论文标题
复合泊松过程的小波可压缩性
Wavelet Compressibility of Compound Poisson Processes
论文作者
论文摘要
在本文中，我们精确地量化了复合泊松过程的小波可压缩性。为此，我们将给定的随机过程扩展到HAAR小波的基础上，并分析其渐近近似特性。通过仅考虑非零小波系数达到给定的量表，我们称为贪婪的近似，我们利用了小波膨胀的极端稀疏性，该膨胀源自复合泊松过程的分段恒定性质。更确切地说，我们为复合泊松过程的贪婪近似值的平均平方误差提供了下限和上限。然后，我们能够推断出贪婪的近似误差具有亚指数和超级物质渐近行为。最后，我们提供数值实验，以突出基于小波的词典在实现高度可压缩的复合泊松过程近似值方面具有显着的能力。
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.
