本文選題：壓縮感知　切入點(diǎn)：矩陣補(bǔ)全　出處：《電子科技大學(xué)》2017年博士論文

【摘要】：在電子工程中存在一類非常重要的數(shù)學(xué)問(wèn)題,那就是:在欠采樣情況下滿足觀測(cè)數(shù)據(jù)的可行向量不是唯一的,而是可行解組成的一個(gè)線性子空間,在此類情況下怎樣尋找符合現(xiàn)實(shí)問(wèn)題的唯一可行解。要在一個(gè)線性子空間中確定唯一正確的可行解,則這個(gè)唯一正確的可行解必須具有其他特殊性質(zhì),在電子工程中特殊性質(zhì)研究較多的是稀疏性(非零元素個(gè)數(shù)比較少)。原始問(wèn)題可以描述為:在欠采樣情況下,已知觀測(cè)數(shù)據(jù)求解具備稀疏性的可行向量。隨著問(wèn)題研究的深入,提出壓縮感知的這一嶄新理論,同時(shí)針對(duì)壓縮感知問(wèn)題學(xué)者們提出各種各樣求解壓縮感知問(wèn)題的算法,以及利用壓縮感知理論和相應(yīng)算法求解電子工程中很多棘手的現(xiàn)實(shí)問(wèn)題。壓縮感知理論的研究也拉開(kāi)了低維度結(jié)構(gòu)化成份重構(gòu)問(wèn)題的研究,比如:矩陣作為高維向量的稀疏性以及矩陣奇異值向量組的稀疏性(也稱為矩陣的低秩性)。本文研究的重點(diǎn)放在低秩結(jié)構(gòu)矩陣的重構(gòu)問(wèn)題以及低維度結(jié)構(gòu)化成份重構(gòu)的一般模型穩(wěn)定性研究,主要研究?jī)?nèi)容分為四部分:第一部分重點(diǎn)研究矩陣低秩稀疏重構(gòu)模型——觀測(cè)矩陣是由一個(gè)稀疏矩陣和一個(gè)低秩矩陣疊加組成時(shí),怎樣重構(gòu)觀測(cè)矩陣的稀疏分量以及低秩分量。首先在全空間采樣下情況考慮此類問(wèn)題,并利用凸優(yōu)化理論證明此類問(wèn)題的強(qiáng)凸優(yōu)化模型在一定條件下能準(zhǔn)確且唯一重構(gòu)低秩分量和稀疏分量,并在此基礎(chǔ)上給出算法設(shè)計(jì)中具體參數(shù)的選擇標(biāo)準(zhǔn)。第二部分重點(diǎn)研究矩陣的低秩稀疏重構(gòu)模型比全空間采樣更一般的情況,即研究矩陣低秩稀疏重構(gòu)模型在隨機(jī)子空間采樣下的重構(gòu)問(wèn)題。利用凸優(yōu)化理論證明在隨機(jī)采樣下滿足一定條件時(shí)強(qiáng)凸優(yōu)化模型的最優(yōu)解與凸優(yōu)化模型的最優(yōu)解一致性,并給出算法設(shè)計(jì)中具體參數(shù)選擇標(biāo)準(zhǔn)。第三部分重點(diǎn)研究矩陣低秩稀疏重構(gòu)模型強(qiáng)凸優(yōu)化模型中向隨機(jī)子空間投影矩陣算子。證明在一定條件下此隨機(jī)矩陣算子滿足嚴(yán)格等距性質(zhì),并利用矩陣算子的嚴(yán)格等距性質(zhì)優(yōu)化算法設(shè)計(jì)中參數(shù)選擇標(biāo)準(zhǔn)。第四部分重點(diǎn)研究矩陣低秩稀疏重構(gòu)模型的推廣形式,也就是低維度結(jié)構(gòu)化重構(gòu)問(wèn)題。證明在有界噪聲干擾下低維度結(jié)構(gòu)化重構(gòu)問(wèn)題的凸優(yōu)化模型最優(yōu)解具有穩(wěn)定性。
[Abstract]:There is a very important mathematical problem in electronic engineering, that is, the feasible vector satisfying the observed data in the case of under-sampling is not unique, but a linear subspace composed of feasible solutions. In this case, how to find the only feasible solution of the problem in accordance with the reality, in order to determine the only correct feasible solution in a linear subspace, the only correct feasible solution must have other special properties. In electronic engineering, sparseness (the number of non-zero elements is relatively small) is more studied in electronic engineering. The original problem can be described as: in the case of under-sampling, the known observation data can be used to solve the feasible vector with sparsity. This new theory of compressed perception is put forward, and a variety of algorithms to solve the problem of compressed perception are put forward for the problem of compressed perception. And using the theory of compressed perception and corresponding algorithms to solve many thorny practical problems in electronic engineering. The research of compressed perception theory also opens the research of low-dimensional structural component reconstruction. For example, the sparsity of matrix as high dimensional vector and the sparsity of singular value vector system of matrix (also called low rank of matrix). In this paper, we focus on the reconstruction of low rank structure matrix and the structural composition of low dimension. Research on the stability of the general model of reconfiguration, The main research contents are divided into four parts: the first part focuses on the low-rank sparse reconstruction model of matrix, when the observation matrix is composed of a sparse matrix and a low-rank matrix. How to reconstruct the sparse component and the low rank component of the observation matrix. The convex optimization theory is used to prove that the strong convex optimization model of this kind of problems can reconstruct the low rank and sparse components accurately and uniquely under certain conditions. On this basis, the selection criteria of the specific parameters in the algorithm design are given. In the second part, the low rank sparse reconstruction model of matrix is studied more generally than the full space sampling. By using convex optimization theory, it is proved that the optimal solution of strongly convex optimization model is consistent with that of convex optimization model when it satisfies certain conditions under random sampling. In the third part, we focus on studying the projection matrix operator of the directed random subspace in the strong convex optimization model of the sparse reconstruction model with low rank matrix, and prove the random matrix operator under certain conditions. Satisfying the strict isometric property, In the fourth part, we focus on the generalized form of matrix low rank sparse reconstruction model. It is proved that the optimal solution of the convex optimization model for the low dimensional structured reconstruction problem under bounded noise disturbance is stable.
【學(xué)位授予單位】：電子科技大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O151.21

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