Statistical Invariance of Betti Numbers

# .orange[Statistical Invariance of Betti Numbers .tiny[In the thermodynamic regime]]

### Siddharth Vishwanath .small[.grey[With Kenji Fukumizu, Satoshi Kuriki & Bharath Sriperumbudur]]

---

# Outline

1. .large[.orange[.bolder[Statistics + Topology]]]

1. .large[.bolder[Preliminaries]]

1. .large[.bolder[Invariance : Characterization]]

1. .large[.bolder[Invariance via Topological Groups]]

1. .large[.bolder[Invariance via Excess Mass]]

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# Statistics + Topology

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# Sufficient Statistics may not suffice

<img src="images/anscombe.png" width="650">
.center[Anscombe's quartet]
</div>

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# Sufficient Statistics may not suffice

<div class="centered">
<img align="centered" class="animated-gif" src="images/DinoSequential.gif" width="120%">
.center[Anscombe's quartet on steroids]
</div>

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# Preliminaries
## .orange[Topological Ingredients]

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# `$\beta_2$` 
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## 0
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## 1
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## 1
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## 3
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# Preliminaries
## .orange[Probabilistic Ingredients]

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# Random Topology

* Given a probability space `$(\Omega,\mathcal{F},\mathbb{P})$` and some metric-space `$\mathcal{X}$`

* `$\mathbb{X}_n = \{ \boldsymbol{X}_1, \boldsymbol{X}_2, \dots \boldsymbol{X}_n \} \sim \mathbb{P}$`
  - A fixed probability measure, i.e., observed i.i.d.
  - A random field, e.g., Poisson Process

--
* A simplicial complex, `$\mathcal{K}(\mathbb{X}_n,r)$`, is a random-variable measurable w.r.t. `$\mathbb{P}^{\otimes n}$`

--
* `$\mathbf S : \mathcal X^n \rightarrow \mathcal S$` is a topological summary, e.g., `$\beta_k \left( \mathcal{K}(\mathbb{X}_n,r) \right) : \mathcal{X}^n \rightarrow \mathbb{N}$`

--

* What are the properties of these **.purple[random]** topological summaries?
`\begin{align}
\text{(LLN)} & & \lim\limits_{n\rightarrow \infty}\frac{1}{n}\beta_k\left( \mathcal{K}(\mathbb{X}_n,r) \right) = \color{red}{\gamma_k(\mathbb{P})} \ \ \text{a.s.} \hspace{2cm}\\ \\
\text{(CLT)} & & \lim\limits_{n\rightarrow \infty}\frac{\beta_k\left( \mathcal{K}(\mathbb{X}_n,r) \right) - \mathbb{E}(\beta_k\left( \mathcal{K}(\mathbb{X}_n,r) \right))}{\sqrt{n}}  \sim \color{red}{\mathcal{N}(0,\sigma^2)}
\end{align}`
.tiny[.caption[Bobrowski and Kahle (2018); Kahle and Meckes (2013); Yogeshwaran, Subag, and Adler (2017)]]

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# .left[Asymptotic Regimes]

In the simplicial complex `$\mathcal{K}\left( \mathbb{X}_n, {r_n} \right)$`, `$r_n$` depends on `$n$`

.column.bg-main1[.content[
 
.center[
Dense
<img src="images/dense.gif" width="300",height="300">]
`$nr_n^d \rightarrow \infty$`
]]

.column.bg-main1[.content[
 
.center[
Sparse
<img src="images/sparse.gif" width="300",height="300">]
`$nr_n^d \rightarrow 0$`
]]

.column.bg-main1[.content[
 
.center[
Thermodynamic
<img src="images/thermodynamic.gif" width="300",height="300">]
`$nr_n^d \rightarrow t \in (0,\infty)$`
]]

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# Statistical Invariance
## .orange[Characterization]

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# Statistical Invariance of Betti Numbers

* Consider a family of distributions* `$\mathcal{P}$` on `$\mathcal X \subseteq \mathbb R^D$` with `$\mathbb P, \mathbb Q \in \mathcal P$`

