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Signal sparse representation has attracted much attention in a wide range of application fields. A central aim of signal sparse representation is to find a sparse solution with the fewest nonzero entries from an underdetermined linear system, which leads to various optimization problems. In this paper, we propose an Adaptive Gradient Projection (AGP) algorithm to solve the piecewise convex optimization in signal sparse representation. To find a sparser solution, AGP provides an adaptive stepsize to move the iteration solution out of the attraction basin of a suboptimal sparse solution and enter the attraction basin of a sparser solution. Theoretical analyses are used to show its fast convergence property. The experimental results of real-world applications in compressed spectrum sensing show that AGP outperforms the traditional detection algorithms in low signal-to-noise-ratio environments. [ABSTRACT FROM AUTHOR]

Optimizing the mutual coherence of a learned dictionary plays an important role in sparse representation and compressed sensing. In this paper, a efficient framework is developed to learn an incoherent dictionary for sparse representation. In particular, the coherence of a previous dictionary (or Gram matrix) is reduced sequentially by finding a new dictionary (or Gram matrix), which is closest to the reference unit norm tight frame of the previous dictionary (or Gram matrix). The optimization problem can be solved by restricting the tightness and coherence alternately at each iteration of the algorithm. The significant and different aspect of our proposed framework is that the learned dictionary can approximate an equiangular tight frame. Furthermore, manifold optimization is used to avoid the degeneracy of sparse representation while only reducing the coherence of the learned dictionary. This can be performed after the dictionary update process rather than during the dictionary update process. Experiments on synthetic and real audio data show that our proposed methods give notable improvements in lower coherence, have faster running times, and are extremely robust compared to several existing methods. [ABSTRACT FROM AUTHOR]