Matrix analysis
Atomic Decomposition by Basis Pursuit
SIAM Review
Covariance-Preconditioned Iterative Methods for Nonnegatively Constrained Astronomical Imaging
SIAM Journal on Matrix Analysis and Applications
ICASSP '09 Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing
Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications
Discrete & Computational Geometry
Online Learning for Matrix Factorization and Sparse Coding
The Journal of Machine Learning Research
Learning with l1-graph for image analysis
IEEE Transactions on Image Processing
A Unique “Nonnegative” Solution to an Underdetermined System: From Vectors to Matrices
IEEE Transactions on Signal Processing
Greed is good: algorithmic results for sparse approximation
IEEE Transactions on Information Theory
On the Uniqueness of Nonnegative Sparse Solutions to Underdetermined Systems of Equations
IEEE Transactions on Information Theory
Uncertainty Relations and Sparse Signal Recovery for Pairs of General Signal Sets
IEEE Transactions on Information Theory
Exemplar-Based Sparse Representations for Noise Robust Automatic Speech Recognition
IEEE Transactions on Audio, Speech, and Language Processing
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The non-negative solution to an underdetermined linear system can be uniquely recovered sometimes, even without imposing any additional sparsity constraints. In this paper, we derive conditions under which a unique non-negative solution for such a system can exist, based on the theory of polytopes. Furthermore, we develop the paradigm of combined sparse representations, where only a part of the coefficient vector is constrained to be non-negative, and the rest is unconstrained (general). We analyze the recovery of the unique, sparsest solution, for combined representations, under three different cases of coefficient support knowledge: (a) the non-zero supports of non-negative and general coefficients are known, (b) the non-zero support of general coefficients alone is known, and (c) both the non-zero supports are unknown. For case (c), we propose the combined orthogonal matching pursuit algorithm for coefficient recovery and derive the deterministic sparsity threshold under which recovery of the unique, sparsest coefficient vector is possible. We quantify the order complexity of the algorithms, and examine their performance in exact and approximate recovery of coefficients under various conditions of noise. Furthermore, we also obtain their empirical phase transition characteristics. We show that the basis pursuit algorithm, with partial non-negative constraints, and the proposed greedy algorithm perform better in recovering the unique sparse representation when compared to their unconstrained counterparts. Finally, we demonstrate the utility of the proposed methods in recovering images corrupted by saturation noise.