Combining Persistent Homology and Invariance Groups for Shape Comparison

Abstract

In many applications concerning the comparison of data expressed by 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 Homeo(X) of all selfhomeomorphisms of X, this theory is not tailored to manage the case in which G is a proper subgroup of 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 Ginvariant 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.

Towards a topological–geometrical theory of group equivariant non-expansive operators for data analysis and machine learning

“…Several results in this section and in Section 6 mimic the corresponding results in [3], where the particular case (Φ, G) = (Ψ, H), T = Id : G → H is considered. Note that considering different function spaces and different groups of equivariance is fundamental, as it allows one to compose operators hierarchically, in the same fashion as computational units are linked in an artificial neural network.…”

“…from G × G to R (see Appendix A). 3 The definition of isometry between pseudo-metric spaces can be considered as a special case of isometry between metric spaces. Let (X 1 , d 1 ) and (X 2 , d 2 ) be two pseudo-metric spaces.…”

“…We define the natural pseudo-distance d G on the space Φ [3]. The natural pseudo-distance d G represents the ground truth in our model.…”

“…A. Metric learned with filters of size px on selected and sampled IENEOs on classes (3,8) and (5,7 Figure 4. We reproduced three times the same experiment by varying the size and the number of 1-dimensional Gaussians used to initialise the IENEOs.…”

The persistent homotopy type distance

“…2.3. The importance of choosing a subcategory C of S. The main motivation for considering a subcategory C of S instead of only the category S is to generalize the natural pseudo-distance, whose definition depends on the selection of a set of objects and a set of morphisms that may be respectively smaller than the set of all real-valued continuous functions and the set of all homeomorphisms (cf., e.g., Section 7.1 in [2], [6] and [18]). The following examples show that this choice is fruitful.…”