复合Poisson过程的小波可压缩性
Wavelet Compressibility of Compound Poisson Processes
摘要：
本文对复合Poisson过程的小波压缩性进行了精确的量化。为此，我们扩展给定的随机过程的Haar小波基，我们分析其渐近逼近性质。通过只考虑非零小波系数到一个给定的规模，我们称之为贪婪近似，我们利用极端稀疏的小波扩展，来自分段常数性质的复合泊松过程。更确切地说，我们提供了下和上界的贪婪逼近复合泊松过程的均方误差。然后，我们能够推断出贪婪近似误差具有次指数和超多项式的渐近行为。最后，我们提供了数值实验，突出了显着的能力，基于小波的字典在实现高度可压缩的复合泊松过程的近似。
英文摘要：
In this paper, we precisely quantify the wavelet compressibility of compound Poisson processes. To that end, we expand the given random process over the Haar wavelet basis and we analyse its asymptotic approximation properties. By only considering the nonzero wavelet coefficients up to a given scale, what we call the greedy approximation, we exploit the extreme sparsity of the wavelet expansion that derives from the piecewise-constant nature of compound Poisson processes. More precisely, we provide lower and upper bounds for the mean squared error of greedy approximation of compound Poisson processes. We are then able to deduce that the greedy approximation error has a sub-exponential and super-polynomial asymptotic behavior. Finally, we provide numerical experiments to highlight the remarkable ability of wavelet-based dictionaries in achieving highly compressible approximations of compound Poisson processes.
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