Construction of distributions on complex topological spaces of signals and their application to machine learning and state estimation
Abstract
Robust analysis of complex systems, such as signal classification and clustering or state estimation in topological spaces of signals emitted by a sensor network (SN), requires a colossal engagement of topology, geometry, statistics and machine learning. Signal classification investigates topological features of the point cloud data space, summarized by persistence diagrams (PDs). Threat detection and assessment in a SN requires understanding the network topology by estimating hidden states of PDs, which compress necessary information about the networkÕs connectivity. Previous studies have engaged statistical learning methods for PDs for addressing these problems by adopting artificial assumptions tailored to the application and the data under consideration. A general foundational strategy for analyzing PDs is desperately required for data analysis of complex systems. The PI will alleviate this problem by proposing the construction of distributions of random PDs. Armored with the panoply of these distributions, the PI will establish a Bayesian framework for estimating conditional distributions of PDs associated with SN signals and will discover cutting-edge unsupervised and supervised learning methods. A persistence diagram (PD) is a visualization tool that encapsulates the topological properties of the underlying dynamical space from which signals arise. Each point on a PD represents a topological feature through 3 inputs: a topological dimension (indicating a cluster or a high-dimensional hole) and the birth-death location of the feature. The PI will shed light on the investigation of PDs by constructing their probability distributions via finite set statistics. Decomposing random PDs into a union of singleton sets, its distribution will depend on each single elementÕs birth-death distribution. A closed form of PDs distribution, relying on kernel methods, is analyzed, and a rare event probability estimate corroborates the robustness of our method. Overall, this novel approach will bridge the previous fractured body of recent work. For instance, statistical summaries of PDs like their mean and variance will be computed without relying on special properties of each problemÕs associated data space. Derivation of these summaries will lead to the establishment of innovative (un)supervised machine learning techniques by optimizing appropriate metrics from the distribution or mean of PDs under investigation. Additionally, the PI will investigate an original Bayesian framework for computing conditional distributions of PDs given a fixed PD, associated with transmitted signals from a SN. Based on a family of PDsÕ prior distributions and pertinent likelihoods, these posterior PD distributions will estimate their hidden states and recognize patterns about the topological behavior of the SN. This ingenious research proposal, engaging topology, geometry, statistics and machine learning, addresses the interdisciplinary problem of (un)supervised machine learning methods for signals and their hidden pattern recognition. The proposed framework will enjoy immediate impact in other sciences and engineering. The PI will collaborate with the Army Research Lab (ARL) to apply this novel construction to analyze acoustic signal data, e.g. to robustly classify explosion types. Accurate results will aid military officers in making tactical decisions based on the type of weapon system. Moreover, the PI will potentially partner with ARL and deCervo to analyze brain and muscle signals. This transformative methodology will have a great impact in speech recognition and computer vision. Gained knowledge from this project will be embraced into a one year course which will serve as the theoretical backbone of the new PhD in Data Sciences at UTK, of which the PI is one of the founding fathers.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Oct 11, 2018
- Source ID
- W911NF1710313
Entities
People
- Vasileios Maroulas
Organizations
- Army Contracting Command
- United States Army
- University of Tennessee