TOPOLOGICAL DATA ANALYSIS

Topological Data Analysis

DATA

$\mathbb{X}_n \subset \mathbb{R}^D$

Topological Data Analysis

Let $X$ be a topological space and let $\mathbb{X}_n = \{x_1,...,x_n\}$ be a finite sample of $X$ .
Inference problem: How to estimate topological properties of $X$ from $\mathbb{X}_n$ ?

DATA

$\mathbb{X}_n \subset \mathbb{R}^D$

UNDERLYING SPACE

$X$

Topological Data Analysis

Let $X$ be a topological space and let $\mathbb{X}_n = \{x_1,...,x_n\}$ be a finite sample of $X$ .
Inference problem: How to estimate topological properties of $X$ from $\mathbb{X}_n$ ?

DATA

$\mathbb{X}_n \subset \mathbb{R}^D$

For

$\epsilon>0$ , the $\epsilon$ -thickening of

$\mathbb{X}_n$ :

$\displaystyle U_\epsilon = \bigcup_{x\in \mathbb{X}_n}B_{\epsilon}(x)$

Topological Data Analysis

Let $X$ be a topological space and let $\mathbb{X}_n = \{x_1,...,x_n\}$ be a finite sample of $X$ .
Inference problem: How to estimate topological properties of $X$ from $\mathbb{X}_n$ ?

DATA

$\mathbb{X}_n \subset \mathbb{R}^D$

$\text{Evolving thickenings}$

Topological Data Analysis

Let $X$ be a topological space and let $\mathbb{X}_n = \{x_1,...,x_n\}$ be a finite sample of $X$ .
Inference problem: How to estimate topological properties of $X$ from $\mathbb{X}_n$ ?

DATA

$\mathbb{X}_n \subset \mathbb{R}^D$

$\text{Evolving thickenings}$

$\text{Sequence of combinatorial spaces}$ $\epsilon = 0 ~~~~~~~\epsilon = 0.7 ~~~~~~~\epsilon = 1 ~~~~~~~\epsilon = 2$

Topological Data Analysis

Let $X$ be a topological space and let $\mathbb{X}_n = \{x_1,...,x_n\}$ be a finite sample of $X$ .
Inference problem: How to estimate topological properties of $X$ from $\mathbb{X}_n$ ?

DATA

$\mathbb{X}_n \subset \mathbb{R}^D$

$\text{Sequence of combinatorial spaces}$

$\text{Persistent Homology}$

Persistent homology

Input dataset $X$ .

Combinatorics: construct an increasing family $\{X_\epsilon\}$ of complexes around $X$ , where the indexing $\epsilon$ is a scale parameter in $\mathbb R\geq 0$ .
Algebraic topology: compute the $d$ -th homology vector spaces $H_d(X_\epsilon)$ for scales $\epsilon\in R_{\geq 0}$ and dimensions $d \in Z_{\geq 0}$ .
Representation theory: decompose each family of vector spaces $\{H_d(X_\epsilon)\colon \epsilon \geq 0\}$ into irreducible summands, thus producing a barcode.

Source: B. Rieck

Persistent homology

Input dataset $X$ .

Combinatorics: construct an increasing family $\{X_\epsilon\}$ of complexes around $X$ , where the indexing $\epsilon$ is a scale parameter in $\mathbb R\geq 0$ .
Algebraic topology: compute the $d$ -th homology vector spaces $H_d(X_\epsilon)$ for scales $\epsilon\in R_{\geq 0}$ and dimensions $d \in Z_{\geq 0}$ .
Representation theory: decompose each family of vector spaces $\{H_d(X_\epsilon)\colon \epsilon \geq 0\}$ into irreducible summands, thus producing a barcode.

Source: B. Rieck

Persistent homology

Input dataset $X$ .

Metric geometry: Stability property. Let $(X, d_X)$ , $(Y, d_Y)$ be metric spaces. Then, $d_B(\mathrm{dgm}(X, d_X), \mathrm{dgm}(Y, d_Y)) \leq d_{GH}((X, d_X),(Y, d_Y))$

Source: B. Rieck

Persistent homology

Input dataset $X$ .

Metric geometry: Stability property. Let $(X, d_X)$ , $(Y, d_Y)$ be metric spaces. Then, $d_B(\mathrm{dgm}(X, d_X), \mathrm{dgm}(Y, d_Y)) \leq d_{GH}((X, d_X),(Y, d_Y))$

Source: B. Rieck

Persistent homology

Problem: Persistent homology is not robust in general to noise and outliers, and it might be very sensitive to the embedding in the ambient space.