* Given `$\mathbb{X}_n= \{ \boldsymbol{X}_1, \boldsymbol{X}_2, \dots \boldsymbol{X}_n \} \sim \mathbb{P}$` and `$\mathbb{Y}_m= \{ \boldsymbol{Y}_1, \boldsymbol{Y}_2, \dots \boldsymbol{Y}_m \} \sim \mathbb{Q}$`

.content-box-purple[
`$\mathbf{S}(\mathbb{P}^{\otimes n})\!:= \mathbf{S}(\mathbb{X}_n)\!= \frac{1}{n} \Big( \beta_0\big( \mathcal{K}(\mathbb{X}_n,r_n) \big), \beta_1\big( \mathcal{K}(\mathbb{X}_n,r_n) \big), \dots , \beta_d\big( \mathcal{K}(\mathbb{X}_n,r_n) \big) \Big)$`
]

--
* **.purple[Invariance.]** 
.center[<body> `$\bbox[20px, border: 2px solid orange]{ \text{Can we have that } \lim\limits_{n\rightarrow \infty}\mathbf{S}(\mathbb{P}^{\otimes n}) {=} \lim\limits_{m\rightarrow \infty}\mathbf{S}(\mathbb{Q}^{\otimes m}) \text{ ? } }$`</body>]

--
 
* **Example (1).** Consider `$\color{red}{\mathcal{P} = \{ \mathcal{N}(\theta,\mathbf{I}_d) : \theta \in \mathbb{R}^d \}}$` and `$\color{green}{\mathbf{S}(\mathbb{X}_n) = \overline{\mathbf{X}}_n}$`

.center[
If `$\mathbf X_i \sim \mathbb P_{\theta_1}$` and `$\mathbf Y_i \sim \mathbb P_{\theta_2}$` then

`$\lim\limits_{n\rightarrow \infty}\overline{\mathbf{X}}_n = \theta_1 \neq \theta_2 = \lim\limits_{m\rightarrow \infty}\overline{\mathbf{Y}}_m$`
]
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# Statistical Invariance of Betti Numbers

* Consider a family of distributions* `$\mathcal{P}$` on `$\mathcal X \subseteq \mathbb R^D$` with `$\mathbb P, \mathbb Q \in \mathcal P$`

* **.purple[Invariance.]** 
.center[<body> `$\bbox[20px, border: 2px solid orange]{ \text{Can we have that } \lim\limits_{n\rightarrow \infty}\mathbf{S}(\mathbb{P}^{\otimes n}) {=} \lim\limits_{m\rightarrow \infty}\mathbf{S}(\mathbb{Q}^{\otimes m}) \text{ ? } }$`</body>]

* **Example (2).**  Consider `$\color{red}{\mathcal{P} = \{ \mathcal{N}(\mathbf 0,\boldsymbol{\theta}) : \boldsymbol{\theta} \in \mathcal S^d_{++} \}}$` and `$\color{green}{\mathbf{S}(\mathbb{X}_n) = \overline{\mathbf{X}}_n}$`

.center[
If `$\mathbf X_i \sim \mathbb P_{\theta_1}$` and `$\mathbf Y_i \sim \mathbb P_{\theta_2}$` then

`$\lim\limits_{n\rightarrow \infty}\overline{\mathbf{X}}_n = \mathbf 0 = \lim\limits_{m\rightarrow \infty}\overline{\mathbf{Y}}_m$`
]
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# Statistical Invariance of Betti Numbers

* Consider a family of distributions* `$\mathcal{P}$` on `$\mathcal X \subseteq \mathbb R^D$` with `$\mathbb P, \mathbb Q \in \mathcal P$`

* **Example (3).**  Consider `$\color{red}{\mathcal{P} = \{ \mathcal{N}(\mathbf 0,\boldsymbol{\theta}) : \boldsymbol{\theta} \in \mathcal S^d_{++} \}}$` and `$\color{green}{\mathbf{S}(\mathbb{X}_n) = \text{Cov}(\mathbb{X}_n)}$`

