The paper studies the problem of approximating a general Gaussian location mixture by finite Gaussian mixtures. The key results are as follows:
For compactly supported mixing distributions, the minimum number of components m required to achieve an approximation error ε, measured by various f-divergences (TV, Hellinger, KL, χ2), is shown to be:
For distribution families with exponential tail decay (e.g., sub-Gaussian and sub-exponential), the minimum number of components m required to achieve an approximation error ε satisfies:
The upper bounds are achieved using local moment matching, while the lower bounds are established by relating the approximation error to the low-rank approximation of certain trigonometric moment matrices, followed by a refined spectral analysis.
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by Yun Ma,Yihon... lúc arxiv.org 04-16-2024
https://arxiv.org/pdf/2404.08913.pdfYêu cầu sâu hơn