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# The Haar Wavelet Analysis of Matrices and its Applications

2019-05-08

Academic Report

Title: The Haar Wavelet Analysis of Matrices and its Applications

Reporter: Prof. SHI Xiquan

Time: May 10, 2019 (Thursday) PM 15:30-17:00

Location: A1101# room, Innovation Park Building

Contact: Prof. LEI Fengchun (tel:84708360)

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. Does it has 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 equivalent 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 employee other wavelet transformations and Fourier transformation to obtain similar results.

The brief introduction to the reporter: Shi Xiquan is a lifelong professor of Delaware State University. From 1992 to 2001, he served as an associate professor, professor and doctoral supervisor of Dalian University of Technology. His honors include Alexander von Humboldt-Stiftung, Fok Yingdong University Young Teachers'Research Fund, the Second Prize for Science and Technology Progress of the Ministry of Education (formerly the State Education Commission), and the Top Ten Outstanding Science and Technology Youth in Dalian. Dr. Shi Xiquan has published more than 90 papers on computational geometry, multivariate splines and special functions.