.center[
If `$\mathbf X_i \sim \mathbb P_{\theta_1}$` and `$\mathbf Y_i \sim \mathbb P_{\theta_2}$` then

`$\lim\limits_{n\rightarrow \infty}\text{Cov}(\mathbb X_n) = \boldsymbol \theta_1 \neq \boldsymbol \theta_2 = \lim\limits_{m\rightarrow \infty}\text{Cov}(\mathbb Y_m)$`
]

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# Statistical Invariance of Betti Numbers

* Consider a family of distributions* `$\mathcal{P}$` on `$\mathcal X \subseteq \mathbb R^D$` with `$\mathbb P, \mathbb Q \in \mathcal P$`

* ***Assumptions:** 
1. .small[<body> `$\text{dim}(\mathcal{X}) = d \le D$`. If `$d < D$` then `$\mathcal{X}$` is a compact, `$d$`-dimensional `$\mathcal{C}^1$` manifold </body> ]

1. .small[<body> Each `$\mathbb{P} \in \mathcal P$` admits a density `$f$` such that `$f \in L_p(\mathcal{X})$` for all `$p \in \mathbb{N}$` </body> ]

1. .small[<body> `$\mathcal{K}(\mathbb{X}_n,r_n)$` is the Čech complex constructed on `$\mathbb{X}_n$` with `$r_n \asymp n^{-1/d}$` </body> ]

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# Statistical Invariance of Betti Numbers

* Consider a family of distributions* `$\mathcal{P}$` on `$\mathcal X \subseteq \mathbb R^D$` with `$\mathbb P, \mathbb Q \in \mathcal P$`

* **.purple[Definition.]**

As `$n \rightarrow \infty$` and `$nr_n^d \rightarrow t$`, the **.purple[thermodynamic limit]** is the functional 
`\begin{align}
\mathbf{S}(\mathbb{P}; t) = \lim_{n \rightarrow\infty}\mathbf{S}(\mathbb{P}^{\otimes n})
\end{align}`

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# `$\beta$`-equivalence

.content-box-purple[
**.purple[Definition.]**  `$\mathcal{P}$` admits `$\beta$`-equivalence if `$\mathbf S(\mathbb{P}; t) = \mathbf S(\mathbb{Q}; t)$` for all `$\mathbb P, \mathbb Q \in \mathcal P$`
]

* If `$\mathcal{P} = \{ \mathbb P_\theta : \theta \in \Theta \}$` is a **parametric** family of distributions:

.content-box-purple.center[
`$\mathbf S(\mathbb{P}_\theta; t) = \eta(t)$` for all `$\theta \in \Theta$`
]

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# `$\beta$`-equivalence

* Let `$\color{purple}{\mathcal{F}}$` be the family of probability density functions associated with `$\color{purple}{\mathcal{P}}$`

.content-box-purple[**.purple[Proposition.]** Suppose `$\mathcal{F}$` is a family of distributions such that: 
for all `$f,g \in \mathcal{F}$` with `$\mathbf{X} \sim f$` and `$\mathbf{Y} \sim g$`, it holds that `$f(\mathbf{X}) \stackrel{d}{=} g(\mathbf{Y})$`. 
Then, `$\mathcal{F}$` admits `$\beta$`-equivalence.]

* **.purple[Main idea.]** The SLLN from Yogeshwaran, Subag, and Adler (2017); Trinh (2017); Goel, Trinh, and Tsunoda (2019) are used to express `$\mathbf{S}(\mathbb{P}; t)$` as an expectation

--
* **.purple[Remark.]** Invariance extends to persistent Betti numbers and Euler characteristic

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# Statistical Invariance
## .orange[via Topological Groups]

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# .left[Group Invariance in Statistics]

.column.top[.content[
 
.left[.small[
* `$\mathcal{G} = \{ g_\theta : \theta \in \Theta \}$` is a group acting on `$\mathcal{X}$`

* Consider a map `$T: \mathcal{X} \rightarrow \mathcal{T}$`
 - `$T$` is `$\color{purple}{\mathcal{G}}$`.purple[-invariant] if it is constant on orbits, i.e., .center[ <body> `$T(g_\theta\cdot\mathbf{x}) = T(\mathbf{x})$` </body> ]
 
