Xiquan SHI.The Haar Wavelet Analysis of Matrices and Its Applications[J].数学研究及应用,2017,37(1):19~28 
The Haar Wavelet Analysis of Matrices and Its Applications 
The Haar Wavelet Analysis of Matrices and Its Applications 
投稿时间：20160828 最后修改时间：20160923 
DOI：10.3770/j.issn:20952651.2017.01.002 
中文关键词: wavelet analysis Fourier analysis matrix decomposition $k$means clustering linear equation 
英文关键词:wavelet analysis Fourier analysis matrix decomposition $k$means clustering linear equation 
基金项目: 

摘要点击次数: 1981 
全文下载次数: 1017 
中文摘要: 
It is well known that Fourier analysis or wavelet analysis is a very powerful and useful tool for a function since they convert timedomain problems into frequencydomain problems. Are there similar tools for a matrix? By pairing a matrix to a piecewise function, a Haarlike wavelet is used to set up a similar tool for matrix analyzing, resulting in new methods for matrix approximation and orthogonal decomposition. By using our method, one can approximate a matrix by matrices with different orders. Our method also results in a new matrix orthogonal decomposition, reproducing Haar transformation for matrices with orders of powers of two. The computational complexity of the new orthogonal decomposition is linear. That is, for an $m\times n$ matrix, the computational complexity is $O(mn)$. In addition, when the method is applied to $k$means clustering, one can obtain that $k$means clustering can be equivalently converted to the problem of finding a best approximation solution of a function. In fact, the results in this paper could be applied to any matrix related problems. In addition, one can also employ other wavelet transformations and Fourier transformation to obtain similar results. 
英文摘要: 
It is well known that Fourier analysis or wavelet analysis is a very powerful and useful tool for a function since they convert timedomain problems into frequencydomain problems. Are there similar tools for a matrix? By pairing a matrix to a piecewise function, a Haarlike wavelet is used to set up a similar tool for matrix analyzing, resulting in new methods for matrix approximation and orthogonal decomposition. By using our method, one can approximate a matrix by matrices with different orders. Our method also results in a new matrix orthogonal decomposition, reproducing Haar transformation for matrices with orders of powers of two. The computational complexity of the new orthogonal decomposition is linear. That is, for an $m\times n$ matrix, the computational complexity is $O(mn)$. In addition, when the method is applied to $k$means clustering, one can obtain that $k$means clustering can be equivalently converted to the problem of finding a best approximation solution of a function. In fact, the results in this paper could be applied to any matrix related problems. In addition, one can also employ other wavelet transformations and Fourier transformation to obtain similar results. 
查看全文 查看/发表评论 下载PDF阅读器 


