Dr. Mathieu Carrière
Title: An introduction to Topological Data Analysis
Topological Data Analysis (TDA) is a growing field of research at the intersection of data science and computational geometry and topology. It has encountered key successes in several different applications (ranging from cancer subtype identification in bioinformatics to shape recognition in computer vision, just to name a few), and become the landmark product of several companies in the recent years. Indeed, many data sets nowadays come in the form of point clouds embedded in very large dimensional spaces, yet concentrated around low-dimensional geometric structures that need to be uncovered. Unraveling these structures is precisely the goal of TDA, which can build descriptors that can reliably capture geometric and topological information (connectivity, loops, holes, curvature, etc.) from the data sets without the need for an explicit mapping to lower-dimensional space. This is extremely useful since the hidden, non-trivial topology of many data sets can make it very challenging to perform well for classical techniques in data science and machine learning, such as dimensionality reduction.
In this talk, I will perform a global overview of TDA, by introducing its main descriptors (namely, the Mapper complex and the persistence diagrams), and by presenting the theoretical guarantees that they enjoy. I will also show how they can be efficiently computed in practice with the dedicated, open-source library GUDHI, and describe some applications where TDA proved useful.
Prof. Pierre Alliez
Title: Robust Geometry Processing
The lack of robustness in current geometry processing software makes it impossible to streamline the processing pipeline. Stringent requirements for input data for the outputs prevent the software building blocks from working together seamlessly. The quest for robustness requires devising algorithms that are reliable on real-world computers (which have inherent imperfections due to finite arithmetic precision). In addition, and in response to the increasing trend for geometry to be acquired through measurements, it requires devising algorithms that are resilient to real-world data. This trend means that the common assumptions applied to input data clash with the practical reality of real-world datasets, which are riddled with imperfections.
In the first part of this talk I will briefly review the methods dealing with finite arithmetic precision. I will then review several ideas devised to tackle the resilience to imperfect data, with application to several key problems in geometry processing: reconstruction, approximation, dense correspondence and mesh generation.
Prof. Dr. Bettina Speckmann
Title: Geometric Algorithms in the Field
Computational geometry is the area within algorithms research dealing with the design and analysis of algorithms and data structures for spatial data. It combines clever algorithmic techniques with beautiful geometric concepts to obtain efficient solutions to algorithmic problems involving points, lines, and higher-dimensional geometric objects. In this talk I will present a variety of results – ranging from purely theoretical to experimental – which highlight the use of computational geometry techniques in two different application areas: moving object analysis and digital humanities. Specifically, in the first part of the lecture I will discuss algorithms for the analysis and visualization of groups of moving objects, and in the second part I will discuss an agglomerative clustering problem motivated by geo-visualization for very large library collections.