Structured Sparsity: Discrete and Convex approaches

Compressive sensing (CS) exploits sparsity to recover sparse or compressible signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity is also used to enhance interpretability in machine learning and statistics applications: While the ambient dimension is vast in modern data analysis problems, the relevant information therein typically resides in a much lower dimensional space. However, many solutions proposed nowadays do not leverage the true underlying structure. Recent results in CS extend the simple sparsity idea to more sophisticated {\em structured} sparsity models, which describe the interdependency between the nonzero components of a signal, allowing to increase the interpretability of the results and lead to better recovery performance. In order to better understand the impact of structured sparsity, in this chapter we analyze the connections between the discrete models and their convex relaxations, highlighting their relative advantages. We start with the general group sparse model and then elaborate on two important special cases: the dispersive and the hierarchical models. For each, we present the models in their discrete nature, discuss how to solve the ensuing discrete problems and then describe convex relaxations. We also consider more general structures as defined by set functions and present their convex proxies. Further, we discuss efficient optimization solutions for structured sparsity problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure

Some perspectives on the problem of model selection

Ph.DDOCTOR OF PHILOSOPH

Gridless Multisnapshot Variational Line Spectral Estimation from Coarsely Quantized Samples

Due to the increasing demand for low power and higher sampling rates, low resolution quantization for data acquisition has drawn great attention recently. Consequently, line spectral estimation (LSE) with multiple measurement vectors (MMVs) from coarsely quantized samples is of vital importance in cutting edge array signal processing applications such as range estimation and DOA estimation in millimeter wave radar systems. In this paper, we combine the low complexity gridless variational line spectral estimation (VALSE) and expectation propagation (EP) and propose an MVALSE-EP algorithm to estimate the frequencies from coarsely quantized samples. In addition, the Cram\'{e}r Rao bound (CRB) is derived as a benchmark performance of the proposed algorithm, and insights are provided to reveal the effects of system parameters on estimation performance. It is shown that snapshots benefits the frequency estimation, especially in coarsely quantized scenarios. Numerical experiments are conducted to demonstrate the effectiveness of MVALSE-EP, including real data set

Block-based Collaborative 3-D Transform Domain Modeling in Inverse Imaging

The recent developments in image and video denoising have brought a new generation of filtering algorithms achieving unprecedented restoration quality. This quality mainly follows from exploiting various features of natural images. The nonlocal self-similarity and sparsity of representations are key elements of the novel filtering algorithms, with the best performance achieved by adaptively aggregating multiple redundant and sparse estimates. In a very broad sense, the filters are now able, given a perturbed image, to identify its plausible representative in the space or manifold of possible solutions. Thus, they are powerful tools not only for noise removal, but also for providing accurate adaptive regularization to many ill-conditioned inverse imaging problems. In this thesis we show how the image modeling of the well-known Block-matching 3-D transform domain (BM3D) filter can be exploited for designing efficient algorithms for image reconstruction. First, we formalize the BM3D-modeling in terms of the overcomplete sparse frame representation. We construct analysis and synthesis BM3D-frames and study their properties, making BM3D-modeling available for use in variational formulations of image reconstruction problems. Second, we demonstrate that standard variational formulations based on single objective optimization, such as Basis Pursuit Denoising and its various extensions, cannot be used with the imaging models generating non-tight frames, such as BM3D. We propose an alternative sparsity promoting problem formulation based on the generalized Nash equilibrium (GNE). Finally, using BM3D-frames we develop practical algorithms for image deblurring and super-resolution problems. To the best of our knowledge, these algorithms provide results which are the state of the art in the field

Unsupervised methods for large-scale, cell-resolution neural data analysis

In order to keep up with the volume of data, as well as the complexity of experiments and models in modern neuroscience, we need scalable and principled analytic programmes that take into account the scientific goals and the challenges of biological experiments. This work focuses on algorithms that tackle problems throughout the whole data analysis process. I first investigate how to best transform two-photon calcium imaging microscopy recordings – sets of contiguous images – into an easier-to-analyse matrix containing time courses of individual neurons. For this I first estimate how the true fluorescence signal gets transformed by tissue artefacts and the microscope setup, by learning the parameters of a realistic physical model from recorded data. Next, I describe how individual neural cell bodies may be segmented from the images, based on a cost function tailored to neural characteristics. Finally, I describe an interpretable non-linear dynamical model of neural population activity, which provides immediate scientific insight into complex system behaviour, and may spawn a new way of investigating stochastic non-linear dynamical systems. I hope the algorithms described here will not only be integrated into analytic pipelines of neural recordings, but also point out that algorithmic design should be informed by communication with the broader community, understanding and tackling the challenges inherent in experimental biological science