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Section: New Results

Topological approach for multimodal data processing

On the Stability of Functional Maps and Shape Difference Operators

Participants : Frédéric Chazal, Ruqi Huang, Maks Ovsjanikov.

In this paper, we provide stability guarantees for two frameworks that are based on the notion of functional maps. We consider two types of perturbations in our analysis: one is on the input shapes and the other is on the change in scale. In theory, we formulate and justify the robustness that has been observed in practical implementations of those frameworks. Inspired by our theoretical results, we propose a pipeline for constructing shape difference operators on point clouds and show numerically that the results are robust and informative. In particular, we show that both the shape difference operators and the derived areas of highest distortion are stable with respect to changes in shape representation and change of scale. Remarkably, this is in contrast with the well-known instability of the eigenfunctions of the Laplace-Beltrami operator computed on point clouds compared to those obtained on triangle meshes.

Local Equivalence and Intrinsic Metrics Between Reeb Graphs

Participants : Steve Oudot, Mathieu Carrière.

As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent contributions emphasize this aspect, proposing novel distances such as functional distortion or interleaving that are provably more discriminative than the so-called bottleneck distance, being true metrics whereas the latter is only a pseudo-metric. Their main drawback compared to the bottleneck distance is to be comparatively hard (if at all possible) to evaluate. In [28] we take the opposite view on the problem and show that the bottleneck distance is in fact good enough locally, in the sense that it is able to discriminate a Reeb graph from any other Reeb graph in a small enough neighborhood, as efficiently as the other metrics do. This suggests considering the intrinsic metrics induced by these distances, which turn out to be all globally equivalent. This novel viewpoint on the study of Reeb graphs has a potential impact on applications, where one may not only be interested in discriminating between data but also in interpolating between them.

Structure and Stability of the One-Dimensional Mapper

Participants : Steve Oudot, Mathieu Carrière.

Given a continuous function f:X->R and a cover I of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover f-1(I). Despite its success in applications, little is known about the structure and stability of this construction from a theoretical point of view. As a pixelized version of the Reeb graph of f, it is expected to capture a subset of its features (branches, holes), depending on how the interval cover is positioned with respect to the critical values of the function. Its stability should also depend on this positioning. In [16] we propose a theoretical framework relating the structure of the Mapper to that of the Reeb graph, making it possible to predict which features will be present and which will be absent in the Mapper given the function and the cover, and for each feature, to quantify its degree of (in-)stability. Using this framework, we can derive guarantees on the structure of the Mapper, on its stability, and on its convergence to the Reeb graph as the granularity of the cover I goes to zero.