Posts by Collection

Moduli Spaces of Morse Function for Persistence

Published in Journal of Applied and Computational Topology, 2020

We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning than existing topological invariants, such as persistence barcodes and Reeb graphs. We give a method to relate any two Morse–Smale vector fields on the sphere by a sequence of fundamental moves by considering graph-equivalent Morse functions. We also explore the combinatorially rich world of height-equivalent Morse functions, considered as height functions of embedded spheres in $\mathbb{R}^3$. Their level-set invariant, a poset generated by nested disks and annuli from levels sets, gives insight into the moduli space of Morse functions sharing the same persistence barcode.

Recommended citation: Catanzaro, M. J., Curry, J. M., Fasy, B. T., Lazovskis, J., Malen, G., Riess, H., Wei, B., & Zabka, M. (2020). Moduli spaces of morse functions for persistence. Journal of Applied and Computational Topology , 4(3), 353-385. https://arxiv.org/pdf/1909.10623.pdf

Cellular Sheaves of Lattices and the Tarski Laplacian

Published in arXiv, 2020

This paper initiates a discrete Hodge theory for cellular sheaves taking values in a category of lattices and Galois connections. The key development is the Tarski Laplacian, an endomorphism on the cochain complex whose fixed points yield a cohomology that agrees with the global section functor in degree zero. This has immediate applications in consensus and distributed optimization problems over networks and broader potential applications.

Recommended citation: Ghrist, R. & H. Riess. (2020). Cellular Sheaves of Lattices and the Tarski Laplacian. Homology, Homotopy, and Applications 24(1).pp. 325-245. https://hansriess.com/files/tarski-laplacian.pdf

Quiver Signal Processing

Published in arXiv, 2020

In this paper we state the basics for a signal processing framework on quiver representations. A quiver is a directed graph and a quiver representation is an assignment of vector spaces to the nodes of the graph and of linear maps between the vector spaces associated to the nodes. Leveraging the tools from representation theory, we propose a signal processing framework that allows us to handle heterogeneous multidimensional information in networks. We provide a set of examples where this framework provides a natural set of tools to understand apparently hidden structure in information. We remark that the proposed framework states the basis for building graph neural networks where information can be processed and handled in alternative ways.

Recommended citation: Parada-Mayorga, A., Riess, H., Ribeiro, A., & Ghrist, R. (2020). Quiver Signal Processing (QSP). arXiv preprint. https://arxiv.org/pdf/2010.11525.pdf

Multidimensional Persistence Module Classification via Lattice-Theoretic Convolutions

Published in NeurIPS Workshop: Topological Data Analysis and Beyond, 2020

Multiparameter persistent homology has been largely neglected as an input to machine learning algorithms. We consider the use of lattice-based convolutional neural network layers as a tool for the analysis of features arising from multiparameter persistence modules. We find that these show promise as an alternative to convolutions for the classification of multidimensional persistence modules.

Recommended citation: Riess, H., & Hansen, J. (2020). Multidimensional Persistence Module Classification via Lattice-Theoretic Convolutions. arXiv preprint. https://arxiv.org/pdf/2011.14057.pdf

A Temporal Logic-based Hierarchial Network Connectivity Controler

Published in arXiv, 2021

In this paper, we consider networks of static sensors with integrated sensing and communication capabilities. The goal of the sensors is to propagate their collected information to every other agent in the network and possibly a human operator. Such a task requires constant communication among all agents which may result in collisions and congestion in wireless communication. To mitigate this issue, we impose locally non-interfering communication constraints that must be respected by every agent. We show that these constraints along with the requirement of propagating information in the network can be captured by a Linear Temporal Logic (LTL) framework. Existing temporal logic control synthesis algorithms can be used to design correct-by-construction communication schedules that satisfy the considered LTL formula. Nevertheless, such approaches are centralized and scale poorly with the size of the network. We propose a hierarchical LTL-based algorithm that designs communication schedules that determine which agents should communicate while maximizing network usage. We show that the proposed algorithm is complete and demonstrate its efficiency and scalability through numerical experiments.

Recommended citation: Riess, H., Kantaros, Y., Pappas, G., & Ghrist, R. (2021). A Temporal Logic-based Network Connectivity Controller. To appear in Proceedings of SIAM Control and Applications. https://arxiv.org/pdf/2009.00669.pdf

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Cellular Sheaves and the Tarski Laplacian

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At the kind invitation of Vidit Nanda, I gave a talk to Samson Abramsky and his research group about our paper "Cellular Sheaves of Lattices and the Tarski Laplacian."

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Poster

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A Lattice-Theoretic Laplacian for Cellular Sheaves

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I presented work on the Tarski Laplacian for the mini-symposium Emerging Directions in Computational Topology organized by Brad Nelson and Greg Henselman. I discussed some computational challenges of computing global sections with the Tarski Laplacian.

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Introduction to Probability and Statistics

Masters, University of Pennsylvania, Department of Electrical/Systems Engineering, 2018

I was the teaching assistant (TA) for a course Engineering Mathematics (ENM) 503 for the Summer 2018 term. The course was accelerated, meeting 2 times per week for 3+ hours. My responsibilities included

• Leading weekly 1 hour recitation sessions where I would collaboratively solve homework problems on the chalkboard. I tried hard to foster an friendly classroom environment where all students felt comfortable taking part in solving challenging problems.
• I graded extensive weekly problem sets as well as quizzes. Students would take quizzes at the beggning of the class time and I would provide feedback before the class was finished.
• I held office hours every week. Since the class was so small, I had the opportunity to mentor students 1-on-1, especially in during these office hours.