[2603.25204] A CDF-First Framework for Free-Form Density Estimation

Computer Science > Machine Learning arXiv:2603.25204 (cs) [Submitted on 26 Mar 2026] Title:A CDF-First Framework for Free-Form Density Estimation Authors:Chenglong Song, Mazharul Islam, Lin Wang, Bing Chen, Bo Yang View a PDF of the paper titled A CDF-First Framework for Free-Form Density Estimation, by Chenglong Song and 4 other authors View PDF HTML (experimental) Abstract: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 di...