Sparse Bayesian mass-mapping with uncertainties: hypothesis testing of structure

A crucial aspect of mass-mapping, via weak lensing, is quantification of the uncertainty introduced during the reconstruction process. Properly accounting for these errors has been largely ignored to date. We present results from a new method that reconstructs maximum a posteriori (MAP) convergence maps by formulating an unconstrained Bayesian inference problem with Laplace-type $\ell_1$-norm sparsity-promoting priors, which we solve via convex optimization. Approaching mass-mapping in this manner allows us to exploit recent developments in probability concentration theory to infer theoretically conservative uncertainties for our MAP reconstructions, without relying on assumptions of Gaussianity. For the first time these methods allow us to perform hypothesis testing of structure, from which it is possible to distinguish between physical objects and artifacts of the reconstruction. Here we present this new formalism, demonstrate the method on illustrative examples, before applying the developed formalism to two observational datasets of the Abel-520 cluster. In our Bayesian framework it is found that neither Abel-520 dataset can conclusively determine the physicality of individual local massive substructure at significant confidence. However, in both cases the recovered MAP estimators are consistent with both sets of data

Off-the-Grid Line Spectrum Denoising and Estimation with Multiple Measurement Vectors

Compressed Sensing suggests that the required number of samples for reconstructing a signal can be greatly reduced if it is sparse in a known discrete basis, yet many real-world signals are sparse in a continuous dictionary. One example is the spectrally-sparse signal, which is composed of a small number of spectral atoms with arbitrary frequencies on the unit interval. In this paper we study the problem of line spectrum denoising and estimation with an ensemble of spectrally-sparse signals composed of the same set of continuous-valued frequencies from their partial and noisy observations. Two approaches are developed based on atomic norm minimization and structured covariance estimation, both of which can be solved efficiently via semidefinite programming. The first approach aims to estimate and denoise the set of signals from their partial and noisy observations via atomic norm minimization, and recover the frequencies via examining the dual polynomial of the convex program. We characterize the optimality condition of the proposed algorithm and derive the expected convergence rate for denoising, demonstrating the benefit of including multiple measurement vectors. The second approach aims to recover the population covariance matrix from the partially observed sample covariance matrix by motivating its low-rank Toeplitz structure without recovering the signal ensemble. Performance guarantee is derived with a finite number of measurement vectors. The frequencies can be recovered via conventional spectrum estimation methods such as MUSIC from the estimated covariance matrix. Finally, numerical examples are provided to validate the favorable performance of the proposed algorithms, with comparisons against several existing approaches.Comment: 14 pages, 10 figure

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

Deep Learning for Single Image Super-Resolution: A Brief Review

Single image super-resolution (SISR) is a notoriously challenging ill-posed problem, which aims to obtain a high-resolution (HR) output from one of its low-resolution (LR) versions. To solve the SISR problem, recently powerful deep learning algorithms have been employed and achieved the state-of-the-art performance. In this survey, we review representative deep learning-based SISR methods, and group them into two categories according to their major contributions to two essential aspects of SISR: the exploration of efficient neural network architectures for SISR, and the development of effective optimization objectives for deep SISR learning. For each category, a baseline is firstly established and several critical limitations of the baseline are summarized. Then representative works on overcoming these limitations are presented based on their original contents as well as our critical understandings and analyses, and relevant comparisons are conducted from a variety of perspectives. Finally we conclude this review with some vital current challenges and future trends in SISR leveraging deep learning algorithms.Comment: Accepted by IEEE Transactions on Multimedia (TMM