Research Projects

Topological Signal Processing (TSP) extends classical and graph signal processing to analyze data defined over higher-order structures such as simplicial complexes, which include nodes, edges, triangles, and higher-dimensional elements. Unlike traditional signals defined on regular grids or simple graphs, topological signals must satisfy local-to-global consistency constraints that reflect the intrinsic relationships among these complex structures. These models arise naturally in multi-agent coordination, sensor fusion, distributed estimation, and social or communication networks where information is distributed across multi-way interactions. Our inquiry is rooted in the classical theory of non-uniform and graph signal sampling, but the unique nature of topological signals—governed by higher-order dependencies and global consistency requirements—demands new theoretical development. While there is considerable literature on point-to-point sampling and recovery of topological signals, to the best of our knowledge, there is almost no work addressing distributed reconstruction methods where agents have only local observations and limited communication. Developing such distributed algorithms is the primary focus of this project. This project proposes to develop a rigorous framework for distributed sampling and reconstruc- tion of topological signals, focusing on their often sparse or compressible representations in bases derived from topological decompositions such as harmonic and gradient components. Building upon theories from compressed sensing, inverse problems, and distributed optimization, the project will develop algorithms that allow agents with access to local data to jointly reconstruct consistent global signals. Key challenges include designing message-passing and consensus protocols that preserve sparsity, ensure convergence, and reduce communication demands. The research will study one-shot and iterative reconstruction methods, including Kalman filter-inspired and distributed recursive least-squares algorithms tailored to topological signal domains. In addition, adaptive and event-triggered sampling methods will be developed, in which measurements and communications occur only when necessary, thereby reducing resource use 7 without compromising accuracy. These event-based sampling strategies are especially important for resource-constrained, distributed systems. To evaluate temporal coherence and the timeliness of reconstructed signals under asynchronous updates and network delays, the project will introduce the Age of Gossip metric, which is suited for distributed topological inference. Robustness will be addressed through the use of regularization methods and probabilistic mod- els consistent with the topological structure to handle noise, missing data, and communication failures typical in distributed networks. The theoretical contributions will bring together concepts from algebraic topology, compressed sensing, distributed filtering, and consensus optimization into a unified approach for topological signal recovery. The proposed framework will be validated through simulations of distributed agents operating over networked simplicial complexes. These experiments will assess algorithm performance under various sampling schemes—including uniform, non-uniform, adaptive, and event-triggered sampling—communication limitations, sparsity patterns, and dynamic conditions. Application areas include multi-agent formation control, distributed environmental monitoring, and social dynamics analysis, where signals possess inherently topological features. In summary, this project aims to provide a solid theoretical foundation and practical algo- rithms for processing structured, high-dimensional topological data in networked settings where conventional signal processing approaches are insufficient. The expected outcomes will advance our understanding of distributed processing underlying multi-agent systems, and social networks.

Topological Signals, distributed sampling, fast recovery, event-triggered sampling, social network gossip