--- title: Weighted Total Persistence in TDA url: https://www.emergentmind.com/topics/weighted-total-persistence type: topic --- # Weighted Total Persistence in TDA Searching arXiv for the cited papers to ground the article in current metadata. arXiv search query: 1701.00239 Randomly Weighted d-complexes Minimal Spanning Acycles and Persistence Diagrams Weighted total persistence is a family of aggregate functionals on persistence diagrams that summarize barcode geometry by weighting intervals through their lifetimes or endpoint-dependent kernels. In the degree-$p$ form, it is the sum of persistence lengths $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$ with $\mathrm{pers}(x)=d-b$; in randomly weighted $d$-complexes with births in $H_{d-1}$ at $0$, it becomes $\sum_i D_i^p$ and, for $p=1$, the lifetime sum $L_{d-1}$; in persistent magnitude, it is replaced by the signed exponential weighting $\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i})$; and in weighted persistence diagrams on metric measure spaces, it is the interval-moment functional $T_q(w\text{-}\sigma)=\int_{\bar Q}\operatorname{len}(I)^q\,d\mu_{\bar Q}(I)$ [1601.01741, 1701.00239, 1911.11016, 2504.11694]. These formulations suggest that weighted total persistence is not a single invariant but a class of persistence summaries whose weights may be polynomial, bounded monotone, exponential, or measure-theoretic. ## 1. Definitions and principal formulations A persistence diagram $D$ is a multiset of points $(b,d)\in\mathbb{R}^2$ with $b\le d$, together with the diagonal $\Delta=\{(a,a):a\in\mathbb{R}\}$ counted with infinite multiplicity. The persistence of a point $x=(b,d)$ is $\mathrm{pers}(x)=d-b$ [1601.01741]. | Formulation | Formula | Source | |---|---|---| | Degree-$p$ total persistence | $\mathrm{Pers}_p(D)=\sum_{x\in D}\mathrm{pers}(x)^p$ | [1601.01741] | | Thresholded total persistence | $T_p(f,t)=\sum_{x\in D_q(f):\mathrm{pers}(x)>t}\mathrm{pers}(x)^p$ | [1601.01741] | | Birth-zero lifetime moments | $(L'_{n,d-1})^\alpha:=\sum_i (D'_i)^\alpha$ | [1701.00239] | | General barcode moment | $\mathrm{TotalPersistence}_p(D_k)=\sum_{(B_i,D_i)\in D_k}(D_i-B_i)^p$ | [1701.00239] | | Persistent magnitude | $|M_*|=\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i})$ | [1911.11016] | | Weighted-diagram moment | $T_q(w\text{-}\sigma)=\int_{\bar Q}\operatorname{len}(I)^q\,d\mu_{\bar Q}(I)$ | [2504.11694] | The literature represented by these papers does not impose a single universal definition. One paper explicitly defines degree-$p$ total persistence, one identifies weighted total persistence with sums of powers of death times when births are zero, one introduces a signed endpoint-weighted variant through persistent magnitude, and one places the construction on weighted diagrams endowed with a probability measure on intervals [1601.01741, 1701.00239, 1911.11016, 2504.11694]. A plausible implication is that “weighted total persistence” is best understood as a design pattern for persistence summaries rather than as a uniquely standardized term. ## 2. Degree-$p$ total persistence and analytic bounds In the sublevel-set formulation of persistent homology, for a tame Lipschitz function $f:M\to\mathbb{R}$ on a triangulable compact metric space, the thresholded degree-$p$ total persistence is $$ T_p(f,t)=\sum_{x\in D_q(f):\mathrm{pers}(x)>t}\mathrm{pers}(x)^p, $$ for $0\le t\le \mathrm{amp}(f)$, where $\mathrm{amp}(f)=(\max_M f)-(\min_M f)$. The full degree-$p$ total persistence is $T_p(f)=T_p(f,0)$ [1601.01741]. The analytic importance of this definition is that it admits explicit upper bounds in standard TDA settings. If $M\subset\mathbb{R}^d$ is triangulable and compact, $f$ is tame and Lipschitz with Lipschitz constant $\mathrm{Lip}(f)$, and $N(r)$ is the minimum number of simplices needed in a triangulation with mesh at most $r$, then $$ T_p(f,t)\le t^p N(t/\mathrm{amp}(f)) + p\int_{\lambda=t}^{\mathrm{amp}(f)} N(\lambda/\mathrm{amp}(f))\lambda^{p-1}\,d\lambda. $$ When $M\subset\mathbb{R}^d$ and $p>d$, the estimate $N(r)\le C_M r^{-d}$ yields $$ T_p(f)\le \frac{p}{p-d}C_M\,(\mathrm{amp}(f))^d\,(\mathrm{Lip}(f))^{p-d}. $$ For a finite subset $X\subset M$, the function $f_X(x)=\min_{x_i\in X} d_M(x,x_i)$ satisfies $\mathrm{Lip}(f_X)=1$ and $\mathrm{amp}(f_X)\le \mathrm{diam}(M)$, so $$ T_p(f_X)\le \frac{p}{p-d}C_M\,(\mathrm{diam}(M))^{p-d}. $$ These inequalities are central because they convert geometric control of the underlying space into quantitative control of persistence moments [1601.01741]. The same paper also proves monotonicity in the persistence exponent: if $1\le p\le q<\infty$ and $\mathrm{Pers}_p(D)$ is bounded, then $\mathrm{Pers}_q(D)$ is also bounded. Equivalently, if $v(D)=(\mathrm{pers}(x_i))\in\mathbb{R}^n$, then $\|v(D)\|_q\le \|v(D)\|_p$, hence $$ \mathrm{Pers}_q(D)^{1/q}\le \mathrm{Pers}_p(D)^{1/p}. $$ This places weighted total persistence in the same hierarchy of $\ell^p$-type summaries that underlies many stability and approximation arguments [1601.01741]. ## 3. Minimal spanning acycles, death times, and lifetime sums For a finite simplicial complex $K$, a weighted $d$-complex is a pair $(K,w)$ in which $w:K\to\mathbb{R}$ is monotone in the sense that $w(\sigma)\le w(\tau)$ whenever $\sigma\subset\tau$. The associated filtration is the sublevel filtration $K(t)=w^{-1}((-\infty,t])$. Within this framework, a subset $S\subset F^d$ is spanning if $\beta_{d-1}(K^{d-1}\cup S)=0$, an acycle if $\beta_d(K^{d-1}\cup S)=0$, and a spanning acycle if both hold. The minimal spanning acycle $M_d$ minimizes $w(S)=\sum_{\sigma\in S} w(\sigma)$ among spanning acycles [1701.00239]. The central structural result is the equivalence between minimal spanning acycle weights and persistence death times. If $\beta_{d-1}(K)=0$, $\mathcal{D}$ is the multiset of death times in $H_{d-1}(K)$, $\mathcal{B}$ is the multiset of birth times in $H_d(K)$, and $M$ is a $d$-minimal spanning acycle, then $$ \mathcal{D}=\{w(\sigma):\sigma\in M\},\qquad \mathcal{B}=\{w(\sigma):\sigma\in F^d\setminus M\}. $$ This identification is obtained by comparing the Incremental Persistence Algorithm with the simplicial Kruskal algorithm: a $d$-face is added to the MSA exactly when it is negative, and negative faces are exactly the faces whose weights are recorded as death times in $H_{d-1}$ [1701.00239]. A further identity, attributed there as the Hiraoka–Shirai identity, links lifetime sums to spanning acycle weights: $$ L_{d-1}=\sum_i (D_i-B_i)=w(M_d)+w(M_{d-1})-w(F^{d-1}). $$ In the mean-field random model in which all lower-dimensional faces have weight $0$, births in $H_{d-1}$ occur at $0$ and $w(M_{d-1})=w(F^{d-1})=0$. Consequently, $$ \sum_{(0,D_i)\in D_{d-1}} D_i = L_{d-1}=w(M_d)=\sum_{\sigma\in\mathrm{MSA}_d} w(\sigma), $$ and, more generally, $$ \sum_i D_i^p=\sum_{\sigma\in M_d} w(\sigma)^p \qquad \text{if births are all }0. $$ In this regime, weighted total persistence in degree $d-1$ is literally the total weight, or the moment sequence of weights, of the $d$-dimensional minimal spanning acycle [1701.00239]. ## 4. Stability and random asymptotics in weighted $d$-complexes The stability theorem for weighted $d$-complexes compares two monotone weight functions $f$ and $f'$ on the same finite complex. If $\mathcal{B}_f=\{B_i\}$ and $\mathcal{D}_f=\{D_i\}$ are the birth and death times under $f$, and $\mathcal{B}_{f'}=\{B'_i\}$ and $\mathcal{D}_{f'}=\{D'_i\}$ are the corresponding sets under $f'$, then for any $p\in\{0,1,2,\dots,\infty\}$, $$ \max\left\{ \inf_{\pi\in\Pi_D}\sum_i |D_i-\pi(D_i)|^p,\; \inf_{\pi\in\Pi_B}\sum_i |B_i-\pi(B_i)|^p \right\} \le \sum_{\sigma\in F^d} |f(\sigma)-f'(\sigma)|^p, $$ with the $p=\infty$ case interpreted as a supremum. Via the equivalence between MSA weights and death times, the same bound applies to the multiset of MSA $d$-face weights [1701.00239]. In the mean-field model, where $w(\sigma)=0$ for $\sigma\in F^0,\dots,F^{d-1}$ and $w(\sigma)=\phi(\sigma)+\epsilon_n(\sigma)$ on $F^d$, this stability estimate yields continuity of total persistence moments under perturbations of the $d$-face weights. The paper derives $$ |L^p_{n,d-1}-(L')^p_{n,d-1}| \le p\,\|\epsilon_n\|_p \left[ \left(\sum_i (D'_i)^p\right)^{(p-1)/p} + \left(\sum_i D_i^p\right)^{(p-1)/p} \right], $$ and an accompanying expectation bound. For $p=1$, the inequality simplifies to $$ \inf_{\pi\in\Pi_D}\sum_i |D_i-\pi(D_i)| \le \sum_{\sigma\in F^d}|f(\sigma)-f'(\sigma)|, $$ so $\mathrm{TotalPersistence}_1$ is $1$-Lipschitz with respect to the $\ell_1$ change in $d$-face weights up to optimal matching of death sets [1701.00239]. The random model also supports a detailed extremal theory. If $F$ is Lipschitz continuous and $n\|\epsilon_n\|_\infty\to 0$ in probability, then the scaled point processes of extremal nearest-face distances, extremal death times in $H_{d-1}$, and extremal weights in the $d$-MSA all converge vaguely in distribution to a Poisson point process on $\mathbb{R}$ with intensity $e^{-x}\,dx$. In particular, $$ \mathcal{P}_C\Rightarrow \Pi,\qquad \mathcal{P}_D\Rightarrow \Pi,\qquad \mathcal{P}_M\Rightarrow \Pi. $$ This completely characterizes the extremal points of persistence diagrams and MSAs in the model [1701.00239]. The same stability mechanism transfers classical random spanning-tree asymptotics to noisy higher-dimensional settings. If $F$ is uniform on $[0,1]$ and $$ \sup_{\sigma\in F^d}\mathbb{E}[|\epsilon_n(\sigma)|^p]=o(n^{-(p+1)}), $$ then $$ n^{-(d-p)}\,\mathbb{E}\big[(L'_{n,d-1})^p\big]\to I^p_{d-1}. $$ For $d=1$ and $p=1$, this recovers Frieze’s $\zeta(3)$ limit for random minimal spanning trees under suitable noise; for higher $d$, it gives the corresponding asymptotics for random minimal spanning acycles and therefore for weighted total persistence in $H_{d-1}$ [1701.00239]. ## 5. Persistence-weighted kernels and bounded monotone weighting In the kernel approach of the persistence weighted Gaussian kernel, weighted total persistence appears both as a direct summary and as a control quantity for diagram embeddings. The weight assigned to a point $x=(b,d)$ is $$ w_{\mathrm{arc}}(x)=\arctan\!