Compressive Sensing-Based Detection with Multimodal Dependent Data

Abstract

Detection with high dimensional multimodal data is a challenging problem when there are complex inter- and intra- modal dependencies. While several approaches have been proposed for dependent data fusion (e.g., based on copula theory), their advantages come at a high price in terms of computational complexity. In this paper, we treat the detection problem with compressive sensing (CS) where compression at each sensor is achieved via low dimensional random projections. CS has recently been exploited to solve detection problems under various assumptions on the signals of interest, however, its potential for dependent data fusion has not been explored adequately. We exploit the capability of CS to capture statistical properties of uncompressed data in order to compute decision statistics for detection in the compressed domain. First, a Gaussian approximation is employed to perform likelihood ratio (LR) based detection with compressed data. In this approach, inter-modal dependence is captured via a compressed version of the covariance matrix of the concatenated (temporally and spatially) uncompressed data vector. We show that, under certain conditions, this approach with a small number of compressed measurements per node leads to enhanced performance compared to detection with uncompressed data using widely considered suboptimal approaches. Second, we develop a nonparametric approach where a decision statistic based on the second order statistics of uncompressed data is computed in the compressed domain. The second approach is promising over other related nonparametric approaches and the first approach when multimodal data is highly correlated at the expense of slightly increased computational complexity.