Approximating Persistent Homology in Euclidean Space Through Collapses

The \v{C}ech complex is one of the most widely used tools in applied algebraic topology. Unfortunately, due to the inclusive nature of the \v{C}ech filtration, the number of simplices grows exponentially in the number of input points. A practical consequence is that computations may have to terminate at smaller scales than what the application calls for. In this paper we propose two methods to approximate the \v{C}ech persistence module. Both are constructed on the level of spaces, i.e. as sequences of simplicial complexes induced by nerves. We also show how the bottleneck distance between such persistence modules can be understood by how tightly they are sandwiched on the level of spaces. In turn, this implies the correctness of our approximation methods. Finally, we implement our methods and apply them to some example point clouds in Euclidean space

Topological exploration of artificial neuronal network dynamics

One of the paramount challenges in neuroscience is to understand the dynamics of individual neurons and how they give rise to network dynamics when interconnected. Historically, researchers have resorted to graph theory, statistics, and statistical mechanics to describe the spatiotemporal structure of such network dynamics. Our novel approach employs tools from algebraic topology to characterize the global properties of network structure and dynamics. We propose a method based on persistent homology to automatically classify network dynamics using topological features of spaces built from various spike-train distances. We investigate the efficacy of our method by simulating activity in three small artificial neural networks with different sets of parameters, giving rise to dynamics that can be classified into four regimes. We then compute three measures of spike train similarity and use persistent homology to extract topological features that are fundamentally different from those used in traditional methods. Our results show that a machine learning classifier trained on these features can accurately predict the regime of the network it was trained on and also generalize to other networks that were not presented during training. Moreover, we demonstrate that using features extracted from multiple spike-train distances systematically improves the performance of our method

A Notion of Harmonic Clustering in Simplicial Complexes

We outline a novel clustering scheme for simplicial complexes that produces clusters of simplices in a way that is sensitive to the homology of the complex. The method is inspired by, and can be seen as a higher-dimensional version of, graph spectral clustering. The algorithm involves only sparse eigenproblems, and is therefore computationally efficient. We believe that it has broad application as a way to extract features from simplicial complexes that often arise in topological data analysis

FORLORN: A Framework for Comparing Offline Methods and Reinforcement Learning for Optimization of RAN Parameters

The growing complexity and capacity demands for mobile networks necessitate innovative techniques for optimizing resource usage. Meanwhile, recent breakthroughs have brought Reinforcement Learning (RL) into the domain of continuous control of real-world systems. As a step towards RL-based network control, this paper introduces a new framework for benchmarking the performance of an RL agent in network environments simulated with ns-3. Within this framework, we demonstrate that an RL agent without domain-specific knowledge can learn how to efficiently adjust Radio Access Network (RAN) parameters to match offline optimization in static scenarios, while also adapting on the fly in dynamic scenarios, in order to improve the overall user experience. Our proposed framework may serve as a foundation for further work in developing workflows for designing RL-based RAN control algorithms

Dynamics of CLIMP-63 S-acylation control ER morphology

A key player in the formation of endoplasmic reticulum sheets is CLIMP-63, but mechanistic details remained elusive. Here authors combined cellular experiments and mathematical modelling to show that S-acylation of CLIMP-63 regulates its function by mediating its oligomerisation, turnover, and localisation