Data analysis and representation on a general domain using eigenfunctions of Laplacian |
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Authors: | Naoki Saito |
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Affiliation: | aDepartment of Mathematics, University of California, Davis, CA 95616, USA |
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Abstract: | We propose a new method to analyze and efficiently represent data recorded on a domain of general shape in by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonalize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the computation. We also show that our method is better suited for small sample data than the Karhunen–Loève transform/principal component analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further application, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain. |
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Keywords: | Laplacian eigenfunctions Boundary conditions Green's function Spectral decomposition Karhunen– Loè ve transform Principal component analysis Heat equation |
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