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Data with density

Let $\mathbb{X}_n = \{x_1,...,x_n\}\subseteq \mathbb{R}^D$ data points.

Assume that:

$\mathbb{X}_n$ is a sample of a compact Riemannian manifold $\mathcal M$ of dimension $d$ .
The points are sampled according to a density $f\colon \mathcal M\to \mathbb R$ .

Goal: Infer $'H_\bullet(\mathcal M, f)'$

$~$

X. Fernandez, E. Borghini, G. Mindlin, P. Groisman. 'Intrinsic persistent-homology via density-based metric learning'. Journal of Machine Learning Research. 24 (2023) 1-42.

Density-based geometry

Fermat principle

The path taken by a ray between two given points is the path that can be traversed in the least time.

That is, it is the extreme of the functional $\gamma\mapsto \int_{0}^1\eta(\gamma_t)||\dot{\gamma}_t|| dt$ with $\eta$ is the refraction index.

Density-based geometry

UNDERLYING SPACE

$\mathcal M \subseteq \mathbb{R}^D$ manifold, $f:\mathcal{M}\to \mathbb{R}_{>0}$ density.

For $q>0$ , deformed Riemannian distance in $\mathcal{M}$ $d_{f,q}(x,y) = \inf_{\gamma:x\sim y} \int_{\gamma}\frac{1}{f(\gamma_t)^{q}}||\dot \gamma_t||dt.$

Density-based geometry

UNDERLYING SPACE

$\mathcal M \subseteq \mathbb{R}^D$ manifold, $f:\mathcal{M}\to \mathbb{R}_{>0}$ density.

For $q>0$ , deformed Riemannian distance in $\mathcal{M}$ $d_{f,q}(x,y) = \inf_{\gamma:x\sim y} \int_{\gamma}\frac{1}{f(\gamma_t)^{q}}||\dot \gamma_t||dt.$

DATA

$\mathbb{X}_n = \{x_1,...,x_n\}\subseteq \mathbb{R}^D$ sample.

For $p> 1$ , Fermat distance in $\mathbb{X}_n$ $d_{\mathbb{X}_n, p}(x,y) = \inf_{\gamma:x\sim y} \sum_{i=0}^{r}|x_{i+1}-x_i|^{p}.$

Density-based geometry

Theorem (F., Borghini, Mindlin, Groisman)

$\big(\mathbb{X}_n, C(n,p,d) d_{\mathbb{X}_n,p}\big)\xrightarrow[n\to \infty]{GH}\big(\mathcal{M}, d_{f,q}\big) ~~~ \text{ for } q = (p-1)/d$

Density-based geometry

Theorem (F., Borghini, Mindlin, Groisman)

$\big(\mathbb{X}_n, C(n,p,d) d_{\mathbb{X}_n,p}\big)\xrightarrow[n\to \infty]{GH}\big(\mathcal{M}, d_{f,q}\big) ~~~ \text{ for } q = (p-1)/d$

Let $\mathcal{M}$ be a closed smooth $d$ -dimensional Riemannian manifold embedded in $\mathbb{R}^D$ . Let $\mathbb X_n\subseteq \mathcal{M}$ be a set of $n$ independent sample points with common smooth density $f:\mathcal{M}\to \mathbb{R}_{>0}$ .

Given $p>1$ and $q=(p-1)/d$ , there exists a constant $\mu = \mu(p,d)$ such that for every $\lambda \in \big((p-1)/pd, 1/d\big)$ and $\varepsilon>0$ there exist $\theta>0$ satisfying $\mathbb{P}\left( d_{GH}\left(\big(\mathcal{M}, d_{f,q}\big), \big(\mathbb{X}_n, {\scriptstyle \frac{n^{q}}{\mu}} d_{\mathbb{X}_n, p}\big)\right) > \varepsilon \right) \leq \exp{\left(-\theta n^{(1 - \lambda d) /(d+2p)}\right)}$ for $n$ large enough.

