Signal Processing on Product Spaces

We establish a framework for signal processing on product spaces of simplicial and cellular complexes. For simplicity, we focus on the product of two complexes representing time and space, although our results generalize naturally to products of simplicial complexes of arbitrary dimension. Our framework leverages the structure of the eigenmodes of the Hodge Laplacian of the product space to jointly filter along time and space. To this end, we provide a decomposition theorem of the Hodge Laplacian of the product space, which highlights how the product structure induces a decomposition of each eigenmode into a spatial and temporal component. Finally, we apply our method to real world data, specifically for interpolating trajectories of buoys in the ocean from a limited set of observed trajectories

Analysis of vulnerabilities in Internet firewalls

Firewalls protect a trusted network from an untrusted network by filtering traffic according to a specified security policy. A diverse set of firewalls is being used today. As it is infeasible to examine and test each firewall for all possible potential problems, a taxonomy is needed to understand firewall vulnerabilities in the context of firewall operations. This paper describes a novel methodology for analyzing vulnerabilities in Internet firewalls. A firewall vulnerability is defined as an error made during firewall design, implementation, or configuration, that can be exploited to attack the trusted network that the firewall is supposed to protect. We examine firewall internals, and cross reference each firewall operation with causes and effects of weaknesses in that operation, analyzing twenty reported problems with available firewalls. The result of our analysis is a set of matrices that illustrate the distribution of firewall vulnerability causes and effects over firewall operations. These matrices are useful in avoiding and detecting unforeseen problems during both firewall implementation and firewall testing. Two case studies of Firewall-1 and Raptor illustrate our methodology

ICML 2023 Topological Deep Learning Challenge:Design and Results

This paper presents the computational challenge on topological deep learning that was hosted within the ICML 2023 Workshop on Topology and Geometry in Machine Learning. The competition asked participants to provide open-source implementations of topological neural networks from the literature by contributing to the python packages TopoNetX (data processing) and TopoModelX (deep learning). The challenge attracted twenty-eight qualifying submissions in its two month duration. This paper describes the design of the challenge and summarizes its main findings.</p

ICML 2023 topological deep learning challenge. Design and results

This paper presents the computational challenge on topological deep learning that was hosted within the ICML 2023 Workshop on Topology and Geometry in Machine Learning. The competition asked participants to provide open-source implementations of topological neural networks from the literature by contributing to the python packages TopoNetX (data processing) and TopoModelX (deep learning). The challenge attracted twenty-eight qualifying submissions in its two-month duration. This paper describes the design of the challenge and summarizes its main finding