Compressed Sensing Using Binary Matrices of Nearly Optimal Dimensions

In this paper, we study the problem of compressed sensing using binary measurement matrices and $\ell_1$-norm minimization (basis pursuit) as the recovery algorithm. We derive new upper and lower bounds on the number of measurements to achieve robust sparse recovery with binary matrices. We establish sufficient conditions for a column-regular binary matrix to satisfy the robust null space property (RNSP) and show that the associated sufficient conditions % sparsity bounds for robust sparse recovery obtained using the RNSP are better by a factor of $(3 \sqrt{3})/2 \approx 2.6$ compared to the sufficient conditions obtained using the restricted isometry property (RIP). Next we derive universal \textit{lower} bounds on the number of measurements that any binary matrix needs to have in order to satisfy the weaker sufficient condition based on the RNSP and show that bipartite graphs of girth six are optimal. Then we display two classes of binary matrices, namely parity check matrices of array codes and Euler squares, which have girth six and are nearly optimal in the sense of almost satisfying the lower bound. In principle, randomly generated Gaussian measurement matrices are "order-optimal". So we compare the phase transition behavior of the basis pursuit formulation using binary array codes and Gaussian matrices and show that (i) there is essentially no difference between the phase transition boundaries in the two cases and (ii) the CPU time of basis pursuit with binary matrices is hundreds of times faster than with Gaussian matrices and the storage requirements are less. Therefore it is suggested that binary matrices are a viable alternative to Gaussian matrices for compressed sensing using basis pursuit. \end{abstract}Comment: 28 pages, 3 figures, 5 table

A statistical multiresolution approach for face recognition using structural hidden Markov models

This paper introduces a novel methodology that combines the multiresolution feature of the discrete wavelet transform (DWT) with the local interactions of the facial structures expressed through the structural hidden Markov model (SHMM). A range of wavelet filters such as Haar, biorthogonal 9/7, and Coiflet, as well as Gabor, have been implemented in order to search for the best performance. SHMMs perform a thorough probabilistic analysis of any sequential pattern by revealing both its inner and outer structures simultaneously. Unlike traditional HMMs, the SHMMs do not perform the state conditional independence of the visible observation sequence assumption. This is achieved via the concept of local structures introduced by the SHMMs. Therefore, the long-range dependency problem inherent to traditional HMMs has been drastically reduced. SHMMs have not previously been applied to the problem of face identification. The results reported in this application have shown that SHMM outperforms the traditional hidden Markov model with a 73% increase in accuracy

EEG sleep stages identification based on weighted undirected complex networks

Sleep scoring is important in sleep research because any errors in the scoring of the patient's sleep electroencephalography (EEG) recordings can cause serious problems such as incorrect diagnosis, medication errors, and misinterpretations of patient's EEG recordings. The aim of this research is to develop a new automatic method for EEG sleep stages classification based on a statistical model and weighted brain networks. Methods each EEG segment is partitioned into a number of blocks using a sliding window technique. A set of statistical features are extracted from each block. As a result, a vector of features is obtained to represent each EEG segment. Then, the vector of features is mapped into a weighted undirected network. Different structural and spectral attributes of the networks are extracted and forwarded to a least square support vector machine (LS-SVM) classifier. At the same time the network's attributes are also thoroughly investigated. It is found that the network's characteristics vary with their sleep stages. Each sleep stage is best represented using the key features of their networks. Results In this paper, the proposed method is evaluated using two datasets acquired from different channels of EEG (Pz-Oz and C3-A2) according to the R&K and the AASM without pre-processing the original EEG data. The obtained results by the LS-SVM are compared with those by Naïve, k-nearest and a multi-class-SVM. The proposed method is also compared with other benchmark sleep stages classification methods. The comparison results demonstrate that the proposed method has an advantage in scoring sleep stages based on single channel EEG signals. Conclusions An average accuracy of 96.74% is obtained with the C3-A2 channel according to the AASM standard, and 96% with the Pz-Oz channel based on the R&K standard

Phase Diagram and Approximate Message Passing for Blind Calibration and Dictionary Learning

We consider dictionary learning and blind calibration for signals and matrices created from a random ensemble. We study the mean-squared error in the limit of large signal dimension using the replica method and unveil the appearance of phase transitions delimiting impossible, possible-but-hard and possible inference regions. We also introduce an approximate message passing algorithm that asymptotically matches the theoretical performance, and show through numerical tests that it performs very well, for the calibration problem, for tractable system sizes.Comment: 5 page