 - `$T$` is `$\color{red}{\mathcal{G}}$`.red[-maximal invariant] if it is constant **only** on orbits, i.e.,
 
 .center[ <body> `$T(\mathbf{x}) = T(\mathbf{y})$` if and only if `$\mathbf{y} \in \mathcal{G}\mathbf{x}$` </body>]
 
* If `$\mathbf{X} \sim f$` be a random variable
 - The action of `$\mathcal{G}$` on `$\mathbf{X}$` *induces* `$g_\theta \cdot \mathbf{X} \sim f_\theta$`
]]]]

.column.bg-main1.top[.content[
 
.center[<img src="images/group2.svg" width="400",height="400">]]
]

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`$\mathcal{G} = \{ g_\theta : \theta \in \Theta \}$` *generates* a family of distributions `$\mathcal{F} = \{f_\theta : \theta \in \Theta\}$`

.tiny.left[.caption[Eaton (1989); Wijsman (1990)]]

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`$\mathbf{X_\theta} \sim f_\theta \ \ \  \longrightarrow  \ \ \ Z_\theta = f_\theta(\mathbf X_\theta) \ \ \ \longrightarrow \ \ \ f_{Z_\theta}$` is `$\mathcal G$`-invariant

.tiny.left[.caption[Eaton (1989); Wijsman (1990)]]

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# Statistical Invariance - I

.content-box-purple[**.purple[Theorem.]** 
* Define a family of distributions `$\mathcal{F} = \{ f_\theta : \theta \in \Theta \}$` by
`\begin{align}
\smash{f_\theta(x) := \xi\big( g_\theta \circ \Psi(x) \big),}
\end{align}`
where 
  - `$\Psi:\mathcal{X} \rightarrow \mathcal{Y}$` is differentiable 
  - `$\mathcal{G} = \{ g_\theta : \theta \in \Theta \}$` is a group of isometries on `$\mathcal{Y}$`
  - `$T:\mathcal{Y} \rightarrow \mathcal{T}$` is `$\mathcal{G}$`-maximal invariant
  - `$\xi$` ensures `$f_\theta$` is a valid density function.

* Then, `$\mathcal{F}$` admits `$\beta$`-equivalence if and only if there exists some `$\zeta:\mathcal{T} \rightarrow \mathbb{R}$`
`\begin{align}
\text{det}(J_{\Psi^{-1}}(y)) = \zeta(T(y))
\end{align}`
]

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# Statistical Invariance - I : .red[Example]

.content-box-red[
$$
f_\theta(x_1,x_2) = \big( \cos(\theta) \Phi^{-1}(x_1) + \sin(\theta )\Phi^{-1}(x_2) \big)^2 \hspace{0.5cm} \mathbf{1}(0 \le x_1,x_2 \le 1)
$$
]

--
* .small[ <body> `$\mathcal{X} = [0,1]^2, \hspace{0.2cm} \mathcal{Y} = \mathbb{R}^2,$` and `$\Psi : \mathcal{X} \rightarrow \mathcal{Y}$` such that `$(x_1,x_2) \mapsto (\Phi^{-1}(x_1),\Phi^{-1}(x_2))$` </body> ]

* .small[ <body> `$\mathcal G = S\mathcal{O}(2)$` for which `$g_\theta = \smash{\begin{pmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{pmatrix}}$` and `$T(\mathbf{y}) = ||\mathbf{y}||$` </body> ]

* .small[ <body> `$\xi:\mathcal{Y} \rightarrow \mathbb{R}$` is given by `$\xi(\mathbf{y}) = \big( (1,0)^\top\mathbf{y} \big)^2$` </body>]

--
* .small[ <body> Then we can express `$\bbox[2pt,#ff9166]{f_\theta(x_1,x_2) = \xi(g_\theta \circ \Psi(x_1,x_2))}$` </body> ]

