Section: New Results

Topological approach for multimodal data processing

Persistence-based Pooling for Shape Pose Recognition

Participants : Thomas Bonis, Frédéric Chazal, Steve Oudot, Maksim Ovsjanikov.

We propose a novel pooling approach for shape classification and recognition using the bag-of-words pipeline, based on topological persistence, a recent tool from Topological Data Analysis. Our technique extends the standard max-pooling, which summarizes the distribution of a visual feature with a single number, thereby losing any notion of spatiality. Instead, we propose to use topological persistence, and the derived persistence diagrams, to provide significantly more informative and spatially sensitive characterizations of the feature functions, which can lead to better recognition performance. Unfortunately, despite their conceptual appeal, persistence diagrams are difficult to handle, since they are not naturally represented as vectors in Euclidean space and even the standard metric, the bottleneck distance is not easy to compute. Furthermore, classical distances between diagrams, such as the bottleneck and Wasserstein distances, do not allow to build positive definite kernels that can be used for learning. To handle this issue, we provide a novel way to transform persistence diagrams into vectors, in which comparisons are trivial. Finally, we demonstrate the performance of our construction on the Non-Rigid 3D Human Models SHREC 2014 dataset, where we show that topological pooling can provide significant improvements over the standard pooling methods for the shape pose recognition within the bag-of-words pipeline [23].

Structure and Stability of the 1-Dimensional Mapper

Participants : Steve Oudot, Mathieu Carrière.

Given a continuous function $f:X\to ℝ$ and a cover $ℐ$ of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover $f^{-1}\left(ℐ\right)$. 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. We propose a theoretical framework that relates the structure of the Mapper to the one 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 [25].

Decomposition of exact pfd persistence bimodules

Participants : Steve Oudot, Jérémy Cochoy.

We characterize the class of persistence modules indexed over ${ℝ}^2$ that are decomposable into summands whose support have the shape of a block—i.e. a horizontal band, a vertical band, an upper-right quadrant, or a lower-left quadrant. Assuming the modules are pointwise finite-dimensional (pfd), we show that they are decomposable into block summands if and only if they satisfy a certain local property called exactness. Our proof follows the same scheme as the proof of decomposition for pfd persistence modules indexed over $ℝ$, yet it departs from it at key stages due to the product order not being a total order on ${ℝ}^2$, which leaves some important gaps open. These gaps are filled in using more direct arguments. Our work is motivated primarily by the stability theory for zigzags and interlevel-sets persistence modules, in which block-decomposable bimodules play a key part. Our results allow us to drop some of the conditions under which that theory holds, in particular the Morse-type conditions [39].