Global hard thresholding algorithms for joint sparse image representation and denoising

Sparse coding of images is traditionally done by cutting them into small patches and representing each patch individually over some dictionary given a pre-determined number of nonzero coefficients to use for each patch. In lack of a way to effectively distribute a total number (or global budget) of nonzero coefficients across all patches, current sparse recovery algorithms distribute the global budget equally across all patches despite the wide range of differences in structural complexity among them. In this work we propose a new framework for joint sparse representation and recovery of all image patches simultaneously. We also present two novel global hard thresholding algorithms, based on the notion of variable splitting, for solving the joint sparse model. Experimentation using both synthetic and real data shows effectiveness of the proposed framework for sparse image representation and denoising tasks. Additionally, time complexity analysis of the proposed algorithms indicate high scalability of both algorithms, making them favorable to use on large megapixel images.

Fast and Effective Algorithms for Symmetric Nonnegative Matrix Factorization

Symmetric Nonnegative Matrix Factorization (SNMF) models arise naturally as simple reformulations of many standard clustering algorithms including the popular spectral clustering method. Recent work has demonstrated that an elementary instance of SNMF provides superior clustering quality compared to many classic clustering algorithms on a variety of synthetic and real world data sets. In this work, we present novel reformulations of this instance of SNMF based on the notion of variable splitting and produce two fast and effective algorithms for its optimization using i) the provably convergent Accelerated Proximal Gradient (APG) procedure and ii) a heuristic version of the Alternating Direction Method of Multipliers (ADMM) framework. Our two algorithms present an interesting tradeoff between computational speed and mathematical convergence guarantee: while the former method is provably convergent it is considerably slower than the latter approach, for which we also provide significant but less stringent mathematical proof regarding its convergence. Through extensive experiments we show not only that the efficacy of these approaches is equal to that of the state of the art SNMF algorithm, but also that the latter of our algorithms is extremely fast being one to two orders of magnitude faster in terms of total computation time than the state of the art approach, outperforming even spectral clustering in terms of computation time on large data sets.