Adaptive Markov Random Fields for Structured Compressive Sensing

Abstract

Compressive sensing (CS) has underpinned recent developments in data compression and signal acquisition systems. The goal of CS is to recover a high dimensional sparse signal from a few measurements. Recent progress in CS has attempted to further reduce the measurements by employing signal structures. This thesis presents a novel structured sparsity model, namely, adaptive Markov random field (MRF) to effectively extract the signal structures. The adaptive MRF achieves two desirable properties: flexibility—the ability to represent a wide range of structures—and adaptability—being adaptive to any structures. However, most existing work can only achieve one of these two properties. Previous MRF-based methods offer high flexibility but cannot adapt to new signal structures, while the data-adaptive based methods assume limited signal structures. Therefore, the contribution of this thesis is the novel and efficient signal recovery methods for CS. We propose to leverage the adaptability of the MRF by refining the MRF parameters based on a point estimate of the latent sparse signal, and then the sparse signal is estimated based on the resulting MRF. This method is termed Two-step-Adaptive MRF. To maximize the adaptability, we also propose a new sparse signal estimation method that estimates the sparse signal, support, and noise parameters jointly. The point estimation of the latent sparse signals underpins the performance of MRF parameter estimation, but it cannot depict the statistical uncertainty of the latent sparse signals, which can lead to inaccurate parameter estimations, and thus limit the ultimate signal recovery performance. Therefore, we reformulate the parameter estimation problem to offer better generalization over the latent sparse signals. We propose to obtain the MRF parameters from given measurements by solving a maximum marginal likelihood (MML) problem. The resulting MML problem allows the MRF parameters to be estimated from measurements directly in one step; thus, we term this method One-step-Adaptive MRF. To solve the MML problem efficiently, we propose to approximate the MRF model with the product of two simpler distributions which enables closed-form solutions for all unknown variables with low computational cost. Extensive experiments on three real-world datasets demonstrate the promising performance of Two-steps-Adaptive MRF. One-step-Adaptive MRF further improves over the state-of-the-art methods. Motivated by this, we apply One-step-Adaptive MRF to collaborative-representation based classifications (CRCs) to extract the underlying information that can help identify the class label of the corresponding query sample. CRCs have offered state-of-the-art performance in wearable sensor-based human activity recognition when training samples are limited. Existing work is based on the shortest Euclidean distance to a query sample, which can be susceptible to noise and correlation in the training samples. To improve robustness, we employ the adaptive MRF to extract the underlying structure of a representation vector directly from the query sample to improve discriminative power, because the underlying structure is unique to its corresponding query sample and independent of the quality of the training samples. The adaptive MRF can be customized to further reduce to the correlation in the training samples. Extensive experiments on two real-world datasets demonstrate the promising performance of the proposed method.