We consider the problem of computing persistent homology (PH) for large-scale Euclidean point cloud data, aimed at downstream machine learning tasks, where the exponential growth of the most widely-used Vietoris-Rips complex imposes serious computational limitations. Although more scalable alternatives such as the Alpha complex or sparse Rips approximations exist, they often still result in a prohibitively large number of simplices. This poses challenges in the complex construction and in the subsequent PH computation, prohibiting their use on large-scale point clouds. To mitigate these issues, we introduce the Flood complex, inspired by the advantages of the Alpha and Witness complex constructions. Informally, at a given filtration value $r\geq 0$, the Flood complex contains all simplices from a Delaunay triangulation of a small subset of a point cloud $X$ that are fully covered by the union of balls of radius $r$ emanating from $X$, a process we call flooding. Our construction allows for efficient PH computation, possesses several desirable theoretical properties, and is amenable to GPU parallelization. Scaling experiments on 3D point cloud data show that we can compute PH of up to dimension 2 on several millions of points. Importantly, when evaluating object classification performance on real-world and synthetic data, we provide evidence that this scaling capability is needed, especially if objects are geometrically or topologically complex, yielding performance superior to other PH-based methods and neural networks for point cloud data. Source code and datasets are available on GitHub: https://github.com/plus-rkwitt/flooder

We consider the problem of learning the dynamics in the topology of time-evolving point clouds, the prevalent spatiotemporal model for systems exhibiting collective behavior, such as swarms of insects and birds or particles in physics. In such systems, patterns emerge from (local) interactions among self-propelled entities. While several well-understood governing equations for motion and interaction exist, they are notoriously difficult to fit to data, as most prior work requires knowledge about individual motion trajectories, i.e., a requirement that is challenging to satisfy with an increasing number of entities. To evade such confounding factors, we investigate collective behavior from a _topological perspective_, but instead of summarizing entire observation sequences (as done previously), we propose learning a latent dynamical model from topological features _per time point_. The latter is then used to formulate a downstream regression task to predict the parametrization of some a priori specified governing equation. We implement this idea based on a latent ODE learned from vectorized (static) persistence diagrams and show that a combination of recent stability results for persistent homology justifies this modeling choice. Various (ablation) experiments not only demonstrate the relevance of each model component but provide compelling empirical evidence that our proposed model -- _Neural Persistence Dynamics_ -- substantially outperforms the state-of-the-art across a diverse set of parameter regression tasks.

We consider the problem of variational Bayesian inference in a latent variable model where a (possibly complex) observed stochastic process is governed by the unobserved solution of a latent stochastic differential equation (SDE). Motivated by the challenges that arise when trying to learn a latent SDE in $\mathbb{R}^n$ from large-scale data, such as efficient gradient computation, we take a step back and study a specific subclass instead. In our case, the SDE evolves inside a homogeneous latent space and is induced by stochastic dynamics of the corresponding (matrix) Lie group. In the context of learning problems, SDEs on the $n$-dimensional unit sphere are arguably the most relevant incarnation of this setup. For variational inference, the sphere not only facilitates using a uniform prior on the initial state of the SDE, but we also obtain a particularly simple and intuitive expression for the KL divergence between the approximate posterior and prior process in the evidence lower bound. We provide empirical evidence that a latent SDE of the proposed type can be learned efficiently by means of an existing one-step geometric Euler-Maruyama scheme. Despite restricting ourselves to a less diverse class of SDEs, we achieve competitive or even state-of-the-art performance on a collection of time series interpolation and classification benchmarks.

We study the excess capacity of deep networks in the context of supervised classification. That is, given a capacity measure of the underlying hypothesis class - in our case, empirical Rademacher complexity - to what extent can we (a priori) constrain this class while retaining an empirical error on a par with the unconstrained regime? To assess excess capacity in modern architectures (such as residual networks), we extend and unify prior Rademacher complexity bounds to accommodate function composition and addition, as well as the structure of convolutions. The capacity-driving terms in our bounds are the Lipschitz constants of the layers and a (2,1) group norm distance to the initializations of the convolution weights. Experiments on benchmark datasets of varying task difficulty indicate that (1) there is a substantial amount of excess capacity per task, and (2) capacity can be kept at a surprisingly similar level across tasks. Overall, this suggests a notion of compressibility with respect to weight norms, complementary to classic compression via weight pruning. Source code is available at https://github.com/rkwitt/excess_capacity.

