Flagifying the Dowker Complex

Marius Huber, Patrick Schnider

Keywords: topological data analysis, tda, persistent homology, dowker complex

TL;DR: We introduce the Dowker-Rips complex, a flagified version of the Dowker complex, which is much more practical to compute while remaining equally useful in practical applications.

Abstract:

The Dowker complex $\text{D}\_{R}(X,Y)$ is a simplicial complex capturing the topological interplay between two finite sets $X$ and $Y$ under some relation $R\subseteq X\times Y$.
While its definition is asymmetric, the famous Dowker duality states that $\text{D}\_{R}(X,Y)$ and $\text{D}\_{R}(Y,X)$ have homotopy equivalent geometric realizations.
We introduce the Dowker-Rips complex $\text{DR}\_{R}(X,Y)$, defined as the flagification of the Dowker complex or, equivalently, as the maximal simplicial complex whose $1$-skeleton coincides with that of $\text{D}\_{R}(X,Y)$.
This is motivated by applications in topological data analysis, since as a flag complex, the Dowker-Rips complex is less expensive to compute than the Dowker complex.
While the Dowker duality does not hold for Dowker-Rips complexes in general, we show that one still has that $\text{H}\_{i}(\text{DR}\_{R}(X,Y))\cong\text{H}\_{i}(\text{DR}\_{R}(Y,X))$ for $i=0,1$.
We further show that this weakened duality extends to the setting of persistent homology, and quantify the "failure" of the Dowker duality in homological dimensions higher than $1$ by means of interleavings.
This makes the Dowker-Rips complex a less expensive, approximate version of the Dowker complex that is usable in topological data analysis.
Indeed, we provide a Python implementation of the Dowker-Rips complex and, as an application, we show that it can be used as a drop-in replacement for the Dowker complex in a tumor microenvironment classification pipeline.
In that pipeline, using the Dowker-Rips complex leads to increase in speed while retaining classification performance.

Primary Area: other topics in machine learning (i.e., none of the above)

Submission Number: 19852