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 修订日期：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 
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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. 
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