$~$

Density-based geometry

Theorem (F., Borghini, Mindlin, Groisman)

$\big(\mathbb{X}_n, C(n,p,d) d_{\mathbb{X}_n,p}\big)\xrightarrow[n\to \infty]{GH}\big(\mathcal{M}, d_{f,q}\big) ~~~ \text{ for } q = (p-1)/d$

Density-based Persistent Homology

Theorem (F., Borghini, Mindlin, Groisman)

$\mathrm{dgm}\Big(\big(\mathbb{X}_n, C(n,p,d) d_{\mathbb{X}_n,p}\big)\Big)\xrightarrow[n\to \infty]{B}\mathrm{dgm}\Big(\big(\mathcal{M}, d_{f,q}\big)\Big) ~~~ \text{ for } q = (p-1)/d$

Density-based Persistent Homology

Robustness to outliers

Density-based Persistent Homology

Robustness to outliers

Prop (F., Borghini, Mindlin, Groisman, 2023)

Let $\mathbb{X}_n$ be a sample of $\mathcal{M}$ and let $Y\subseteq \mathbb{R}^D\smallsetminus \mathcal{M}$ be a finite set of outliers. Let $\delta = \displaystyle \min\Big\{\min_{y\in Y} d_E(y, Y\smallsetminus \{y\}), ~d_E(\mathbb X_n, Y)\Big\}$ .
Then, for all $k>0$ and $p>1$ , $\mathrm{dgm}_k(\mathrm{Rips}_{<\delta^p}(\mathbb{X}_n \cup Y, d_{\mathbb{X}_n\cup Y, p})) = \mathrm{dgm}_k(\mathrm{Rips}_{<\delta^p}(\mathbb{X}_n, d_{\mathbb{X}_n, p}))$ where $\mathrm{Rips}_{<\delta^p}$ stands for the Rips filtration up to parameter $\delta^{p}$ and $\mathrm{dgm}_k$ for the persistent homology of deg $k$ .

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APPLICATIONS

1. Topological methods in time series Analysis

X. Fernandez, E. Borghini, G. Mindlin, P. Groisman. 'Intrinsic persistent-homology via density-based metric learning'. Journal of Machine Learning Research. 24 (2023) 1-42.

1. Topological methods in time series Analysis

Anomaly detection en ECG

Delay embedding:
Given $T$ the time delay and $D$ the embedding dimension, $\{\big(\varphi(t), \varphi(t+T), \varphi(t+2 T) \dots, \varphi(t+(D-1)T)\big): t\in \mathbb R\}\subseteq \mathbb{R}^D$

1. Topological methods in time series Analysis

Anomaly detection in ECG

Delay embedding:
Given $T$ the time delay and $D$ the embedding dimension, $\{\big(\varphi(t), \varphi(t+T), \varphi(t+2 T) \dots, \varphi(t+(D-1)T)\big): t\in \mathbb R\}\subseteq \mathbb{R}^D$

1. Topological methods in time series Analysis

Anomaly detection in ECG

1. Topological methods in time series Analysis

Anomaly detection in ECG

$t\mapsto \mathrm{dgm}_t$ Approximate Derivative:

$\frac{d_B(\mathrm{dgm}_t, \mathrm{dgm}_{t'})}{|t'-t|}$

1. Topological methods in time series Analysis

Pattern detection in birdsongs

Source data: Private experiments. Laboratory of Dynamical Systems, University of Buenos Aires.

1. Topological methods in time series Analysis

Pattern detection in birdsongs

1. Topological methods in time series Analysis

Pattern detection in birdsongs

1. Topological methods in time series Analysis

Pattern detection in birdsongs

1. Topological methods in time series Analysis

Pattern detection in birdsongs

$t\mapsto \mathrm{dgm}_t$ Approximate Derivative

2. Topological Methods in Biomedicine

Epileptic seizure detection

$~~~~~$ X. Fernandez, D. Mateos. Topological biomarkers for real-time epileptic seizures. Preprint arXiv:2211.02523 (2024)

2. Topological Methods in Biomedicine

Epileptic seizure detection

Problem: Given a physiological recording of the brain activity, algorithmically detect in real time epileptic seizures.

