Combining persistent homology and invariance groups for shape comparison
This work addresses the need for more tailored topological data analysis in applications requiring invariance to specific transformations, such as shape comparison, but it is incremental as it builds on existing persistent homology methods.
The paper tackles the problem of adapting persistent homology to achieve invariance with respect to specific groups of self-homeomorphisms, rather than the full homeomorphism group, by proposing a dual approach based on G-invariant non-expanding operators. It presents theoretical results and demonstrates the method in experiments comparing 1D-signals under various invariance groups and for image comparison.
In many applications concerning the comparison of data expressed by $\mathbb{R}^m$-valued functions defined on a topological space $X$, the invariance with respect to a given group $G$ of self-homeomorphisms of $X$ is required. While persistent homology is quite efficient in the topological and qualitative comparison of this kind of data when the invariance group $G$ is the group $\mathrm{Homeo}(X)$ of all self-homeomorphisms of $X$, this theory is not tailored to manage the case in which $G$ is a proper subgroup of $\mathrm{Homeo}(X)$, and its invariance appears too general for several tasks. This paper proposes a way to adapt persistent homology in order to get invariance just with respect to a given group of self-homeomorphisms of $X$. The main idea consists in a dual approach, based on considering the set of all $G$-invariant non-expanding operators defined on the space of the admissible filtering functions on $X$. Some theoretical results concerning this approach are proven and two experiments are presented. An experiment illustrates the application of the proposed technique to compare 1D-signals, when the invariance is expressed by the group of affinities, the group of orientation-preserving affinities, the group of isometries, the group of translations and the identity group. Another experiment shows how our technique can be used for image comparison.