\big(C\,\mathrm{pers}(x)^p\big),\qquad C>0,\;p>0, $$ which is increasing in persistence. Near-diagonal points with small persistence receive small weight, while large-persistence points receive larger weight, and the saturation of $\arctan$ prevents unbounded influence [1601.01741]. The weighted atomic measure associated with a diagram is $$ \mu_D^w=\sum_{x\in D} w(x)\delta_x, $$ and the Gaussian RKHS embedding is $$ E(D)=\sum_{x\in D} w(x)\,k_G(x,\cdot), \qquad k_G(x,y)=\exp\!\left(-\frac{\|x-y\|^2}{2\sigma^2}\right). $$ This yields the linear kernel $$ K_L(D,E)=\sum_{x\in D}\sum_{y\in E} w(x)w(y)\exp\!\left(-\frac{\|x-y\|^2}{2\sigma^2}\right), $$ the RKHS distance $d_{k_G}^w(D,E)$, and the nonlinear Gaussian kernel on the RKHS $$ K_G(D,E)=\exp\!\left(-\frac{d_{k_G}^w(D,E)^2}{2\tau^2}\right). $$ The analytic role of total persistence is explicit in the stability estimate $$ d_{k_G}^{w_{\mathrm{arc}}}(D,E)\le L(D,E;C,p,\sigma)\,d_B(D,E), $$ with $$ L(D,E;C,p,\sigma)= C\left\{ \frac{\sqrt{2}}{\sigma}\,\mathrm{Pers}_p(D) + 2p\,\mathrm{Pers}_{p-1}(D) + (2p+1)\,\mathrm{Pers}_{p-1}(E) \right\}. $$ Thus the continuity of the weighted embedding is governed by degree-$p$ and degree-$(p-1)$ total persistence [1601.01741]. For finite point clouds $X,Y\subset M\subset\mathbb{R}^d$, the data-level theorem states that if $p>d+1$, then $$ d_{k_G}^{w_{\mathrm{arc}}}(D_q(X),D_q(Y))\le L(M,d;C,p,\sigma)\,d_H(X,Y), $$ where the constant is independent of $X$ and $Y$. The reason for the condition $p>d+1$ is that uniform control of $\mathrm{Pers}_{p-1}$ and $\mathrm{Pers}_p$ over diagrams arising from finite subsets in $M$ requires $p-1>d$ [1601.01741]. The method also has an explicit computational approximation via random Fourier features. Exact Gram-matrix computation among $n$ diagrams of size at most $m$ costs $O(m^2n^2)$ kernel evaluations, while the random-feature approximation reduces this to $O(mnM+n^2M)$ for $M$ sampled frequencies. The same paper reports that, in a synthesized XOR-like classification problem, PWGK with the RKHS Gaussian kernel achieved approximately $83\%$ test accuracy, compared to approximately $55\%$ for PSSK and approximately $70\%$ for an unweighted Gaussian embedding; in oxide-glass change-point detection it identified a change point converging to $\ell=37$; and in two protein classification tasks it achieved $100\%$ and $88.9\%$ cross-validation accuracies [1601.01741]. ## 6. Persistent magnitude as signed exponential-weighted total persistence Persistent magnitude defines a different weighting paradigm for barcodes. For a finitely presented graded persistence module $M_*$ with half-open bars $[a_i,b_i)$ appearing in degrees $d_i$, the persistent magnitude is $$ |M_*|=\sum_i (-1)^{d_i}(e^{-a_i}-e^{-b_i}), $$ with the convention $e^{-\infty}=0$. Because $$ e^{-a}-e^{-b}=\int_a^b e^{-t}\,dt, $$ each bar contributes an exponentially weighted length, and the sign $(-1)^d$ inserts Euler-characteristic cancellation across homological degrees [1911.11016]. The magnitude function introduces a scaling parameter: $$ |tM_*|=\sum_i (-1)^{d_i}(e^{-a_it}-e^{-b_it}). $$ This admits two equivalent interpretations. First, in terms of the associated graded functor, $$ |M|=|Gr_0(M)|-|Gr_1(M)|. $$ Second, in terms of the Laplace transform, if $\chi_{M_*}(s)=\sum_d (-1)^d \operatorname{rank}(M_d(s))$ is the Euler characteristic curve, then $$ |tM_*|=\mathcal{L}\{\chi'_{M_*}\}(t)=\int_{\mathbb{R}} e^{-st}\,d\chi_{M_*}(s). $$ A discrete breakpoint formula is $$ |tM_*|=\sum_{j=1}^n \chi_{M_*}(r_j)\big(e^{-r_j t}-e^{-r_{j+1} t}\big), $$ where $r_1<\cdots