Compressed Sensing: Novel Applications, Challenges, and Techniques

Compressed Sensing (CS) is a widely used technique for efficient signal acquisition, in which a very small number of (possibly noisy) linear measurements of an unknown signal vector are taken via multiplication with a designed ?sensing matrix? in an application-specific manner, and later recovered by exploiting the sparsity of the signal vector in some known orthonormal basis and some special properties of the sensing matrix which allow for such recovery. We study three new applications of CS, each of which poses a unique challenge in a different aspect of it, and propose novel techniques to solve them, advancing the field of CS. Each application involves a unique combination of realistic assumptions on the measurement noise model and the signal, and a unique set of algorithmic challenges. We frame Pooled RT-PCR Testing for COVID-19 ? wherein RT-PCR (Reverse Transcription Polymerase Chain Reaction) tests for COVID-19 (COronaVIrus Disease of 2019) are saved in a low infection rate setting by performing them on a small number of pooled samples instead of individual samples and inferring the viral loads of the individual samples from the viral loads of the pools ? as a CS problem. Our proposed Tapestry algorithm effectively combines traditional group testing (GT) algorithms with some conventional CS algorithms, in such a way that the combination exhibits superior performance to other standalone GT or standalone CS techniques. The combination exploits the inherent heteroscedasticity of the noise in RT-PCR measurements, in particular the fact that negative pooled tests are always noiseless in RT-PCR, unlike positive pooled tests. We demonstrate the superiority of our method over traditional GT and CS via in-silico experiments, validate it in wet lab experiments with oligomers, and prove theoretical guarantees for it. For Efficient Automated Image Moderation, we bring CS methods to the domains of imbalanced binary classification and outlier detection for images. The first task is to use a neural network to classify images as objectionable or not. We propose the quantitative matrix-pooled neural network, which takes a superposition of multiple images as input and efficiently outputs the counts of objectionable images in a small number of pools of these images specified by the rows of a binary matrix. From these (possibly noisy) counts, the classification of the images, as being objectionable or not, is inferred via CS decoding if only a small fraction of input images are objectionable. We empirically demonstrate that computation is saved relative to a network which processes the images separately, while maintaining sufficient accuracy. Our extension of this method to deep outlier detection infers the count of outlier images in a pool by comparing it with a pre-learned distribution of pool-level feature vectors extracted from our network, and is applied to the problem of moderation of off-topic images on topical forums. Lastly, we propose the Compressive Perturbed Graph Recovery problem, in which the signal vector to be recovered from compressive measurements is sparse in the domain of the eigenvectors of the Laplacian matrix of an undirected, unweighted graph known only up to a few edge perturbations. This makes signal recovery challenging due to uncertainty in our knowledge of the orthonormal sparsifying basis. Our method solves this by performing joint signal and graph recovery, using cross-validation error on a held-out set of measurements to disambiguate between candidate graphs generated via a greedy edge selection strategy. We extend our method to solve the problem of recovery of images containing sharp edges (whose locations are not known a priori) from compressive measurements, generating candidate graphs in a structured manner from hypothesis linear image edges. We demonstrate the efficacy of our methods via extensive experiments, and prove theoretical guarantees for a brute-force version of our algorithms.