About me
In Algebraic Topology, persistent homology is a methodology to compute the homology groups of a simplicial complexes endowed with a filtration. The simplicial complex is usually constructed from a topological space and homology groups persisting over a large range of the filtration can be interpreted as meaningful features of the underlying topological space. Datasets often have interesting structure, but due to high dimensionality and the discrete nature this structure is difficult to quantify. Through the construction of a Vietoris-Rips complex and a computation of the persistent homology one is able to make qualitative statements on the structure, or loosly speaking shape, of the underlying topological space from which the original dataset was sampled. The resulting so-called persistence diagram can be interpreted as the number of connected components, loops and voids in the dataset at various scales, providing valuable information to practitioners. We aim to study how the topological information of datasets can be exploited to create deep learning architectures that are both sparser and more expressive. We hope to apply this to expand and combine on the work done and apply results to biomedical data.
Interests
Bonbons
Exploring different flavour combinations that speak to the imagination and invoke conversation is a passion of mine. Chocolate does not only works amazingly well with a large variety spices, such as nutmeg, pepper, paprika and kardamom, but also with herbs and aromatics such as rozemary, lemongrass and rozes.
Photography
Photography drives the photographer to reevaluate the way we look at things, finding the unique angles or hidden detail we often overlook. It is this sort of cat and mouse game and the sheer depth one can find in seemingly simple objects that is truly exhilarating
Hiking
During the weekends I love to explore the surrounding mountains just south of Munich. The nature is wonderful and lets you revitalize during the weekends.
Research
Differentiable Euler Characteristic Transforms for Shape Classification
The Euler Characteristic Transform (ECT) has proven to be a powerful representation, combining geometrical and topological characteristics of shapes and graphs. However, the ECT was hitherto unable to learn task-specific representations. We overcome this issue and develop a novel computational layer that enables learning the ECT in an end-to-end fashion. Our method, the Differentiable Euler Characteristic Transform (DECT), is fast and computationally efficient, while exhibiting performance on a par with more complex models in both graph and point cloud classification tasks. Moreover, we show that this seemingly simple statistic provides the same topological expressivity as more complex topological deep learning layers.
Normal form for maps with nilpotent linear part
The normal form for an n-dimensional map with irreducible nilpotent linear part is determined using sl2-representation theory. We sketch by example how the reducible case can also be treated in an algorithmic manner. The construction (and proof) of the sl2-triple from the nilpotent linear part is more complicated than one would hope for, but once the abstract sl2 theory is in place, both the description of the normal form and the computational splitting to compute the generator of the coordinate transformation can be handled explicitly in terms of the nilpotent linear part without the explicit knowledge of the triple. Where in the vector field case one runs into invariant theoretical problems when the dimension gets larger if one wants to describe the general form of the normal form, for maps we obtain results without any restrictions on the dimension. In the literature only the 2-dimensional nilpotent case has been described sofar, as far as we know.
Generative Topology for Shape Synthesis
The Euler Characteristic Transform (ECT) is a powerful invariant for assessing geometrical and topological characteristics of a large variety of objects, including graphs and embedded simplicial complexes. Although the ECT is invertible in theory, no explicit algorithm for general data sets exists. In this paper, we address this lack and demonstrate that it is possible to learn the inversion, permitting us to develop a novel framework for shape generation tasks on point clouds. Our model exhibits high quality in reconstruction and generation tasks, affords efficient latent-space interpolation, and is orders of magnitude faster than existing methods.
MANTRA: The Manifold Triangulations Assemblage
The rising interest in leveraging higher-order interactions present in complex systems has led to a surge in more expressive models exploiting high-order structures in the data, especially in topological deep learning (TDL), which designs neural networks on high-order domains such as simplicial complexes. However, progress in this field is hindered by the scarcity of datasets for benchmarking these architectures. To address this gap, we introduce MANTRA, the first large-scale, diverse, and intrinsically high order dataset for benchmarking high-order models, comprising over 43,000 and 249,000 triangulations of surfaces and three-dimensional manifolds, respectively. With MANTRA, we assess several graph- and simplicial complex-based models on three topological classification tasks. We demonstrate that while simplicial complex-based neural networks generally outperform their graph-based counterparts in capturing simple topological invariants, they also struggle, suggesting a rethink of TDL. Thus, MANTRA serves as a benchmark for assessing and advancing topological methods, leading the way for more effective high-order models.
A Diffusion Model Predicts 3D Shapes from 2D Microscopy Images
Diffusion models are a special type of generative model, capable of synthesising new data from a learnt distribution. We introduce DISPR, a diffusion-based model for solving the inverse problem of three-dimensional (3D) cell shape prediction from two-dimensional (2D) single cell microscopy images. Using the 2D microscopy image as a prior, DISPR is conditioned to predict realistic 3D shape reconstructions. To showcase the applicability of DISPR as a data augmentation tool in a feature-based single cell classification task, we extract morphological features from the red blood cells grouped into six highly imbalanced classes. Adding features from the DISPR predictions to the three minority classes improved the macro F1 score. We thus demonstrate that diffusion models can be successfully applied to inverse biomedical problems, and that they learn to reconstruct 3D shapes with realistic morphological features from 2D microscopy images.