An Error Bound of Signal Recovery for Penalized Programs in Linear Inverse Problems
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摘要: 惩罚优化问题常常用于在有噪声的条件下用较少的观测个数来求解线性逆问题。目前，对惩罚优化问题恢复误差的研究主要存在以下两点不足：一是对权重参数往往有要求；二是噪声的方向对误差的影响未知。针对这两个问题，该文研究了当存在有界噪声时，惩罚优化问题恢复的误差界。首先，该文从问题的几何出发，给定了一个几何条件。当这一条件满足时，就能够推导出惩罚优化问题恢复的一个明确的误差界。这个误差界保证了恢复的解是稳定的，也就是说，恢复误差不会超过观测误差的常数倍。同时，这一误差界对于任意的正权重参数都成立，并且揭示了恢复误差以及最优的权重选择与观测噪声的方向之间的联系。进一步地，当观测矩阵是一个高斯矩阵时，依据这一几何条件可以得到高概率稳定恢复所需的观测次数。仿真实验证明了理论结果的正确性。Abstract: Penalized programs are widely used to solve linear inverse problems in the presence of noise. For now, the study of the performance of panelized programs has two disadvantages. First, the results have some limitations on the tradeoff parameters. Second, the effect of the direction of the noise is not clear. This paper studies the performance of penalized programs when bounded noise is presented. A geometry condition which is used to study the noise-free problems and constrained problems is provided. Under this condition, an explicit error bound which guarantees stable recovery (i.e., the recovery error is bounded by the observation noise up to some constant factor) is proposed. The results are different from many previous studies in two folds. First, the results provide an explicit bound for all positive tradeoff parameters, while many previous studies require that the tradeoff parameter is sufficiently large. Second, the results clear the role of the direction of the observation noise playing in the recovery error, and reveal the relationship between the optimal tradeoff parameters and the noise direction. Furthermore, if the sensing matrix has independent standard normal entries, the above geometry condition can be studied using Gaussian process theory, and the measurement number needed to guarantee stable recovery with high probability is obtained. Simulations are provided to verify the theoretical results.
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Key words:
- Linear inverse problem /
- Compressed sensing /
- Stable recovery /
- Penalized program /
- Tradeoff
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表 1 本文符号说明
符号 说明 ${\text{y}}$ 观测信号 ${\text{A}}$ 观测矩阵 ${{\text{x}}^ * }$ 真实信号 ${\text{z}}$ 观测噪声 $\varepsilon $ 观测噪声${\text{z}}$的幅度 $m$ 观测个数(观测信号${\text{y}}$的维度) $n$ 真实信号${{\text{x}}^ * }$的维度 $f$ 真实信号${{\text{x}}^ * }$的结构函数 $\lambda $ 惩罚优化问题中的权重调节参数 $\partial f({{\text{x}}^ * })$ 函数$f$在点${{\text{x}}^ * }$的次梯度 $D(f,{{\text{x}}^ * })$ 函数$f$在点${{\text{x}}^ * }$的下降锥 $N(f,{{\text{x}}^ * })$ 函数$f$在点${{\text{x}}^ * }$的法锥 $w(D(f,{{\text{x}}^ * }) \cap {{\text{S}}^{n - 1}})$ 下降锥$D(f,{{\text{x}}^ * })$的球面高斯宽度 
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