In this paper, we study the factorization of bi-orthogonal Laurent polynomial wavelet matrices with degree one into simple blocks. A conjecture about advanced factorization is given.
In this paper, we study the factorization of bi-orthogonal Laurent polynomial wavelet matrices with degree one into simple blocks. A conjecture about advanced factorization is given.
A background removal method based on two-dimensional notch filtering in the frequency domain for polarization interference imaging spectrometers（PIISs） is implemented. According to the relationship between the spati...A background removal method based on two-dimensional notch filtering in the frequency domain for polarization interference imaging spectrometers（PIISs） is implemented. According to the relationship between the spatial domain and the frequency domain, the notch filter is designed with several parameters of PIISs, and the interferogram without a background is obtained. Both the simulated and the experimental results demonstrate that the background removal method is feasible and robust with a high processing speed. In addition, this method can reduce the noise level of the reconstructed spectrum, and it is insusceptible to a complicated background, compared with the polynomial fitting and empirical mode decomposition（EMD） methods.展开更多
Partial eigenvalue decomposition(PEVD) and partial singular value decomposition(PSVD) of large sparse matrices are of fundamental importance in a wide range of applications, including latent semantic indexing, spectra...Partial eigenvalue decomposition(PEVD) and partial singular value decomposition(PSVD) of large sparse matrices are of fundamental importance in a wide range of applications, including latent semantic indexing, spectral clustering, and kernel methods for machine learning. The more challenging problems are when a large number of eigenpairs or singular triplets need to be computed. We develop practical and efficient algorithms for these challenging problems. Our algorithms are based on a filter-accelerated block Davidson method.Two types of filters are utilized, one is Chebyshev polynomial filtering, the other is rational-function filtering by solving linear equations. The former utilizes the fastest growth of the Chebyshev polynomial among same degree polynomials; the latter employs the traditional idea of shift-invert, for which we address the important issue of automatic choice of shifts and propose a practical method for solving the shifted linear equations inside the block Davidson method. Our two filters can efficiently generate high-quality basis vectors to augment the projection subspace at each Davidson iteration step, which allows a restart scheme using an active projection subspace of small dimension. This makes our algorithms memory-economical, thus practical for large PEVD/PSVD calculations. We compare our algorithms with representative methods, including ARPACK, PROPACK, the randomized SVD method, and the limited memory SVD method. Extensive numerical tests on representative datasets demonstrate that, in general, our methods have similar or faster convergence speed in terms of CPU time, while requiring much lower memory comparing with other methods. The much lower memory requirement makes our methods more practical for large-scale PEVD/PSVD computations.展开更多
基金The Work was Partially Supported by NSFC# 69735 0 2 0
文摘In this paper, we study the factorization of bi-orthogonal Laurent polynomial wavelet matrices with degree one into simple blocks. A conjecture about advanced factorization is given.
基金The work was partially supported by NSFC # 69735052
文摘In this paper, we study the factorization of bi-orthogonal Laurent polynomial wavelet matrices with degree one into simple blocks. A conjecture about advanced factorization is given.
基金supported by the Major Program of the National Natural Science Foundation of China(No.41530422)the National Science and Technology Major Project of the Ministry of Science and Technology of China(No.32-Y30B08-9001-13/15)+1 种基金the National Natural Science Foundation of China(Nos.61275184,61540018,61405153,and 60278019)the National High Technology Research and Development Program of China(No.2012AA121101)
文摘A background removal method based on two-dimensional notch filtering in the frequency domain for polarization interference imaging spectrometers（PIISs） is implemented. According to the relationship between the spatial domain and the frequency domain, the notch filter is designed with several parameters of PIISs, and the interferogram without a background is obtained. Both the simulated and the experimental results demonstrate that the background removal method is feasible and robust with a high processing speed. In addition, this method can reduce the noise level of the reconstructed spectrum, and it is insusceptible to a complicated background, compared with the polynomial fitting and empirical mode decomposition（EMD） methods.
基金supported by National Science Foundation of USA (Grant Nos. DMS1228271 and DMS-1522587)National Natural Science Foundation of China for Creative Research Groups (Grant No. 11321061)+1 种基金the National Basic Research Program of China (Grant No. 2011CB309703)the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences
文摘Partial eigenvalue decomposition(PEVD) and partial singular value decomposition(PSVD) of large sparse matrices are of fundamental importance in a wide range of applications, including latent semantic indexing, spectral clustering, and kernel methods for machine learning. The more challenging problems are when a large number of eigenpairs or singular triplets need to be computed. We develop practical and efficient algorithms for these challenging problems. Our algorithms are based on a filter-accelerated block Davidson method.Two types of filters are utilized, one is Chebyshev polynomial filtering, the other is rational-function filtering by solving linear equations. The former utilizes the fastest growth of the Chebyshev polynomial among same degree polynomials; the latter employs the traditional idea of shift-invert, for which we address the important issue of automatic choice of shifts and propose a practical method for solving the shifted linear equations inside the block Davidson method. Our two filters can efficiently generate high-quality basis vectors to augment the projection subspace at each Davidson iteration step, which allows a restart scheme using an active projection subspace of small dimension. This makes our algorithms memory-economical, thus practical for large PEVD/PSVD calculations. We compare our algorithms with representative methods, including ARPACK, PROPACK, the randomized SVD method, and the limited memory SVD method. Extensive numerical tests on representative datasets demonstrate that, in general, our methods have similar or faster convergence speed in terms of CPU time, while requiring much lower memory comparing with other methods. The much lower memory requirement makes our methods more practical for large-scale PEVD/PSVD computations.
