用神经网络构造可逆变换，实现高效条件采样。
Constructive conditional normalizing flows
- 基于拉格朗日插值的极坐标分解，分离压缩与不可压缩分量。
- 通过剪切流实现不可压缩部分，保持变换可逆性与计算效率。
- 适用于高维数据的条件生成，特别适合光滑映射场景。
针对条件采样应用，给定概率测度 μ 及微分同胚 ϕ，本文研究如何同时逼近 ϕ 及其推送测度 ϕ#μ。方法基于连续性方程的流，速度场为带分段常数权重的感知机神经网络。提出显式构造：利用 ϕ 的拉格朗日插值的极坐标类似分解，其中压缩分量由特定凸函数的梯度精确实现，不可压缩分量经排列近似后，通过连续性方程固有的剪切流实现。对于更光滑的映射（如 Knöthe-Rosenblatt 重排），提出基于 Maurey 经验法的替代概率构造，其权重间断点数量不随环境维度反比增长。
原文摘要 · Abstract (English)
Motivated by applications in conditional sampling, given a probability measure $μ$ and a diffeomorphism $ϕ$, we consider the problem of simultaneously approximating $ϕ$ and the pushforward $ϕ_{\#}μ$ by means of the flow of a continuity equation whose velocity field is a perceptron neural network with piecewise constant weights. We provide an explicit construction based on a polar-like decomposition of the Lagrange interpolant of $ϕ$. The latter involves a compressible component, given by the gradient of a particular convex function, which can be realized exactly, and an incompressible component, which -- after approximating via permutations -- can be implemented through shear flows intrinsic to the continuity equation. For more regular maps $ϕ$ -- such as the Knöthe-Rosenblatt rearrangement -- we provide an alternative, probabilistic construction inspired by the Maurey empirical method, in which the number of discontinuities in the weights doesn't scale inversely with the ambient dimension.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。