$~~~~~$ X. Fernandez, D. Mateos. Topological biomarkers for real-time epileptic seizures. Preprint arXiv:2211.02523 (2024)

2. Topological Methods in Biomedicine

Epileptic seizure detection

Let $\varphi_1, \varphi_2, \dots \varphi_N:[0,T]\to \mathbb{R}$ a set of signals (EEG, MEG, etc.)
Given a window size $W$ , compute the sliding window embedding for every instant of time $t_0$ $\big\{\big(\varphi_1(t), \varphi_2(t), \dots, \varphi_N(t)\big):t\in [t_0-W, t_0]\big\}\subseteq \mathbb{R}^N.$

2. Topological Methods in Biomedicine

Epileptic seizure detection

Let $\varphi_1, \varphi_2, \dots \varphi_N:[0,T]\to \mathbb{R}$ a set of signals (EEG, MEG, etc.)
Given a window size $W$ , compute the sliding window embedding $\Big\{\big(\varphi_1(t), \dots, \varphi_N(t)\big): t\in [t_0-W, t_0]\Big\}\subseteq \mathbb R^N$
Compute the persistent homology of the time evolving point clouds.

2. Topological Methods in Biomedicine

Epileptic seizure detection

Let $\varphi_1, \varphi_2, \dots \varphi_N:[0,T]\to \mathbb{R}$ a set of signals (EEG, MEG, etc.)
Given a window size $W$ , compute the sliding window embedding $\Big\{\big(\varphi_1(t), \dots, \varphi_N(t)\big):t\in [t_0-W, t_0]\Big\}\subseteq \mathbb R^N$
Compute the persistent homology of the time evolving point clouds.
Estimate the derivative of the path of persistence diagrams.

2. Topological Methods in Biomedicine

Epileptic seizure detection

Let $\varphi_1, \varphi_2, \dots \varphi_N:[0,T]\to \mathbb{R}$ a set of signals (EEG, MEG, etc.)
Given a window size $W$ , compute the sliding window embedding $\Big\{\big(\varphi_1(t), \dots, \varphi_N(t)\big):t\in [t_0-W, t_0]\Big\}\subseteq \mathbb R^N$
Compute the persistent homology of the time evolving point clouds.
Estimate the derivative of the path of persistence diagrams.

2. Topological Methods in Biomedicine

Epileptic seizure detection

EEG (CH-MIT Database)

3. Topological Methods in Audio ID

Research collaboration with Spotify

W. Reise, X. Fernandez, M. Dominguez, H.A. Harrington, M. Beguerisse-Diaz. Topological fingerprints for audio identification. SIAM Journal on Mathematics of Data Science Vol. 6, Iss. 3 (2024)

3. Topological Methods in Audio ID

Research collaboration with Spotify

Problem: Given two audio tracks, identify whether they correspond to the same audio content.

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W. Reise, X. Fernandez, M. Dominguez, H.A. Harrington, M. Beguerisse-Diaz. Topological fingerprints for audio identification. SIAM Journal on Mathematics of Data Science Vol. 6, Iss. 3 (2024)

3. Topological Methods in Audio ID

Research collaboration with Spotify

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3. Topological Methods in Audio ID

Research collaboration with Spotify

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3. Topological Methods in Audio ID

Research collaboration with Spotify

3. Topological Methods in Audio ID

Research collaboration with Spotify

3. Topological Methods in Audio ID

Research collaboration with Spotify

Given

$s$ an audio track, we associate topological fingerprints via local Betti curves.

$~$ $~$ $~$ $~$ $~$ $~~$ $~$ $~$ $~$ $~$

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$~~~~~~~~~~~t_0~~~~~~~~~~~~~~~~~t_1~~~~~~~~~~~~~~~~~~t_2~~~~~~~~~~~~~~~~~t_3~~~~~~~~~~~~~~~~~t_4~~~~~~\dots~~~~~~~~~~~t'_0~~~~~~~~~~~~~~~~~t'_1~~~~~~~~~~~~~~~~~t'_2~~~~~~~~~~~~~~~~t'_3~~~~~~~~~~~~~~~~t'_4~~\dots$

Correlation: 'Smells like teen spirit'. (30sec-30sec): 0.9896

3. Topological Methods in Audio ID

Research collaboration with Spotify

Rigid distortions

Topological distortions

Future projects

Network science

Kuramoto models and synchronization

Kuramoto model

$\frac{d\theta_i}{dt} = \omega_i + \sum_{j=1}^{N} \omega_{ij}\sin(\theta_j - \theta_i), \quad i = 1, 2, \dots, N$

$~~~~~~$ where:

$\theta_i(t) \in [0, 2\pi)$ is the phase of the $i$ -th oscillator at time $t$ ,
$\omega_i$ is the natural frequency of the $i$ -th oscillator,
$\omega_{ij}$ strength of the coupling between oscillators $i$ and $j$ ,
$N$ is the total number of oscillators.