The problem of (point) forecasting univariate time series is considered. Most approaches, ranging from traditional statistical methods to recent learning-based techniques with neural networks, directly operate on raw time series observations. As an extension, we study whether local topological properties, as captured via persistent homology, can serve as a reliable signal that provides complementary information for learning to forecast. To this end, we propose topological attention, which allows attending to local topological features within a time horizon of historical data. Our approach easily integrates into existing end-to-end trainable forecasting models, such as N-BEATS, and, in combination with the latter exhibits state-of-the-art performance on the large-scale M4 benchmark dataset of 100,000 diverse time series from different domains. Ablation experiments, as well as a comparison to recent techniques in a setting where only a single time series is available for training, corroborate the beneficial nature of including local topological information through an attention mechanism.

We study regularization in the context of small sample-size learning with over-parametrized neural networks. Specifically, we shift focus from architectural properties, such as norms on the network weights, to properties of the internal representations before a linear classifier. Specifically, we impose a topological constraint on samples drawn from the probability measure induced in that space. This provably leads to mass concentration effects around the representations of training instances, i.e., a property beneficial for generalization. By leveraging previous work to impose topological constrains in a neural network setting, we provide empirical evidence (across various vision benchmarks) to support our claim for better generalization.

We propose an approach to learning with graph-structured data in the problem domain of graph classification. In particular, we present a novel type of readout operation to aggregate node features into a graph-level representation. To this end, we leverage persistent homology computed via a real-valued, learnable, filter function. We establish the theoretical foundation for differentiating through the persistent homology computation. Empirically, we show that this type of readout operation compares favorably to previous techniques, especially when the graph connectivity structure is informative for the learning problem.

End-user involvement in software engineering is an ambivalent topic. However, novel paradigms such as service-oriented computing suggest more active end-user involvement to gather individual needs for software personalization. In this paper, we present a mobile requirements elicitation tool which enables end-users to blog needs in situ without analysts' facilitation.

Trace links between requirements and code are essential for many software development and maintenance activities. Despite significant advances in traceability research, creating links remains a human-intensive activity and surprisingly little is known about how humans perform basic tracing tasks. We investigate fundamental research questions regarding the effort and quality of recovering traces between requirements and code. Our paper presents two exploratory experiments conducted with 100 subjects who recovered trace links for two open source software systems in a controlled environment. In the first experiment, subjects recovered trace links between the two systems' requirements and classes of the implementation. In the second experiment, trace links were established between requirements and individual methods of the implementation. In order to assess the validity of the trace links cast by subjects, key developers of the two software systems participated in our research and provided benchmarks. Our study yields surprising observations: trace capture is surprisingly fast and can be done within minutes even for larger classes; the quality of the captured trace links, while good, does not improve with higher trace effort; and it is not harder though slightly more expensive to recover the trace links for larger, more complex classes.

Eliciting requirements on-site is challenging as analysts have to simultaneously observe the environment, interact with people and operate RE tools. We have been exploring the use of context-aware technologies to provide better guidance and support for on-site analysts. Our context-aware Mobile Scenario Presenter tool guides analysts by automatically highlighting scenario events relevant to the currently observed work task.

Researchers highlight end-user involvement in system design as an important concept for developing useful and usable solutions. However, end-user involvement in software engineering is still an open-ended topic. Novel paradigms such as service-oriented computing strengthen the need for more active end-user involvement in order to provide systems that are tailored to individual end-user needs. Our work is based on the fact that the majority of end-users are familiar with mobile devices and use an increasing number of mobile applications. A mobile tool enabling end-user led requirements elicitation could be just one of many applications installed on end-users' mobile devices. In this paper, we present a framework of end-user involvement in requirements elicitation which motivates our research. The main contribution of our research is a tool-supported requirements elicitation approach allowing end-users to document needs in situ. Furthermore, we present first evaluation results to highlight the feasibility of on-site end-user led requirements elicitation.

Scenarios are widely and successfully used to discover requirements for software-intensive systems. Recent advances in mobile computing technologies mean that mobile tools can support scenario-based techniques in the workplace, with potential benefits to requirements processes. To explore this idea we have been developing the Mobile Scenario Presenter (MSP) tool and used it successfully in a number of evaluation studies and projects. The MSP supports analysts working in the field to better understand the work context of a future system 's end users. The interactive tool demo allows RE experts to actively use the MSP to explore its capabilities and benefits.

In this paper different methods for the statistical modelling of time variant processes for interference analysis in radio network planning tools are presented. These methods will be pointed out exemplarily for two network features, frequency hopping (FH) and base station power control (BTS power control). The methods are presently being implemented in a radio network planning tool used for planning the GSM network of Mannesmann Mobilfunk GmbH.