TDA

• The \textbf{Hilbert-function} of a pointwise finite dimensional (p.f.d) persistence module $\mathbb{U}$ indexed by $P$ send $p\in P$ to $dim U_p\in \mathbb{N}$. It can be thought of as a 'homological heatmap'.
• The \textbf{rank-invariant} of $\mathbb{U}$ sends a pair $(p,p')$, $p\perceqp'$ to the rank of the map $u_{p,p'}: U_p \rightarrow U_{p'}$, $rank(u_{p,p'}$. Note that two modules with $\mathbb{U} \simeq \mathbb{V}$ have the same rank invariant. For modules indexed by $\mathbb{R}$ this implication also.
• By restricting a bifiltration to a line $L\in\mathbb{R}^2$, where $L : t \mapsto ta + b$ has non negative slope, we get a 1-parameter filtration. The barcode of this restriction is called the \textit{fibered barcode along $L$}. The collection from all such lines forms the \textbf{fibered barcode} of the module. The idea can be extended to lines in $\mathbb{R}^n$.

As we have seen, the persistence modules arising from the Čech or Vietoris-Rips complexes are stable under small perturbations of the underlying data, but not robust.

• Upside: Multiparameter persistence modules are robust to noise and outliers and preserves features that arise from dense clusters of data (greater analytical power). (Robustness
• Downside: Loses simple, complete, and interpretable barcode structure from the 1-parameter case; We can decompose the poset P uniquely into indecomposables, but there is no way to classify or parametrize these indecomposables in general. (Representability)

• \textbf{Degree-Rips bifiltration}: Let $X$ be a finite metric space. Then the family $(\mathbb{VR}^r_d(X))_{r\in\mathbb{R},d\in\mathbb{N}}$ defined above is a filtration indexed by $\mathbb{R}\times \mathbb{N}^{op}$. Notice that $r$ denotes the scale, $d$ denotes the degree. By indexing over $\mathbb{R}\times\mathbb{N}^{op}$, $r$ increases while $d$ decreases. For the single param filtration, the Vietoris-Rips complex is not robust to outliers. The bifiltration counteracts this by introducing a \textbf{density parameter $d$}, which requires that simplices be made of only "dense enough" points. So by indexing over $\mathbb{R}\times \mathbb{N}^{op}$ we get: As $r$ increases, we allow longer edges (i.e. looser geometric structure) while $d$ decreases, allowing points with lower connectivity (i.e. less density). Note: Outliers tend to have low degree, so they are filtered out at high $d$.
• \textbf{Multicover Bifiltration}: Given a finite point cloud $X\subesteq\mathbb{R}^n$, define: \[MC^d_r(X) = \{x\in\mathbb{R}^n\,:\,|B(x,r)\cap X| > d\}.\] This is the union of balls centered at points of $X$ with radius $r$, keeping only those that intersect $> d$ data points. The filtration $(MC_r^d(X))_{(r,d)}$ is also indexed over $\mathbb{R}\times\mathbb{N}^{op}$. This is a density-aware generalization of the Cech filtration. Notice that for large $d$, only regions near densely clustered points survive. So with decreasing $d$, the regions of radius $r$ (increasing) become less dense.
• \textbf{Normalized multicover bifiltrations}: Let $X \subseteq \mathbb{R}^n$ be a finite point cloud. For $r, \rho \in \mathbb{R}$, the sets \[NMC^r_\rho(X) := \{x\in\mathbb{R}^n: |B(x,r)\cap X|\geq \rho|X|\}= \{x\in\mathbb{R}^n:\mu_X(B(x,r)\capX)\geq \rho\}\] form a bifiltration over $\mathbb{R}\times\mathbb{R}^n$ called the \textit{normalized multicover bifiltration.} This persistence module turns out to be stable w.r.t Prohorov distance (i.e. it is stable to outliers). As opposed to the non-normalized version, this bifiltration scales well when wanting to compare datasets of different sizes (MC's density threshold is an absolute count $d\in\mathbb{N}, NMC's density threshold is defined proportionally to the size of the set $\rho|X|$).

