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:2016-08-28  最后修改时间:2016-09-23
DOI:10.3770/j.issn:2095-2651.2017.01.002
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:
作者单位
Xiquan SHI Department of Mathematical Sciences, Delaware State University, Dover 19901, USA 
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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 time-domain problems into frequency-domain problems. Are there similar tools for a matrix? By pairing a matrix to a piecewise function, a Haar-like 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 time-domain problems into frequency-domain problems. Are there similar tools for a matrix? By pairing a matrix to a piecewise function, a Haar-like 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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