MINIMIZATION OF l1-2 FOR COMPRESSED SENSING
MINIMIZATION OF l1-2 FOR COMPRESSED SENSING
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DOI：
10.1137/140952363
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发表时间：
2015-01-01
影响因子：
3.1
通讯作者：
Xin, Jack
中科院分区：
文献类型：
--
作者：
Yin, Penghang;Lou, Yifei;Xin, Jack
我们研究将\(l(1)\)范数和\(l(2)\)范数之差最小化作为一种非凸且李普希茨连续的度量，用于解决有约束和无约束的压缩感知问题。对于有约束问题，我们在受限等距性质（RIP）条件下建立了精确（稳定）的稀疏恢复结果，以及限制在稀疏解的支撑集上的感知矩阵的满秩定理。我们基于凸函数之差算法提出了一种用于\(l(1 - 2)\)最小化的迭代方法，并证明它收敛到满足一阶最优性条件的驻点。我们提出了一种具有非高斯随机扰动的面向稀疏性的模拟退火过程，并证明了组合算法（DCASA）几乎必然收敛到全局最小值。关于稀疏解恢复成功率的计算实例表明，如果感知矩阵是病态的（不满足RIP），那么我们的方法比文献中现有的非凸压缩感知求解器更好。同样，在磁共振成像（MRI）幻影图像恢复问题中，\(l(1 - 2)\)用八个投影就取得了成功。无论感知矩阵的条件如何，在稀疏信号和MRI幻影图像恢复问题中，\(l(1 - 2)\)都优于\(l(1)\)。
We study minimization of the difference of l(1) and l(2) norms as a nonconvex and Lipschitz continuous metric for solving constrained and unconstrained compressed sensing problems. We establish exact (stable) sparse recovery results under a restricted isometry property (RIP) condition for the constrained problem, and a full-rank theorem of the sensing matrix restricted to the support of the sparse solution. We present an iterative method for l(1-2) minimization based on the difference of convex functions algorithm and prove that it converges to a stationary point satisfying the first-order optimality condition. We propose a sparsity oriented simulated annealing procedure with non-Gaussian random perturbation and prove the almost sure convergence of the combined algorithm (DCASA) to a global minimum. Computation examples on success rates of sparse solution recovery show that if the sensing matrix is ill-conditioned (non RIP satisfying), then our method is better than existing nonconvex compressed sensing solvers in the literature. Likewise in the magnetic resonance imaging (MRI) phantom image recovery problem, l(1-2) succeeds with eight projections. Irrespective of the conditioning of the sensing matrix, l(1-2) is better than l(1) in both the sparse signal and the MRI phantom image recovery problems.