Satellites are leaving the realms of niche use, extending our day-to-day networked infrastructure to space - thereby forcing a generalization of network architectures. The Delay Tolerant Networking (DTN) protocol is being developed to give rise to this new Solar System Internet. Predominantly, DTNs in space use globally-distributed contact tables to compute routes. In this paper, we propose and analyze a novel optimized approach for route computations that improves upon traditional approaches. As the general DTN will always include some scheduled links, our new algorithm enables greater scalability and practicality of DTN routing. These contact tables include windows when two nodes can communicate and were classically organized into a contact graph, where the vertices represent contact opportunities. Because the complexity of a contact graph grows with the number of contacts, pathfinding on it does not scale. A new structure using multigraphs with the same data is proposed. We show that a multigraph-based approach, which we call contact multigrapb routing, exhibits performance superior to routing based on contact graphs, allowing greater scaling to schedule-based routing. In this paper, the multigraph-based algorithm is detailed and a proof is included showing it outperforms the previous algorithm given the same input. Pseudocode is included, as are simulation results. We conclude with suggested future work.

Communications networks involving space systems are complex and evolve over time. Orbital mechanics, light propagation, and asset availability are some characteristics that dictate the state of the space network over time, which means that traditional routing methods relying on e.g. Open Shortest Path First (OSPF) cannot be expected to work. This is mainly because traditional routing algorithms assume that the network parameters are not themselves dependent on time, and make assumptions regarding end-to-end connectivity. In order for a routing algorithm to work in a space network, the algorithm must, among other things, account for the fact that the links are constantly changing over time; rather than forwarding and dropping, routers must also be able to store data. In this paper, the authors propose a new method for computing optimal routes through a space network using tropical geometry. This method determines the times at which the single-source shortest-path (SSSP) tree changes using the Joswig algorithm [1]. The efficacy of this method is demonstrated through simulation using orbital analysis software.

Point clouds and sets are input data-types which pose unique problems to deep learning. Since sets can have variable cardinality and are unchanged by permutation, the input space for these problems naturally form infinite-dimensional non-Euclidean spaces. Despite these mathematical difficulties, PointNet (Qi et al. 2017) and Deep Sets (Zaheer et al. 2017) introduced foundational neural network architectures to address these problems. In this paper we present a unified framework to study the expressive power of such networks as well as their extensions beyond point clouds (partially addressing a conjecture on the extendibility of DeepSets along the way). To this end, we demonstrate the crucial role that the Hausdorff and Wasserstein metrics play and prove new cardinality-agnostic universality results to characterize exactly which functions can be approximated by these models. In particular, these results imply that PointNet generally cannot approximate averages of continuous functions over sets (e.g. center-of-mass or higher moments) implying that DeepSets is strictly more expressive than PointNet in the constant cardinality setting. Moreover, we obtain explicit lower-bounds on the approximation error and present a simple method to produce arbitrarily many examples of this failure-mode. Counterintuitively, we also prove that in the unbounded cardinality setting that any function which can be uniformly approximated by both PointNet and normalized-DeepSets must be constant. Finally, we also prove theorems on the Lipschitz properties of PointNet and normalized-DeepSets which shed insight into exploitable inductive bias in these 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.

In recent years, persistent homology has become an attractive method for data analysis. It captures topological features, such as connected components, holes, and voids from point cloud data and summarizes the way in which these features appear and disappear in a filtration sequence. In this project, we focus on improving the performance of Eirene, a computational package for persistent homology. Eirene is a 5000-line open-source software library implemented in the dynamic programming language Julia. We use the Julia profiling tools to identify performance bottlenecks and develop novel methods to manage them, including the parallelization of some time-consuming functions on multicore/manycore hardware. Empirical results show that performance can be greatly improved.

In recent years, persistent homology has become an attractive method for data analysis. It captures topological features, such as connected components, holes, voids, etc., from a point cloud by finding out when these features appear and disappear in the filtration sequence. In this project, we focus on improving the performance of Eirene, a fancy computational persistent homology package. Eirene is a 5000-line open-source software implemented by using the dynamic programming language Julia. We use the Julia profiling tools to identify the performance bottlenecks and develop different methods to manage the bottlenecks, including the parallelization of some time-consuming functions on the multicore/manycore hardware. The empirical results show that the performance can be greatly improved.