投影Nesterov近似梯度算法在凸约束稀疏信号重构中的应用
Projected Nesterov's Proximal-Gradient Algorithm for Sparse Signal Reconstruction with a Convex Constraint
摘要：
我们开发了一个投影Nesterov的近似梯度（PNPG）稀疏信号重建的方法，结合自适应步长与Nesterov的动量加速。我们希望最小化的目标函数是凸可微数据保真度（负对数似然（NLL））项和凸正则化项之和。我们应用稀疏信号正则化，其中信号属于NLL域的封闭内的封闭凸集;凸集约束有利于灵活的NLL域和准确的信号恢复。信号稀疏性分别使用信号的线性变换系数或梯度图上的$\ell_1$范数惩罚来施加。PNPG方法采用了投影Nesterov的加速步骤与重新启动和内部迭代计算邻近映射。我们提出了一个自适应的步长选择方案，以获得一个良好的本地优化功能的NLL和减少时间花费回溯。由于步长自适应，PNPG不需要NLL梯度的Lipschitz连续性。我们提出了一个完整的推导的动量加速和它的$\mathcal{O}(k^{-2})$收敛速度和收敛性的证明，其中考虑到自适应步长的选择，不精确的迭代邻近映射，和凸集约束。PNPG的调优在很大程度上与应用无关。泊松广义线性和高斯线性测量模型的层析成像和压缩感知重建实验证明了所提出的方法的性能。
英文摘要：
We develop a projected Nesterov's proximal-gradient (PNPG) approach for sparse signal reconstruction that combines adaptive step size with Nesterov's momentum acceleration. The objective function that we wish to minimize is the sum of a convex differentiable data-fidelity (negative log-likelihood (NLL)) term and a convex regularization term. We apply sparse signal regularization where the signal belongs to a closed convex set within the closure of the domain of the NLL; the convex-set constraint facilitates flexible NLL domains and accurate signal recovery. Signal sparsity is imposed using the $\ell_1$-norm penalty on the signal's linear transform coefficients or gradient map, respectively. The PNPG approach employs projected Nesterov's acceleration step with restart and an inner iteration to compute the proximal mapping. We propose an adaptive step-size selection scheme to obtain a good local majorizing function of the NLL and reduce the time spent backtracking. Thanks to step-size adaptation, PNPG does not require Lipschitz continuity of the gradient of the NLL. We present an integrated derivation of the momentum acceleration and its $\mathcal{O}(k^{-2})$ convergence-rate and iterate convergence proofs, which account for adaptive step-size selection, inexactness of the iterative proximal mapping, and the convex-set constraint. The tuning of PNPG is largely application-independent. Tomographic and compressed-sensing reconstruction experiments with Poisson generalized linear and Gaussian linear measurement models demonstrate the performance of the proposed approach.
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