A General Proximal Alternating Minimization Method with Application to Nonconvex Nonsmooth 1D Total Variation Denoising

We deal with a class of problems whose objective functions are compositions of nonconvex nonsmooth functions, which has a wide range of applications in signal/image processing. We introduce a new auxiliary variable, and an efficient general proximal alternating minimization algorithm is proposed. This method solves a class of nonconvex nonsmooth problems through alternating minimization. We give a brilliant systematic analysis to guarantee the convergence of the algorithm. Simulation results and the comparison with two other existing algorithms for 1D total variation denoising validate the efficiency of the proposed approach. The algorithm does contribute to the analysis and applications of a wide class of nonconvex nonsmooth problems

Convex and Network Flow Optimization for Structured Sparsity

We consider a class of learning problems regularized by a structured sparsity-inducing norm defined as the sum of l_2- or l_infinity-norms over groups of variables. Whereas much effort has been put in developing fast optimization techniques when the groups are disjoint or embedded in a hierarchy, we address here the case of general overlapping groups. To this end, we present two different strategies: On the one hand, we show that the proximal operator associated with a sum of l_infinity-norms can be computed exactly in polynomial time by solving a quadratic min-cost flow problem, allowing the use of accelerated proximal gradient methods. On the other hand, we use proximal splitting techniques, and address an equivalent formulation with non-overlapping groups, but in higher dimension and with additional constraints. We propose efficient and scalable algorithms exploiting these two strategies, which are significantly faster than alternative approaches. We illustrate these methods with several problems such as CUR matrix factorization, multi-task learning of tree-structured dictionaries, background subtraction in video sequences, image denoising with wavelets, and topographic dictionary learning of natural image patches.Comment: to appear in the Journal of Machine Learning Research (JMLR

Sparse variational regularization for visual motion estimation

The computation of visual motion is a key component in numerous computer vision tasks such as object detection, visual object tracking and activity recognition. Despite exten- sive research effort, efficient handling of motion discontinuities, occlusions and illumina- tion changes still remains elusive in visual motion estimation. The work presented in this thesis utilizes variational methods to handle the aforementioned problems because these methods allow the integration of various mathematical concepts into a single en- ergy minimization framework. This thesis applies the concepts from signal sparsity to the variational regularization for visual motion estimation. The regularization is designed in such a way that it handles motion discontinuities and can detect object occlusions

Imaging and uncertainty quantification in radio astronomy via convex optimization : when precision meets scalability

Upcoming radio telescopes such as the Square Kilometre Array (SKA) will provide sheer amounts of data, allowing large images of the sky to be reconstructed at an unprecedented resolution and sensitivity over thousands of frequency channels. In this regard, wideband radio-interferometric imaging consists in recovering a 3D image of the sky from incomplete and noisy Fourier data, that is a highly ill-posed inverse problem. To regularize the inverse problem, advanced prior image models need to be tailored. Moreover, the underlying algorithms should be highly parallelized to scale with the vast data volumes provided and the Petabyte image cubes to be reconstructed for SKA. The research developed in this thesis leverages convex optimization techniques to achieve precise and scalable imaging for wideband radio interferometry and further assess the degree of confidence in particular 3D structures present in the reconstructed cube. In the context of image reconstruction, we propose a new approach that decomposes the image cube into regular spatio-spectral facets, each is associated with a sophisticated hybrid prior image model. The approach is formulated as an optimization problem with a multitude of facet-based regularization terms and block-specific data-fidelity terms. The underpinning algorithmic structure benefits from well-established convergence guarantees and exhibits interesting functionalities such as preconditioning to accelerate the convergence speed. Furthermore, it allows for parallel processing of all data blocks and image facets over a multiplicity of CPU cores, allowing the bottleneck induced by the size of the image and data cubes to be efficiently addressed via parallelization. The precision and scalability potential of the proposed approach are confirmed through the reconstruction of a 15 GB image cube of the Cyg A radio galaxy. In addition, we propose a new method that enables analyzing the degree of confidence in particular 3D structures appearing in the reconstructed cube. This analysis is crucial due to the high ill-posedness of the inverse problem. Besides, it can help in making scientific decisions on the structures under scrutiny (e.g., confirming the existence of a second black hole in the Cyg A galaxy). The proposed method is posed as an optimization problem and solved efficiently with a modern convex optimization algorithm with preconditioning and splitting functionalities. The simulation results showcase the potential of the proposed method to scale to big data regimes

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum