbb{R} \) is said to be \(\epsilon\)-increasing on \( J \subseteq \{ t_1, ..., t_N \} \) iff \( \frac{\theta(t') - \theta(t)}{t' - t} \ge \epsilon \) for all \(t, t' \in J\) whenever \(t' > t\). |
Def. | A lifted coordinate function \( \theta: \{ t_1, ..., t_N \} \rightarrow \mathbb{R} \) is said to be \(\epsilon\)-cycling on an interval \( (a, b) \subseteq \mathbb{R} \) for \( a,b \in \{ t_1, ..., t_N \} \) if \( \theta \) is \(\epsilon\)-increasing on \( \{t_1, ..., t_N\} \big|_{(a,b)} \) and \( \theta(b) - \theta(a) \ge 1\). |
Def. | A cube \( \xi \) in the cover is said to be \(\epsilon\)-cycling w.r.t. the function \(f: \bar{X} \rightarrow \mathbb{R}\) and the boxed trajectory \( \gamma: \{t_1, ..., t_N\} \rightarrow \bar{X} \) if there is some interval \( (a,b) \subseteq \mathbb{R} \) s.t. \( \gamma((a,b)) \ni \xi \) where \( f \circ \gamma: \{ t_1, ..., t_N \} \rightarrow \mathbb{R} \) is \( \epsilon \)-cycling on the interval \( (a, b) \). |
Def. | A cube \( \xi \) in the cover is said to be \(\epsilon\)-transient w.r.t. the function \(f: \bar{X} \rightarrow \mathbb{R}\) and the boxed trajectory \( \gamma: \{t_1, ..., t_N\} \rightarrow \bar{X} \) if for a maximal interval \( (a,b) \subseteq \mathbb{R} \) s.t. \( \gamma((a,b)) \ni \xi \) and \( f \circ \gamma: \{ t_1, ..., t_N \} \rightarrow \mathbb{R} \) is \( \epsilon \)-cycling on the interval \( (a, b) \) we have that \( f(\xi) - f(\gamma(b)) \le 1 \), i.e. the box is in the last cycle of the interval . |
Def. | A dynamical system is some system with time-dependent state, i.e. some system \( S \) with a transition map \(f: S \rightarrow S\) or a continuous-time evolution map \(f: \mathbb{R} \times S \rightarrow S\). |
In our case, our state-space consists of \( \mathbb{R}^n \) and our time-evolution map is provided by an ODE.
We want to decompose the state-space of a given dynamical system into regions where we can understand behaviour and then study how the system transitions between these regions.
Def. | Let \(f, g: X \rightarrow Y\) be two maps between topological spaces. \(f\) and \(g\) are said to be homotopy-equivalent or homotopic if there exists a continuous map \(H: X \times \lbrack 0, 1 \rbrack \rightarrow Y\) such that \(H \big|_{X \times \{0\}} = f \) and \( H \big|_{X \times \{1\}} = g \). |
Def. | Two topological spaces \(X\) and \(Y\) are said to be homotopy-equivalent or homotopic if there exist continuous maps \(f : X \rightarrow Y \) and \(g: Y \rightarrow X\) such that \(f \circ g: Y \rightarrow Y\) is homotopic to the identity map on \(Y\) and \(g \circ f: X \rightarrow X\) is homotopic to the identity map on \(X\). |
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