For the definition of persistence modules over multiple parameters, we introduce a framework based on \textit{poset-indexed persistence modules}.\\

• \textbf{Poset}: A poset (partially order set) is a tuple $(P, \preceq)$ consisting of a set $P$ and a binary relation $\preceq$ on $P$ satisfying: \begin{itemize} \item $a\preceq a$ for all $a \in P$. \item $a \preceq b$ and $b \preceq a$ implies $a = b$ for all $a,b\in P$. \item $a \preceq b$ and $b \preceq c$ implies $a \preceq c$ for all $a,b,c \in P$. \end{itemize} We will sometimes simply write $P$ if the (partial) ordering $\preceq$ is clear from context.
• \textbf{$P$-filtration}: A $P$-filtration is a family of topological spaces $(X_p)_{p\in P}$ such that: $p\perceq p' \implies X_p \subseteq X_{p'}.$
• \textbf{$P$-Persistence Module}: Given a poset $(P, \perceq)$, a $P$-persistence module over a field $\mathbb{F}$ consists of: \begin{itemize} \item A vector space $U_p$ for each $p\in P$. \item A linear map $u_{p,p'}: U_p \rightarrow U_{p'}$ for all $p\perceq p'$. \item These maps satisfy composition: $u_{p_2,p_3}\circ u_{p_1,p_2}= u_{p_1,p_3} for all $p_1, p_2, p_3 \in P$ with $p_1\perceq p_2\perceq p_3$.

Note: A \textbf{single-parameter} persistence module is a special case of a multiparameter persistence module where the poset is totally ordered (for all $a,b\in P$ either $a\perceq b$ or $b \perceq a$, eg. $(\mathbb{R},\leq)$ is a \textit{totally ordered set}.

see page 85/86

see p. 84/85

Let f : X → Z be well-behaved, and U be a (finite) open cover of Z. Then the Mapper is defined as \(M(\mathcal{U}, f) := N(f ^∗ (\mathcal{U}))\). (N is ther nerve)

Let X, Z be topological spaces. Then we call f : X → Z well-behaved if for all path-connected open sets U ⊆ Z,\( f ^{−1} (U)\) has finitely many path-connected components.

Two Reeb graphs \(R_f\), \(R_g\) are ϵ-interleaved, if there exists a pair of function preserving maps \(φ : R_f → S_ϵ(R_g)\), \(ψ : R_g → S_ϵ(R_f)\) such that the diagram commutes. (see diagram p.82)

For a Reeb graph \(R_f\) consider a function\( f_ϵ : (R_f)_ϵ → \mathbb{R}\) such that \((x, t) \mapsto f(x) + t\). The ϵ-smoothing of \(R_f\), denoted by \(S_ϵ(R_f)\) is the Reeb graph of \((R_f)_ϵ\) with regards to \(f_ϵ\).

An ϵ-thickening \(X_ϵ \) of some topological space X is given by \(X_ϵ := X × [−ϵ, +ϵ].\)

A p-th homology class \(h ∈ H_p(X)\) is called horizontal if there is a finite set of values \(A = \{a_1, . . . , a_k\}\) such that \( h ∈ im(ι_∗)\) where \( ι_∗\) is the map \(H_p( \bigcup_{ a∈A} X_a) → H_p(X)\) induced by inclusion, where \(X_a = f ^{−1} (a).\)

The vertical homology group of X with respect to f is the group \(\overset{\vee}{H_p}(X) := H_p(X)/\overline{H_p(X)}\)

Given a simplicial complex K with m faces and a piece-wise linear function \(f : |K| → \mathbb{R}\) on it, we can compute the augmented Reeb graph \(R_f\) of K with respect to f in time O(m log m). (proof p.78)

We define the augmented Reeb graph of a simplicial complex with a piecewise-linear-function, as the discretization of the Reeb graph by placing a vertex for every connected component in all the levelsets for the values a for which f(v) = a for some vertex of the simplicial complex.

Let X be some topological space, and f a function \(f : X → \mathbb{R}\). Two points x, y are called equivalent (x ∼ y), iff f(x) = f(y) = α and x and y are in the same path-connected component of \(f^{−1} (α)\). The Reeb graph \(R_f \)is the quotient space X/ ∼. 

\(d_b(Dgm_p(\mathbb{C}(P)), Dgm_p(\mathbb{C}(Q))) ⩽ d_H(P, Q)\) for all p ⩾ 0. (Proof p.72)

Let A, B ⊆ X be compact sets. Then the Hausdorff distance between A and B is defined as \(d_H(A, B) := max\{max_{a∈A} d(a, B), max_{ b∈B} d(b, A)\}\).

