arXiv:2603.25204cs.LG2026-03

先学累积分布函数,再推概率密度,更稳定准确。

A CDF-First Framework for Free-Form Density Estimation

  • 用累积分布函数(CDF)替代直接估计概率密度
  • 在多个任务上超越现有最优密度估计方法
  • 适合需要精准建模复杂分布的科研与工程场景

条件密度估计(CDE)旨在建模完整的条件分布 $\mathbb{P}(\mathbf{y} \mid \mathbf{x})$,而不仅是点预测。自由形式密度估计需捕捉多模态、非对称或拓扑复杂的分布,但现有方法直接估计概率密度函数(PDF)存在数学病态性:对经验分布求导会放大有限数据中的随机波动,因此依赖强归纳偏置,限制表达能力且易失效。本文提出一种以累积分布函数(CDF)为先的框架,通过学习稳定且良定义的CDF,再通过对平滑学习到的CDF求导恢复PDF。采用光滑极小-极大(SMM)网络参数化CDF,保证生成的PDF合法,支持可计算的近似似然训练,并能保留复杂分布形态。对于多维输出,使用带有SMM因子的自回归分解。实验表明,该方法在多种单变量和多变量任务中均优于当前最优密度估计器。

原文摘要 · Abstract (English)

Conditional density estimation (CDE) is a fundamental task in machine learning that aims to model the full conditional law $\mathbb{P}(\mathbf{y} \mid \mathbf{x})$, beyond mere point prediction (e.g., mean, mode). A core challenge is free-form density estimation, capturing distributions that exhibit multimodality, asymmetry, or topological complexity without restrictive assumptions. However, prevailing methods typically estimate the probability density function (PDF) directly, which is mathematically ill-posed: differentiating the empirical distribution amplifies random fluctuations inherent in finite datasets, necessitating strong inductive biases that limit expressivity and fail when violated. We propose a CDF-first framework that circumvents this issue by estimating the cumulative distribution function (CDF), a stable and well-posed target, and then recovering the PDF via differentiation of the learned smooth CDF. Parameterizing the CDF with a Smooth Min-Max (SMM) network, our framework guarantees valid PDFs by construction, enables tractable approximate likelihood training, and preserves complex distributional shapes. For multivariate outputs, we use an autoregressive decomposition with SMM factors. Experiments demonstrate our approach outperforms state-of-the-art density estimators on a range of univariate and multivariate tasks.

密度估计概率建模CDFSMM

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