--
* .small[ <body> **.red[Jacobian condition:]** For `$\mathbf{y} = (y_1,y_2) \in \mathcal{Y},$` observe `$\Psi^{-1}(y_1,y_2) = (\Phi(y_1),\Phi(y_2))$` </body> ]
`\begin{align}
\text{det}(J_{\Psi^{-1}}(\mathbf{y})) = \phi(y_1)\cdot\phi(y_2) = \exp\Big( -\frac{y_1^2}{2} -\frac{y_2^2}{2} \Big) = \exp\Big( -\frac{1}{2}T(\mathbf{y})^2 \Big)
\end{align}`

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# Statistical Invariance - I : .red[Example]

.content-box-red[
$$
f_\theta(x_1,x_2) = \big( \cos(\theta) \Phi^{-1}(x_1) + \sin(\theta )\Phi^{-1}(x_2) \big)^2 \hspace{0.5cm} \mathbf{1}(0 \le x_1,x_2 \le 1)
$$
]

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# Statistical Invariance
## .orange[via Excess Mass]

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# Statistical Invariance - II

- For `$\mathbb{P}$` with pdf `$f$`, the **.purple[excess mass function]** is given by 
`\begin{align}
\hat{f}(t) := \mathbb{P}\big( \{ \mathbf{x} \in \mathcal{X} : f(\mathbf{x}) \ge t \} \big)
\end{align}`

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# Statistical Invariance - II

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# Statistical Invariance - II

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# Statistical Invariance - II

.content-box-purple.center[
.large[] `$\hspace{0.5cm}$` If `$\mathbf{X} \sim f$` and `$\mathbf{Y} \sim g$`, then `$f(\mathbf{X}) \stackrel{d}{=} g(\mathbf{Y}) \iff \hat{f} = \hat{g}$`
]

- Given a measure `$\mu$` on `$\mathcal{X}$`, a function `$\Psi(\cdot)$` is the .purple[modular character of] `$\color{purple}{\mu}$` if for each `$\phi \in \text{Diff}(\mathcal{X})$` and `$\mathbf{y} = \phi(\mathbf{x})$`, `$\hspace{0.25cm}\mu(d\mathbf{y}) = \Psi(|\text{det}J_\phi|)\mu(d\mathbf{x})$`

`$\hspace{1cm}$` **Example.** When `$\mathcal{X} = \mathbb{R}^d$` and `$\mu = \nu_d$` (Lebesgue), `$\Psi(x) = x$`

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# Statistical Invariance - II

**.purple[Theorem.]**

Suppose `$g$` is a probability density on `$\mathcal{X}$` w.r.t. `$\hspace{0.2cm} \nu$` with modular character `$\Psi$` 
 `$\hspace{0.5cm}$` (1) `$\text{supp}(g) = K \subset \mathcal{X}$`, 
 `$\hspace{0.5cm}$` (2) Let `$\{\phi_1 \dots \phi_n\}$` be maps such that `$K_i := \phi_i(K)$` and `$\nu(K_i \cap K_j) = 0$`.

Define the density `$f_\phi$` by
`\begin{align}
f_\phi(\mathbf{x}) = \smash{\sum_{i=1}^n{g(\phi_i(\mathbf{x}))}}\mathbf{1}(\mathbf{x} \in K_i)
\end{align}`

Then, `$\mathcal{F} = \{ f_\phi : \phi \in \Phi \}$` admits `$\beta$`-equivalence for 
.center[ <body> `$\Phi := \Big\{ \phi_1 \dots \phi_n : \sum\limits_{i=1}^n{\Psi(|\text{det}J_{\phi^{-1}_i}|)}=1 \Big\}$` </body> ] ]

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# Statistical Invariance - II : .red[Example]
.small[ <body> Given a density `$g$` on `$\mathbb{R}_+$` and `$\Theta = \{ (a,b) : \frac{1}{a} + \frac{1}{b} = 1 \}$`, define `$\{ f_\theta\!: \theta\!\in\!\Theta \}$` on `$\mathcal{X} = \mathbb{R}$` </body>]
.small[.center[.content-box-red[
$$
f_\theta(x) = \color{red}{g(-bx) \mathbf{1}(x < 0)} + \color{blue}{g(ax) \mathbf{1}(x \ge 0)}
$$
]]]
.center[<img src="images/Gamma.gif" width="450",height="450">]

