Error Bounds for Compressed Sensing Algorithms With Group Sparsity: A Unified Approach

In compressed sensing, in order to recover a sparse or nearly sparse vector from possibly noisy measurements, the most popular approach is $\ell_1$-norm minimization. Upper bounds for the $\ell_2$- norm of the error between the true and estimated vectors are given in [1] and reviewed in [2], while bounds for the $\ell_1$-norm are given in [3]. When the unknown vector is not conventionally sparse but is "group sparse" instead, a variety of alternatives to the $\ell_1$-norm have been proposed in the literature, including the group LASSO, sparse group LASSO, and group LASSO with tree structured overlapping groups. However, no error bounds are available for any of these modified objective functions. In the present paper, a unified approach is presented for deriving upper bounds on the error between the true vector and its approximation, based on the notion of decomposable and $\gamma$-decomposable norms. The bounds presented cover all of the norms mentioned above, and also provide a guideline for choosing norms in future to accommodate alternate forms of sparsity.Comment: 28 pages, final version of 1401.6623, accepted for publication. arXiv admin note: substantial text overlap with arXiv:1401.662

An approach to one-bit compressed sensing based on probably approximately correct learning theory

In this paper, the problem of one-bit compressed sensing (OBCS) is formulated as a problem in probably approximately correct (PAC) learning. It is shown that the VapnikChervonenkis (VC-) dimension of the set of half-spaces in Rn generated by k-sparse vectors is bounded below by k(blg(n=k)c + 1) and above by b2k lg(en)c. By coupling this estimate with well-established results in PAC learning theory, we show that a consistent algorithm can recover a k-sparse vector with O(k lg n) measurements, given only the signs of the measurement vector. This result holds for all probability measures on Rn. The theory is also applicable to the case of noisy labels, where the signs of the measurements are flipped with some unknown probability

A crowdsourced analysis to identify ab initio molecular signatures predictive of susceptibility to viral infection

The response to respiratory viruses varies substantially between individuals, and there are currently no known molecular predictors from the early stages of infection. Here we conduct a community-based analysis to determine whether pre- or early post-exposure molecular factors could predict physiologic responses to viral exposure. Using peripheral blood gene expression profiles collected from healthy subjects prior to exposure to one of four respiratory viruses (H1N1, H3N2, Rhinovirus, and RSV), as well as up to 24 h following exposure, we find that it is possible to construct models predictive of symptomatic response using profiles even prior to viral exposure. Analysis of predictive gene features reveal little overlap among models; however, in aggregate, these genes are enriched for common pathways. Heme metabolism, the most significantly enriched pathway, is associated with a higher risk of developing symptoms following viral exposure. This study demonstrates that pre-exposure molecular predictors can be identified and improves our understanding of the mechanisms of response to respiratory viruses

A community challenge to evaluate RNA-seq, fusion detection, and isoform quantification methods for cancer discovery

The accurate identification and quantitation of RNA isoforms present in the cancer transcriptome is key for analyses ranging from the inference of the impacts of somatic variants to pathway analysis to biomarker development and subtype discovery. The ICGC-TCGA DREAM Somatic Mutation Calling in RNA (SMC-RNA) challenge was a crowd-sourced effort to benchmark methods for RNA isoform quantification and fusion detection from bulk cancer RNA sequencing (RNA-seq) data. It concluded in 2018 with a comparison of 77 fusion detection entries and 65 isoform quantification entries on 51 synthetic tumors and 32 cell lines with spiked-in fusion constructs. We report the entries used to build this benchmark, the leaderboard results, and the experimental features associated with the accurate prediction of RNA species. This challenge required submissions to be in the form of containerized workflows, meaning each of the entries described is easily reusable through CWL and Docker containers at https://github.com/SMC-RNA-challenge. A record of this paper's transparent peer review process is included in the supplemental information

An approach to one-bit compressed sensing based on probably approximately correct learning theory

This paper builds upon earlier work of the authors in formulating the one-bit compressed sensing (OBCS) problem as a problem in probably approximately correct (PAC) learning theory. It is shown that the solution to the OBCS problem consists of two parts. The first part is to determine the statistical complexity of OBCS by determining the Vapnik-Chervonenkis (VC-) dimension of the set of half-spaces generated by sparse vectors. The second is to determine the algorithmic complexity of the problem by developing a consistent algorithm. In this paper, we generalize the earlier results of the authors by deriving both upper and lower bounds on the VC-dimension of half-spaces generated by sparse vectors, even when the separating hyperplane need not pass through the origin. As with earlier bounds, these bounds grow linearly with respect to with the sparsity dimension and logarithmically with the vector dimension