本文選題：矩陣填充 + Tbeplitz矩陣�。� 參考：《太原理工大學》2017年碩士論文

【摘要】：矩陣填充問題就是從有限的信息中重構(gòu)低秩或者近似低秩矩陣的問題,是近年來國際國內(nèi)在信號處理領(lǐng)域的研究熱點之一.而在實際應(yīng)用中采樣矩陣有時為普通矩陣有時則具有Hankel及Toeplitz等結(jié)構(gòu)的特殊矩陣,同時Toeplitz矩陣在信號和圖像處理中發(fā)揮著重要的作用.故本文分別對Toeplitz矩陣和普通矩陣的矩陣填充問題進行了較深入的研究.通過對矩陣填充問題的研究總結(jié),我們發(fā)現(xiàn)現(xiàn)有的大多數(shù)算法都需要計算矩陣的奇異值分解,同時通過數(shù)值實驗我們也發(fā)現(xiàn),奇異值閾值是算法中主要的耗時部分.因此,對于Toeplitz陣的填充問題,我們根據(jù)普通矩陣的奇異值分解算法復雜度為O(n3)而Toeplitz矩陣的快速奇異值分解算法的復雜度僅為O(n2logn),并通過在左奇異向量空間中對已知元素的最小二乘逼近,形成了新的可行矩陣,最后利用對角線上的均值化使得迭代后的矩陣保持Toeplitz結(jié)構(gòu),從而減少了奇異向量空間的分解時間.對于普通矩陣的填充問題,我們基于最速下降算法提出了 一種兩階迭代最速下降算法:先在內(nèi)層迭代中找到最佳逼近矩陣或近似最佳逼近矩陣,然后在外層迭代中找到r維最優(yōu)流形.理論上,證明了在一定條件下兩類新算法均是收斂的,并通過大量數(shù)值實驗驗證了兩類新算法的合理性和優(yōu)越性.通過對實驗結(jié)果的比較,我們可以發(fā)現(xiàn)新算法降低了計算所需的CPU時間,這將有利于求解大規(guī)模的普通矩陣和Toeplitz矩陣的填充問題,在實際應(yīng)用中節(jié)約了時間降低了成本.
[Abstract]:Matrix filling problem is the problem of reconstruction of low rank matrix or approximate low rank matrix from limited information. It is one of the research hotspots in the field of signal processing at home and abroad in recent years. In practical applications, the sampling matrix is sometimes a common matrix, sometimes a special matrix with Hankel and Toeplitz structure, and the Toeplitz matrix plays an important role in signal and image processing. Therefore, the matrix filling problem of Toeplitz matrix and ordinary matrix is studied in this paper. Based on the study of matrix filling problem, we find that most of the existing algorithms need to calculate the singular value decomposition of the matrix. At the same time, through numerical experiments, we also find that the singular value threshold is the main time-consuming part of the algorithm. Therefore, for the filling problem of Toeplitz arrays, According to the complexity of singular value decomposition (SVD) algorithm of ordinary matrix is ON3) and the complexity of fast SVD algorithm of Toeplitz matrix is only ON2lognan, we form a new feasible matrix by the least square approximation of known elements in the left singular vector space. Finally, the Toeplitz structure of the iterative matrix is kept by means of the mean value on the diagonal line, thus reducing the decomposition time of the singular vector space. For the filling problem of ordinary matrices, we propose a two-order iterative steepest descent algorithm based on the steepest descent algorithm: first, we find the best approximation matrix or approximate best approximation matrix in the inner iteration. Then the r-dimensional optimal manifold is found in the outer iteration. In theory, it is proved that the two new algorithms are convergent under certain conditions, and the rationality and superiority of the two new algorithms are verified by a large number of numerical experiments. By comparing the experimental results, we can find that the new algorithm reduces the CPU time required for computation, which is helpful to solve the filling problem of large scale ordinary matrix and Toeplitz matrix, and saves time and cost in practical application.
【學位授予單位】：太原理工大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O151.21

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