Topological Data Analysis

I have wanted to study topology for over two years now, and have finally carved out time to begin. I had felt called in this direction for several reasons (ehem, manifolds and network topology), but it wasn’t until recently that I found words to articulate why I am eager to get topology in my toolkit.

Essentially, what little I’ve already learned of topology is already giving me fresh scaffolding upon which to grow my understanding of the relationships between individuals and groups of all stripes (particles, people, computers, planets). What elements are connected? Are those connections maintained as the system changes? What are the zones of influence/sharing/exchange? Math can forge new practices of imagination that you can then bring to bare on questions you feel compelled to ask.

I realized that if I didn’t root my study of topology into some concrete questions I would start to drift into its vast ocean, and run the risk of never coalescing my imagination in something fresh and useful, so I’ve decided to focus my attention on how I can apply Topological Data Analysis (TDA) in my work, and use that to as my true north.

Preliminary definitions give us the colors we can fingerpaint with, so let’s check them out.

Topological Spaces

If a space walks up to us and claims itself to be topological, we can verify whether it is telling us the truth or not by asking these four questions:

• Do you contain the empty set?
• Do you contain all your elements?
• Do you contain subsets of all the intersections of your subsets?
• Do you contain subsets of all the unions of all your subsets?

It’s important to note that the elements can be any mathemetical object. Points. Sets. Topological spaces. Topologies of topologies? Now we’re talkin’!

Neighbourhoods

A math poem is perhaps best to introduce this color.

If $X$ is a topological space and $p$ is a point in $X$, a neighborhood of $p$ is a subset $V$ of $X$ that includes an open set $U$ containing $p$. So, $p \in U \subsetneq V$.

Topological Data Analysis

Topological Data Analysis (TDA) seeks to reveal the complex topological and geometrical structures underlying data. One benefit of knowing this structure is that it can be used to narrow in on the core features “running the show” in the system, or at least the features playing a structural role. In my mind, TDA can play a role in EDA and in priming the canvas for other ML models. There is still a lot of work to be done in connecting the theory with practice, but I’m excited about the possibilities.

As with much in ML, the system of orientation and measurement we chose plays a big role in how we process and make meaning of our data. In TDA, data is typically treated as a discrete metric space, or a sample from a metric space. Metric spaces are spaces were distances between all members of the set are defined. My mind jumps to Euclidean space, but there are metric spaces elliptic and hyperbolic geometries as well. Metric spaces were named by good ‘ol Felix Hausdorff, but first introduced by Maurice Frechet. And to be more specific, it is the Euclidean distance of Euclidean n-space that satifies the metric space specs, as does Banach spaces (complete normed vector space).

Equipped with our understanding of metric spaces, we can venture forth to calculate Hausdorff distances, which tell us how far two subsets of a metric space are from each other (the greatest distance from a point in one set to the closest point in the other set). We can use these distances to gain an understanding the proximity of different data sets in our metric space, or comparisons between pairs of compact metric spaces (which can be accomplished with the Gromov-Hausdorff distance). Specifically, the Gromov-Hausdorff distance tells us how far two compact metric spaces are from being isometric, which is to say bijective. But what of this compactness I keep referring to? Compactness refers to a subset of Euclidean space that is both closed (including its limit points) and bounded (having a fixed length between all its points).

I got into simplicial complexes into greater detail elsewhere, but suffice it to say they play an important role in TDA. In the great intro paper Chazal & Michel 2017, the authors explore how one can go about building simplicial complexes from data. An abstract simplicial complex with vertex set $V$ is a set $K$ of finite subsets of V such that the elements of V belong to K, and for any $\omega \in K$, any subset of $\omega$ belongs to $K$. The elements of $K$ are the faces, simplices, of $K$. A geometic simplicial complex K is a collection of simplices such that 1) any face of a siplex of K is a simplex of K, and 2) the intersection of any two simplices of K is either empty or a common face of both.

The paper introduces two specific complexes, Vietoris-Rips and Cech. The Vietoris-Rips complex $R_{\alpha}(\mathbb{X})$ is the set of simplices $[x_{0}, …, x_{k}]$ such that $d_{\mathbb{X}}(x_{i}, x_{j}) \leq \alpha$ for all $(i, j)$. The Cech complex is the set of simplices $[x_{0}, …, x_{k}]$ such that the $k + 1$ closed balls $B(x_{i}, \alpha)$ have a non-empty intersection.

Next, a brief introduction to the concept of a cover. A cover of a set X is a collection of sets whose union contains X as a subset. In this case we say “C covers X”. A subcover of C is a subset of C that covers X, and we deem C an “open cover” if each of its members is an open set (contained in T, the topology of X). And here’s a term whose name I LOVE. A refiniement of C is a cover D of X such that every set in D is contained in some set of C.

Oh my, you’re probably thinking “when will this list of definitions stop”, but I like to think of each concept as an imgination color we can paint with. I promise, it all comes together in some pretty dazzling fireworks displays. For now, let’s turn our attention to the Nerve Theorem:

Let $\mathcal{U}=(U_{i})_{i \in I}$ be a cover of a topological space $X$ by open sets such that the intersection of any subcollection of the $U_{i}$’s is either empty or contractibe. Then X and the nerve $\mathcal{C}(\mathcal{U})$ are homotopy equivalent. From Chazal & Michel (2021) we learn that “given a cover of a data set, where each set of the cover can be, for example, a local cluster or a grouping of data points sharing some common properties, its nerve provides a compact and global combinatorial description of the relationship between these sets through their intersection patterns”. That sets up a good intuition of how we might leverage nerves to encode the topology of continuous spaces.