Frechet means for distributions of persistence diagrams
Consider the measure \(\rho = 1/m \sum_{i=1}^m \delta_{Z_i}\) on the space of persistence diagrams. The population Frechet mean is defined by
\[ Y = \arg\min_Z \int_{\mathcal{D}} d(X,Z)^2 {\rm d}\rho(X) \]
Let \(X_1,\cdots, X_n\) be iid samples from \(\rho\). Let \(\rho_n = 1/n\sum_{i=1}^n \delta_{X_i}\). The empirical Frechet mean is defined by
\[ Y_n = \arg\min_Z \int_{\mathcal{D}} d(X,Z)^2 {\rm d}\rho_n(X) \]
There exists a \(Y_n\) such that with probability greater than \(1-\delta\) \[ d(Y,Y_n)^2\le \frac{m^2F(Y)}{n}\log(\frac{m}{\delta}) \] for \(n\ge 8m\log(m/\delta)\) and the right hand side is less than \(r^2\) where \(r\) characterizes the separation between the local minima of \(F\).
Convergence rates for persistence diagrams estimation in topological data analysis
Let \((M,\rho)\) be a metric space and \(\mu\) is a measure on it with support \(M_\mu\). Further, assume that \(\mu\) satisfies the \((a,b)\)-standard assumption: for any \(x\) in \(M_\mu\) and \(r>0\), \(\mu(B(x,r))\ge \min(ar^b,1)\) (usually \(b=d\) if in \(\mathbb{R}^d\)). Let \(\mathbb{X}=\{X_1,\cdots,X_n\}\) be iid samples from \(\mu\). Then we have
\(\mathbb{E}[d_\infty(\text{Dgm}(M_\mu),\text{Dgm}(\mathbb{X}))]\le C(\frac{\log n}{n})^{1/b}\) where \(C\) is a constant depending on \(a\) and \(b\).
Essentially the result comes from the stability theorem of persistence diagrams and convergence for random sets with respect to Hausdorff distance. In fact, we have
for any \(\epsilon>0\), \(\mathbb{P}[d_H(M_\mu,\mathbb{X})]\le \frac{2^b}{a\epsilon^b}\exp(-na\epsilon^b)\wedge 1\)
Subsampling methods for persistent homology
Let \((M,\rho,\mu)\) be the metric measure space satisfying the \((a,b)\)-standard assumption. Let \(S_1,\cdots,S_n\) be \(n\) iid samples of size \(m\) from \(\mu\). Define the empirical average landscape by
\[ (\hat{\lambda}) = \frac{1}{n}\sum_{i=1}^n\lambda_{S_i} \]
The empirical average landscape is supposed to converge to the population average landscape which is
\[ \mathbb{E}_{(\mu^{\otimes m})_*}[\lambda] \]
The bias part is controlled by the following
Let \(r_m = 2(\frac{\log m}{am})^{1/b}\). If \(\mu\) satisfies the \((a,b,r_0)\) assumption, then \(\|\lambda_{M_\mu}-\mathbb{E}[\lambda]\|_\infty\le r_0+r_m 1_{(r_0,\infty)}(r_m)+C(a,b)\frac{r_m}{(\log m)^2}\)
The variance part is controlled by
\(\mathbb{E}\|\hat{\lambda}-\mathbb{E}[\lambda]\|_\infty\le O(1/\sqrt{n})\)
Estimation and quantization of expected persistence diagrams
Let \(\Omega=\{(x,y)|y>x\}\) be the open half-plane, and \(\partial\Omega=\{(x,x)|x\in\mathbb{R}\}\) be the boundary of \(\Omega\). Let \(\mathcal{M}^p\) be the space of Radon measures supported on \(\Omega\) that have finite total \(p\)-persistence, i.e. \(\int \|x-\partial\Omega\|^p {\rm d}\mu(x)<\infty\). Define the distance between two measures \(\mu\) and \(\nu\) by
\[ \textrm{OT}_p = \inf_{\pi}\left(\int_{\bar{\Omega}\times\bar{\Omega}}\|x-y\|^p {\rm d}\pi(x,y)\right) \]
where \(\pi\) ranges all measures supported on \(\bar{\Omega}\times\bar{\Omega}\) whose first marginal coincides with \(\mu\) and second marginal coincides with \(\nu\). Let \(P\) be a probability distribution supported on \((\mathcal{M}^p,\textrm{OT}_p)\). Let \(\mathbf{E}(P)\) be the measure defined by, for \(A\subset \Omega\) compact,
\[ \mathbb{E}(P)[A] = \mathbb{E}_P[\mu(A)] \]
\(\mathbf{E}(P)\) is called the expected persistence diagram. If \(P\) is a distribution supported on the space of persistence diagrams, it is proved under mild assumptions, \(\mathbb{E}(P)\) admits a density with respect to the Lebesgue measure on \(\Omega\).
Given iid samples \(\mu_1,\cdots,\mu_n\sim P\), the empirical expected persistence diagram is defined by \(\bar{\mu}_n=\frac{1}{n}\sum \mu_i\). We have \(\bar{\mu}_n\rightarrow\mathbf{E}(P)\) under \(\textrm{OT}_p\) almost surely. Specifically,
\[ \mathbb{E}[\textrm{OT}_p^p(\bar{\mu}_n, \mathbf{E}(P))] = O(\frac{1}{n^{1/2}}+\frac{a_p(n)}{n^{p-q}}) \]
where \(1\le p<\infty\), \(0\le q< p\), and \(P\) is a distribution with 'some' restrictions.