Merging Discrete Morse Vector Fields: A Case of Stubborn Geometric Parallelization

We address the basic question in discrete Morse theory of combining discrete gradient fields that are partially defined on subsets of the given complex. This is a well-posed question when the discrete gradient field V is generated using a fixed algorithm which has a local nature. One example is ProcessLowerStars, a widely used algorithm for computing persistent homology associated to a grey-scale image in 2D or 3D. While the algorithm for V may be inherently local, being computed within stars of vertices and so embarrassingly parallelizable, in practical use, it is natural to want to distribute the computation over patches Pi, apply the chosen algorithm to compute the fields Vi associated to each patch, and then assemble the ambient field V from these. Simply merging the fields from the patches, even when that makes sense, gives a wrong answer. We develop both very general merging procedures and leaner versions designed for specific, easy-to-arrange covering patterns.

ES-ImageNet: A Million Event-Stream Classification Dataset for Spiking Neural Networks

With event-driven algorithms, especially spiking neural networks (SNNs), achieving continuous improvement in neuromorphic vision processing, a more challenging event-stream dataset is urgently needed. However, it is well-known that creating an ES-dataset is a time-consuming and costly task with neuromorphic cameras like dynamic vision sensors (DVS). In this work, we propose a fast and effective algorithm termed Omnidirectional Discrete Gradient (ODG) to convert the popular computer vision dataset ILSVRC2012 into its event-stream (ES) version, generating about 1,300,000 frame-based images into ES-samples in 1,000 categories. In this way, we propose an ES-dataset called ES-ImageNet, which is dozens of times larger than other neuromorphic classification datasets at present and completely generated by the software. The ODG algorithm implements image motion to generate local value changes with discrete gradient information in different directions, providing a low-cost and high-speed method for converting frame-based images into event streams, along with Edge-Integral to reconstruct the high-quality images from event streams. Furthermore, we analyze the statistics of ES-ImageNet in multiple ways, and a performance benchmark of the dataset is also provided using both famous deep neural network algorithms and spiking neural network algorithms. We believe that this work shall provide a new large-scale benchmark dataset for SNNs and neuromorphic vision.

Ripser: efficient computation of Vietoris–Rips persistence barcodes

We present an algorithm for the computation of Vietoris–Rips persistence barcodes and describe its implementation in the software Ripser. The method relies on implicit representations of the coboundary operator and the filtration order of the simplices, avoiding the explicit construction and storage of the filtration coboundary matrix. Moreover, it makes use of apparent pairs, a simple but powerful method for constructing a discrete gradient field from a total order on the simplices of a simplicial complex, which is also of independent interest. Our implementation shows substantial improvements over previous software both in time and memory usage.

Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

The combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from specific coordinate systems and does so robustly to noise. Moreover, the geometric content of a discrete gradient vector field is very useful for visualization purposes. The specific case of multivariate data still demands for further investigations, on the one hand, for computational reasons, it is important to reduce the necessary amount of data to be processed. On the other hand, for analysis reasons, the multivariate case requires the detection and interpretation of the possible interdepedance among data components. To this end, in this paper we introduce and study a notion of perfectness for discrete gradient vector fields with respect to multi-parameter persistent homology, called relative-perfectness. As a natural generalization of usual perfectness in Morse theory for homology, relative-perfectness entails having the least number of critical cells relevant for multi-parameter persistence. As a first contribution, we support our definition of relative-perfectness by generalizing Morse inequalities to the filtration structure where homology groups involved are relative with respect to subsequent sublevel sets. In order to allow for an interpretation of critical cells in 2-parameter persistence, our second contribution consists of two inequalities bounding Betti tables of persistence modules from above and below, via the number of critical cells. Our last result is the proof that existing algorithms based on local homotopy expansions allow for efficient computability over simplicial complexes up to dimension 2.

Numerical methods preserving multiple Hamiltonians for stochastic Poisson systems

<p style='text-indent:20px;'>In this paper, we propose a class of numerical schemes for stochastic Poisson systems with multiple invariant Hamiltonians. The method is based on the average vector field discrete gradient and an orthogonal projection technique. The proposed schemes preserve all the invariant Hamiltonians of the stochastic Poisson systems simultaneously, with possibility of achieving high convergence orders in the meantime. We also prove that our numerical schemes preserve the Casimir functions of the systems under certain conditions. Numerical experiments verify the theoretical results and illustrate the effectiveness of our schemes.</p>