Introduction to Persistent Homology

This video presents an introduction to persistent homology, aimed at a viewer who has mathematical aptitude but not necessarily knowledge of algebraic topology. Persistent homology is an algebraic method of discerning the topological features of complex data, which in recent years has found applications in fields such as image processing and biological systems. Using smooth animations, the video conveys the intuition behind persistent homology, while giving a taste of its key properties, applications, and mathematical underpinnings

Topological image modification for object detection and topological image processing of skin lesions

We propose a new method based on Topological Data Analysis (TDA) consisting of Topological Image Modification (TIM) and Topological Image Processing (TIP) for object detection. Through this newly introduced method, we artificially destruct irrelevant objects, and construct new objects with known topological properties in irrelevant regions of an image. This ensures that we are able to identify the important objects in relevant regions of the image. We do this by means of persistent homology, which allows us to simultaneously select appropriate thresholds, as well as the objects corresponding to these thresholds, and separate them from the noisy background of an image. This leads to a new image, processed in a completely unsupervised manner, from which one may more efficiently extract important objects. We demonstrate the usefulness of this proposed method for topological image processing through a case-study of unsupervised segmentation of the ISIC 2018 skin lesion images. Code for this project is available on https://bitbucket.org/ghentdatascience/topimgprocess

Topological descriptors for 3D surface analysis

We investigate topological descriptors for 3D surface analysis, i.e. the classification of surfaces according to their geometric fine structure. On a dataset of high-resolution 3D surface reconstructions we compute persistence diagrams for a 2D cubical filtration. In the next step we investigate different topological descriptors and measure their ability to discriminate structurally different 3D surface patches. We evaluate their sensitivity to different parameters and compare the performance of the resulting topological descriptors to alternative (non-topological) descriptors. We present a comprehensive evaluation that shows that topological descriptors are (i) robust, (ii) yield state-of-the-art performance for the task of 3D surface analysis and (iii) improve classification performance when combined with non-topological descriptors.Comment: 12 pages, 3 figures, CTIC 201

Characterizing the Shape of Activation Space in Deep Neural Networks

The representations learned by deep neural networks are difficult to interpret in part due to their large parameter space and the complexities introduced by their multi-layer structure. We introduce a method for computing persistent homology over the graphical activation structure of neural networks, which provides access to the task-relevant substructures activated throughout the network for a given input. This topological perspective provides unique insights into the distributed representations encoded by neural networks in terms of the shape of their activation structures. We demonstrate the value of this approach by showing an alternative explanation for the existence of adversarial examples. By studying the topology of network activations across multiple architectures and datasets, we find that adversarial perturbations do not add activations that target the semantic structure of the adversarial class as previously hypothesized. Rather, adversarial examples are explainable as alterations to the dominant activation structures induced by the original image, suggesting the class representations learned by deep networks are problematically sparse on the input space

Granular Response to Impact: Topology of the Force Networks

Impact of an intruder on granular matter leads to formation of mesoscopic force networks seen particularly clearly in the recent experiments carried out with photoelastic particles, e.g., Clark et al., Phys. Rev. Lett., 114 144502 (2015). These force networks are characterized by complex structure and evolve on fast time scales. While it is known that total photoelastic activity in the granular system is correlated with the acceleration of the intruder, it is not known how the structure of the force network evolves during impact, and if there is a dominant features in the networks that can be used to describe intruder's dynamics. Here, we use topological tools, in particular persistent homology, to describe these features. Persistent homology allows quantification of both structure and time evolution of the resulting force networks. We find that there is a clear correlation of the intruder's dynamics and some of the topological measures implemented. This finding allows us to discuss which properties of the force networks are most important when attempting to describe intruder's dynamics. Regarding temporal evolution of the networks, we are able to define the upper bound on the relevant time scale on which the networks evolve

TopologyNet: Topology based deep convolutional neural networks for biomolecular property predictions

Although deep learning approaches have had tremendous success in image, video and audio processing, computer vision, and speech recognition, their applications to three-dimensional (3D) biomolecular structural data sets have been hindered by the entangled geometric complexity and biological complexity. We introduce topology, i.e., element specific persistent homology (ESPH), to untangle geometric complexity and biological complexity. ESPH represents 3D complex geometry by one-dimensional (1D) topological invariants and retains crucial biological information via a multichannel image representation. It is able to reveal hidden structure-function relationships in biomolecules. We further integrate ESPH and convolutional neural networks to construct a multichannel topological neural network (TopologyNet) for the predictions of protein-ligand binding affinities and protein stability changes upon mutation. To overcome the limitations to deep learning arising from small and noisy training sets, we present a multitask topological convolutional neural network (MT-TCNN). We demonstrate that the present TopologyNet architectures outperform other state-of-the-art methods in the predictions of protein-ligand binding affinities, globular protein mutation impacts, and membrane protein mutation impacts.Comment: 20 pages, 8 figures, 5 table