.small[
 
`$\phi_1(x) = ax \text{ and } \phi_2(x) = -bx,$` then `$\Psi(|\text{det}J_{\phi^{-1}_1}|)+\Psi(|\text{det}J_{\phi^{-1}_2}|) = \frac 1 a + \frac 1 b = 1.$`
]
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# .left[Statistical Invariance - II]

* `$\mathscr{X}=(\mathcal X, \pi, \mathcal Y, \mathcal Z)$` is a smooth fiber bundle with local trivialization `$\{ U_\alpha,\psi_\alpha \}$`

.column.bg-main1[.content[
 
.center[<img src="images/bundle.svg" width="400",height="400">]]
]

.column[.content[
 
.left[.small[
* `$\mu:$` A probability measure on `$\mathcal Z$` 
* `$\nu:$` Measure on `$\mathcal Y$` with modular character `$\Psi$` 
* `$\mathbb P:$` Distribution on `$\mathcal Y$` with density `$g$` w.r.t `$\nu$` 
* `$\phi:\mathcal Y \times \mathcal Z \rightarrow \mathcal Y$` is a map such that
 * `$\phi(\cdot,z) \in \text{Diff}(\mathcal Y)$` for each `$z \in \mathcal Z$` 
* For each `$x \in \mathcal X$` with local coordinates `$x=(z,y_z)$`, `$y=\psi^{-1}_{\alpha,z}(y_z)$` 
* Define `$f_\phi(x) = g(\phi(z,y))$`
]]]]

---

---

---

.content-box-purple[.small[
**.purple[Theorem.]** `$\{ f_\phi\!:\!\phi\!\in\!\Phi \}$` admits `$\beta$`-equivalence for 
`$\Phi := \Big\{ \phi\!:\!\!\displaystyle\int_{\mathcal Z}\!{\!\Psi\left( \big\lvert \text{det}J_{\phi_z^{-1}} \big\rvert  \right)\!\mu(dz)}\!=\!1 \Big\}$`
]]

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# Verifying `$\beta$`-equivalence

* Given a family of distributions `$\mathcal F$`, can we check if `$\mathcal F$` admits `$\beta$`-equivalence?

--
.content-box-purple[
**.purple[Theorem.]**
`$\mathcal F = \{ f_\theta : \theta \in \Theta \}$` admits `$\beta$`-equivalence if and only if for all `$k \in \mathbb N$`

`\begin{align}
\nabla_{\theta} \left(\int_{\mathcal X}{f_\theta^{k+1}(\mathbf{x}) d\mathbf{x}}\right) = \mathbf 0
\end{align}`
]

--
* Under standard stochastic regularity assumptions, it leads to a geometric constraint on `$f_\theta$` 
.content-box-purple[
`$\mathcal F = \{ f_\theta : \theta \in \Theta \}$` admits `$\beta$`-equivalence if and only if for all `$k \in \mathbb N$`

`\begin{align}
\big\langle S_{\theta},f_\theta^k \big\rangle_{L_2(\mathcal X)} = \mathbf 0,
\end{align}`
where `$S_{\theta} = \nabla_{\theta}\log f_\theta$` is the **score function** used in statistical inference
]

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# Summary

*  `$\mathcal{P}$` (equiv. `$\mathcal F$`) admits `$\beta$`-equivalence if `$\mathbf S(\mathbb P; t) = \mathbf S(\mathbb Q; t)$` for all `$\mathbb P, \mathbb Q \in \mathcal P$`

--
* `$\color{red}{f(\mathbf X) \stackrel{d}{=} g(\mathbf{Y}), \ \forall f,g \in \mathcal F \text{ s.t. } \mathbf X \sim f, \mathbf Y \sim g}$` `$\Rightarrow$` `$\color{green} {\mathcal P}$` .green[admits] `$\color{green}\beta$`.green[-equivalence] 
 1. via topological groups
 1. by preserving excess mass
 1. `$\big\langle S_\theta, f^k_\theta \big\rangle = \mathbf 0$` for all `$k = 0,1,2,\dots$`

