The problem ofnonnegative least squares(NNLS) has numerous applications in signal analysis. Recently, the algorithm unrolling has gained significant attention due to its superior approximation results compared to iterative methods. In this paper, we discuss the NNLS probelm in an interpretable data-driven set up usingunrolled proximal gradient descent method(UPGDM), and establish its analytical guarantees. An advantage of this method over its conventional counterparts is that it provides a faster and better inference for an input data, once the network is trained. In particular, this paper provides convergence guarantees of the network by bounding the number of training samples for zero training error. Further, it demonstrates relevance of UPGDM through an application in Electrical Impedance Tomography.

Linear Inverse Problems (LIPs) with non-negative sparse constraints on the target signal are critical in numerous applications across various fields. Among the recently proposed methods dealing with LIPs, model-based deep learning methods, particularly deep unrolling, have gained popularity due to their interpretability, efficiency, and superior performance compared to traditional iterative methods. In this work, we propose a model-driven deep learning method by unrolling the recently developed ReLU-based Hard Thresholding (RHT) algorithm for non-negative sparse signal recovery. Specifically, we employ soft thresholding activation instead of hard thresholding in the unrolling of RHT, which enhances the model performance during backpropagation. The primary advantages of the proposed Learned ReLU-based Soft Thresholding (LRST) include interpretability and faster inference after the network is trained. Our numerical experiments demonstrate that the proposed LRST outperforms its classical counterparts, showcasing its potential for more effective non-negative sparse signal recovery.

The non-negative least squares (NNLS) aims at finding a non-negative approximation of a matrix system. Such an approximation has been realized in the literature via some iterative methods. Recently, unrolling algorithms have gained significant attention due to their superior approximation results compared to data-driven methods. With a view to obtaining a better approximation, in this work, we deal with the NNLS in a deep learning framework by unrolling the Proximal Gradient Descent Method (PGDM). In the proposed method (referred to as unrolled or learned PGDM), we project the points to the non-negative orthant, which naturally gives rise to the ReLU function in the network as a proximal operator. The network is trained end-to-end in a deep-learning framework. We demonstrate empirically that the Unrolled PGDM (UPGDM) performs better in signal recovery than its classical counterpart involving iterative PGDM.

The construction of binary matrices has attained significance due to its potential for hardware-friendly implementation and appealing applications in compressed sensing (CS). A class of binary matrices with low coherence and flexible row sizes can be constructed from Euler Squares (ES). In this paper, we introduce a generalization of the ES concept, namely, Generalized Euler Square (GES). We show that the binary matrices designed from GES provide significant improvements in column size compared to the ones constructed from Euler square. Exploiting the properties of GES, we obtain that such constructed binary matrices possess block orthogonal structure. As a result, such binary matrices are suitable for the recovery of block sparse signals.

Frames with a large minimum angle between any two distinct frame vectors are desirable in many present day applications. For a unit norm frame, the absolute value of the cosine of the minimum frame angle is also known as coherence. Two frames are equivalent if one can be obtained from the other via left action of an invertible linear operator. Frame angles can change under the action of a linear operator. Most of the existing works solve different optimization problems to find an optimal linear operator that maximizes the minimal frame angle (in other words, minimizes the coherence). In the present work, nevertheless, we consider the question: Is it always possible to find an equivalent frame with smaller coherence for a given frame. In this paper, we derive properties of the initial unit norm frame that can ensure an equivalent frame with strictly larger minimal frame angle compared to the initial one. It turns out that the nullspace property of a certain matrix obtained from the initial frame can guarantee such an equivalent frame. We also present the numerical results that support our theoretical claims.

In the recent past, various methods have been proposed to construct deterministic compressed sensing (CS) matrices. Of interest has been the construction of binary sensing matrices as they are useful for multiplierless and faster dimensionality reduction. In most of these binary constructions, the matrix size depends on primes or their powers. In this study, we propose a composition rule which exploits sparsity and block structure of existing binary CS matrices to construct matrices of general size. We also show that these matrices satisfy optimal theoretical guarantees and have similar density compared to matrices obtained using Kronecker product. Simulation work shows that the synthesized matrices provide comparable results against Gaussian random matrices.

Can reliable telecardiology be achieved at low bandwidth cost? In response, we propose a detector at the user end so that only beats found to be anomalous are transmitted to a diagnostic center, where all received beats are correctly (re)classified. In this framework, high reliability is achieved by detectors with high sensitivity. Having laid the design framework, we then realize desired high-sensitivity detection using a dictionary learning approach. Specifically, using patient records from the MIT-BIH arrhythmia database, we detect ventricular ectopic beats (VEBs), which are known to be precursors to various serious arrhythmic conditions in the heart. In particular, we achieve a reliability of one undetected VEB in one thousand while saving 78.2% bandwidth using dictionaries with 240 atoms. With larger dictionaries with 420 atoms, we achieve an even higher bandwidth savings of 79.2% while allowing no (less than one in 1766) undetected VEB. Finally, we compare our results with performances a large set of reported heartbeat classifiers, and demonstrate the suitability of our approach in the context of telecardiology.

Telecardiology systems, involving remote diagnosis of cardiac anomaly based on ECG signals, generally acquire such signals at the Nyquist rate, and transmits the data to diagnostic facilities. Such systems are not designed under either power or bandwidth constraints. However, in certain scenarios involving remote communities in developing and underdeveloped world, both the above constraints could be acute. The present paper takes a first step towards a constrained design keeping such scenarios in view. Specifically, we propose a system where automated classification is performed on the ECG signals, and only anomalous signals are transmitted for further diagnosis and intervention, thereby saving bandwidth. Additionally, we propose compressive sampling as a low-power alternative to traditional Nyquist sampling method, which also lowers bandwidth requirement. Finally, we illustrate our method by designing such a compressive classifier using ECG signals from the widely used PhysioNet database. Specifically, we demonstrate that an average down sampling factor of three leads to desirable classification performance in terms of both sensitivity and specificity while substantially saving both power and bandwidth.

In this paper, we present an approach that simultaneously clusters images and learns dictionaries from the clusters. The method learns dictionaries and clusters images in the radon transform domain. The main feature of the proposed approach is that it provides both in-plane rotation and scale invariant clustering, which is useful in numerous applications, including content-based image retrieval (CBIR). We demonstrate the effectiveness of our rotation and scale invariant clustering method on a series of CBIR experiments. Experiments are performed on the Smithsonian isolated leaf, Kimia shape, and Brodatz texture datasets. Our method provides both good retrieval performance and greater robustness compared to standard Gabor-based and three state-of-the-art shape-based methods that have similar objectives.

In this paper, we present an approach that simultaneously clusters database members and learns dictionaries from the clusters. The method learns dictionaries in the Radon transform domain, while clustering in the image domain. Themain feature of the proposed approach is that it provides rotation invariant clustering which is useful in Content Based Image Retrieval (CBIR). We demonstrate through experimental results that the proposed rotation invariant clustering provides better retrieval performance than the standard Gabor-based method that has similar objectives.