If U, V are q-tame persistence modules over \(\mathbb{R}\), then \(d_b(DgmU, DgmV) = d_I(U, V)\)

\(d_I(U, V) := inf\{ϵ | \text{U and V are ϵ-interleaved}\}\)

Let U and V be persistence modules over \(\mathbb{R}\) and let ϵ ⩾ 0. We say that U and V are ϵ-interleaved if there exist two families of linear maps, \(φ_a : U_a → V_{a+ϵ}\) and \(ψ_a : V_a → U_{a+ϵ}\) such that the four diagrams are commutative. (diagrams on p.68)

A persistence module V over \(\mathbb{R}\) is a collection \( V = \{V_a\}_{a∈\mathbb{R}}\) of vector spaces \(V_a\) together with linear maps \(v_{a,a′} : V_a → V_{a′ }\) for a ⩽ a ′ , such that \(v_{a,a} = id\) and\( v_{b,c} ◦ v_{a,b} = v_{a,c}\) for all a ⩽ b ⩽ c.

Let \( f, g : K → \mathbb{R}\) be simplex-wise monotone functions. Then, ∀p ⩾ 0 we have \(d_b(Dgm_p(\mathcal{F}_f), Dgm_p(\mathcal{F}_g)) ⩽ ||f − g||_\infty\). Proof/Illustration on p. 62.

For p ⩾ 0, and q ⩾ 1, the q-Wasserstein distance is defined as \(d_{W,q}(Dgm_p(\mathcal{F}), Dgm_p(\mathcal{G})) := [inf_{π∈Π} ( \sum_{ x∈Dgm_p}(\mathcal{F}) (||x − π(x)||_\infty)^q ) ]1/q\)

Let \(Π = \{π : Dgm_p(\mathcal{F}) → Dgm_p(\mathcal{G}) | π \text{ is bijective}\}\) be the set of all bijections between \(Dgm_p(\mathcal{F})\) and \(Dgm_p(\mathcal{G})\). Then, the Bottleneck distance is defined as \(d_b(Dgm_p(\mathcal{F}), Dgm_p(\mathcal{G})) := inf_{π∈Π} sup_{x∈Dgm_p(\mathcal{F})} ||x − π(x)||_\infty\).

Let \( G^r (P)\) be the graph on P where pq is an edge if and only if d(p, q) ⩽ 2r. The parameterized graph induced complex \(\mathbb{G}^r (Q, P)\) is defined as \(\mathbb{G}(Q, G^r (P))\).

Given two finite point sets Q, P in \( \mathbb{R}^ d\), as well as a graph G(P) with vertices in P, we define v : P → Q by sending each point in P to its closest point in Q. The graph induced complex \(\mathbb{G}(Q, G(P)) \) contains a simplex \( σ = \{q_0, . . . , q_k\} ⊂ Q\) if and only if there is a clique \(\{p_0, . . . , p_k\}\) in G(P) for which \(v(p_i) = q_i\).

Given a finite point set Q and a point set P ⊃ Q in some metric space as well as a real number radius r > 0, the parameterized Witness complex \(\mathbb{W}^r (Q, P)\) is a simplicial complex on Q defined as follows: Every point p ∈ Q defines a vertex in \(\mathbb{W}^r (Q, P)\). Further, an edge pq is in \(\mathbb{W}^r (Q, P)\) if it is weakly witnessed by x ∈ P \ Q and d(p, x) ⩽ r and d(q, x) ⩽ r. A simplex σ ⊆ Q is in \(\mathbb{W}^r (Q, P)\) if all its edges are.

The Witness complex W(Q, P) is the collection of simplices on Q for which every face is weakly witnessed by some point in P \ Q.

Given a finite point set Q and a point set P ⊃ Q in some metric space, we say that a simplex σ ⊆ Q is weakly witnessed by x ∈ P \ Q, if d(q, x) ⩽ d(p, x) for every q ∈ σ and p ∈ Q \ σ.

Given a finite point set \( P ⊂ \mathbb{R }^d\) in general position as well as a real number radius r > 0, the Alpha complex \(Del^r (P)\) consists of all simplices σ ∈ Del(P) for which the circumscribing ball of σ has radius at most r. 

Given a finite point set \(P ⊂ \mathbb{R}^ d\), the extended Delaunay complex is the simplicial complex where for every face σ, for d ′ ⩽ d, every d ′ -face of σ is a Delaunay simplex.

A Delaunay triangulation Del(P) of P is a geometric simplicial complex with the vertex set P where every simplex is a Delaunay simplex and whose underlying space covers the convex hull of P.

Given a finite point set \(P ⊂ \mathbb{R}^{d}\), a Delaunay simplex is a geometric simplex whose vertices are in P and lie on the boundary of a ball whose interior contains no points of P.