--
* **.purple[Other topological summaries:]** 
 1. *.blue[Euler characteristic:]* `$\hspace{1.3cm}$` `$\mathbf S(\mathbb P^{\otimes n}) = \frac{1}{n} \ \chi\big( \mathcal K(\mathbb X_n, r_n) \big)$` 
 1. *.blue[Persistent Betti numbers:]* `$\mathbf S(\mathbb P^{\otimes n})\! =\!\!\frac{1}{n} \beta^{(r,s)}_i\big( \mathcal K(\mathbb X_n, r_n) \big), \ i\!\!=\!0,\!1 \!\dots d$` 
 1. *.blue[Process level summaries:]* `$\hspace{0.075cm}$` `$\mathbf{S}(\mathbb P; t) \in \mathscr{D}\big([0,\infty)\big)$` 
 1. *.blue[Persistence diagrams:]* `$\hspace{0.8cm}$` `$\mathbf{S}(\mathbb P) = \mathbf{Dgm}(\text{Filt}(\phi_{\mathbb P}))$` 
 - e.g., `$\phi_{\mathbb P} = K_\sigma * f_{\mathbb P}$`, or `$\phi_{\mathbb P} = D^K_{\mathbb P}$`

.tiny[Bobrowski and Mukherjee (2015); Krebs and Polonik (2019); Owada and Thomas (2020); Phillips, Wang, and Zheng (2015); Chazal, Fasy, Lecci, Michel, Rinaldo, and Wasserman (2017)]
  
  
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# Thank you!
### <a href="https://www.arxiv.org/abs/2001.00220"style="font-family:Courier;font-size:32px;color:orange">arxiv.org/abs/2001.00220 </a>

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# .orange[References]

Bobrowski, O. and M. Kahle (2018). "Topology of random geometric
complexes: A survey". In: _Journal of Applied and Computational
Topology_ 1.3-4, pp. 331-–364.

Bobrowski, O. and S. Mukherjee (2015). "The topology of probability
distributions on manifolds". In: _Probability theory and related
fields_ 161.3-4, pp. 651-686.

Chazal, F, B. Fasy, F. Lecci, et al. (2017). "Robust topological
inference: Distance to a measure and kernel distance". In: _The Journal
of Machine Learning Research_ 18.1, pp. 5845-5884.

Eaton, M. L. (1989). _Group Invariance Applications in Statistics_.
Hayward, CA and Alexandria, VA: Institute of Mathematical Statistics
and the American Statistical Association, pp. i-133.

Goel, A, K. D. Trinh, and K. Tsunoda (2019). "Strong law of large
numbers for Betti numbers in the thermodynamic regime". In: _Journal of
Statistical Physics_ 174.4, pp. 865-892.

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# .orange[References]

Kahle, M. and E. Meckes (2013). "Limit theorems for Betti numbers of
random simplicial complexes". In: _Homology Homotopy Appl._ 15.1, pp.
343-374.

Krebs, J. T. and W. Polonik (2019). "On the asymptotic normality of
persistent Betti numbers". In: _arXiv preprint arXiv:1903.03280_.

Owada, T. and A. M. Thomas (2020). "Limit theorems for process-level
Betti numbers for sparse and critical regimes". In: _Advances in
Applied Probability_ 52.1, p. 1–31. DOI:
[10.1017/apr.2019.50](https://doi.org/10.1017%2Fapr.2019.50).

Phillips, J. M., B. Wang, and Y. Zheng (2015). "Geometric inference on
kernel density estimates". In: _31st International Symposium on
Computational Geometry_. Vol. 34. , pp. 857-871.

Trinh, K. D. (2017). "A remark on the convergence of Betti numbers in
the thermodynamic regime". In: _Pacific Journal of Mathematics for
Industry_ 9.1, p. 4.

Wijsman, R. A. (1990). _Invariant Measures on Groups and Their Use in
Statistics_. Hayward, CA: Institute of Mathematical Statistics.