The Haar Wavelet Analysis of Matrices and Its Applications |
Received:August 28, 2016 Revised:September 23, 2016 |
Key Words:
wavelet analysis Fourier analysis matrix decomposition $k$-means clustering linear equation
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Abstract: |
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. |
Citation: |
DOI:10.3770/j.issn:2095-2651.2017.01.002 |
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