分层流用于估计和采样高维概率
Hierarchic Flows to Estimate and Sample High-dimensional Probabilities
AI总结：
本文提出分层流方法，通过小波散射协方差模型估计和采样高维概率分布，用于生成湍流涡旋场和暗物质密度图像。
摘要：
寻找复杂物理场如湍流的低维可解释模型仍然是一个开放性问题，距Kolmogorov先驱工作已有80年。从数据样本估计高维概率分布受到优化和近似维度灾难的困扰。通过从粗到细的分层概率流可以避免这一问题。这种反向重整化群由不同尺度上的条件概率定义，以小波基函数进行重整化。对于φ⁴标量势，采样这些分层模型可避免相变处的临界减慢。在精心选择的小波基下，条件概率可通过低维参数模型捕捉，因为小波系数间的相互作用在空间和尺度上是局部的。另一个突出问题是近似具有长程相互作用的空间和尺度上的非高斯场。我们引入了基于小波条件概率的低维模型，使用散射协方差进行计算。该计算通过第二次小波变换定义了两个尺度层次上的相互作用。我们估计和采样这些小波散射模型以生成湍流的2D涡旋场和暗物质密度图像。
英文摘要：
Finding low-dimensional interpretable models of complex physical fields such as turbulence remains an open question, 80 years after the pioneer work of Kolmogorov. Estimating high-dimensional probability distributions from data samples suffers from an optimization and an approximation curse of dimensionality. It may be avoided by following a hierarchic probability flow from coarse to fine scales. This inverse renormalization group is defined by conditional probabilities across scales, renormalized in a wavelet basis. For a $\vvarphi^4$ scalar potential, sampling these hierarchic models avoids the critical slowing down at the phase transition. In a well chosen wavelet basis, conditional probabilities can be captured with low dimensional parametric models, because interactions between wavelet coefficients are local in space and scales. An outstanding issue is also to approximate non-Gaussian fields having long-range interactions in space and across scales. We introduce low-dimensional models of wavelet conditional probabilities with the scattering covariance. It is calculated with a second wavelet transform, which defines interactions over two hierarchies of scales. We estimate and sample these wavelet scattering models to generate 2D vorticity fields of turbulence, and images of dark matter densities.
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