一般框架下ℓ_(1)-αℓ_(2)最小化的压缩数据分离

Compressed data separation with general frames viaℓ_(1)-αℓ_(2) minimization

本文考虑一般框架下压缩数据分离问题,即重构多模态数据中的不同子成分.基于已有的ABP(the analysis basis pursuit)算法和ADS(the analysis Dantzig selector)算法,结合ℓ_(1)-αℓ_(2)(0<α≤1)最小化,本文提出对偶ℓ_(1)-αℓ_(2)分解ABP算法及对偶ℓ_(1)-αℓ_(2)分解ADS算法.若多模态数据的两个不同子成分f_(1)和f_(2)分别在不同的一般框架D_(1)∈R^(n×d1)和D_(2)∈Rℓ_(1)-αℓ_(2)下(近似)稀疏,则当测量矩阵满足一定的约束等距性条件且框架间满足某个相互相干性条件时,本文提出的两种算法均可保证多模态数据不同子成分的稳定恢复.数值实验表明,本文的算法相对ℓ1最小化具有较高的重构成功率.

In this paper,we deal with the problem of compressed data separation with general frames,i.e.,the reconstruction of different subcomponents of multimodal data.Based on the existing ABP(the analysis basis pursuit)algorithm and ADS(the analysis Dantzig selector)algorithm,combining withℓ_(1)-αℓ_(2)(0<α≤1)minimization,we propose the dual(ℓ_(1)-αℓ_(2))-split ABP algorithm and the dual(ℓ_(1)-αℓ_(2))-split ADS algorithm.If two different components f1 and f2 of multimodal data are(approximately)sparse in terms of two different general frames D12 Rnd1 and D22 R^(n×d2),respectively,the two algorithms can ensure the stable recovery of the different subcomponents of the multimodal data when the measurement matrix satisfies a restricted isometry property and the frames satisfy a mutual coherence condition.Numerical experiments are carried out to show that our algorithm has a higher reconstruction success rate compared withℓ1 minimization.