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

DOI：10.3770/j.issn:2095-2651.2017.01.002

 作者 单位 Xiquan SHI Department of Mathematical Sciences, Delaware State University, Dover 19901, USA

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.

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.