Robust Quadratic Observers for Decision Making from Heterogenous Inputs
Abstract
Research Problem and Objectives: When a single measurement is on the order of millions of elements, metrics which rely on statistical expectations over high-dimensional stochastic processes are nearly impossible to compute. Even when computational resources are limitless the quantity of sample data can be insufficient to form robust sample statistics required for mathematical observers. Estimating covariance/scatter matrices is a ubiquitous challenge without an obvious solution for high dimension, low sample size (HDLSS) problems. For HDLSS scenarios even simple operations, such as pre-whitening, are intractable because of the reliance on a full-rank covariance estimate. Mathematical observers with robust capabilities to convert high-dimensional, heterogeneous data into information needed for hypothesis testing, data-driven discovery, and causal inferences are sought. Our scope includes applications in which both the mean, covariance, and cross-correlation of heterogeneous (e.g. multi-modality) data are relevant. These relevant sample statistics can be temporal, spatial, spectral, or a combine thereof. This research will focus on information regarding the detection, classification, or estimation of a signal embedded in heteroscedastic data from multiple sources. In other words, the framework will assume differences in the covariance within one source of data or covariances between different sources as potentially informative. Linear transforms will be used to form robust estimates of HDLSS covariance, thus increasing the application span of rigorous mathematical observers. We will develop and demonstrate an optimization to compute a linear transforms that reduces dimensionality while preserving quadratic information. Robust quadratic observers will enable non-linear post-processing to benefit from simplified correlation structure, as well as, provide solutions for hardware configurations of capture systems. Technical Approach: Channelized data is related to the original data by an underdetermined linear transform that reduces the quantity of elements. In this channelized representation the estimation of higher-order statistics from finite training data becomes possible when the reduction is great enough.This research proposal investigates the use of channelized data and a quadratic observer for detection and estimation tasks in bothsupervised and semi-supervised environments. This approach demands answering the following question: Which linear transform is best? It would require extensive Markov-chain Monte Carlo evaluations over the high-dimensional data to exhaustively search for the channels which maximize a figure of merit directly related to task performance. To offer a solution, the PI has published a surrogate figure of merit for detection and estimation task performance which is computationally feasible and, in special cases, monotonically related to the performance of the ideal observer. The computationally feasibility is further improved by a closed-form gradient and Hessian. Anticipated Outcomes and Impact on DoD Capabilities: Channelized representations are particularly appealing for combining heterogeneous data since each channel output is a linear combination of the original measurements. By solving for channels that are optimal for specified tasks, the relative utility of each individual measurement can be analyzed. In other words, among all measurements, those that will improve task performance the most will be weighted more heavily by the optimal channels. Therefore, in addition to enabling more accurate estimation and detection for HDLSS problems the optimal channels themselves can lead to insightful new measurement strategies. Multi-modal imaging will benefit from a principled approach to data fusion that is robust to evolving fidelity in each modal data stream, as well as variations in data statistics with changing operational environmental variables and conditions.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jan 24, 2024
- Source ID
- N000142412074
Entities
People
- Meredith Kupinski
Organizations
- Office of Naval Research
- United States Navy
- University of Arizona