A CDF-First Framework for Free-Form Density Estimation

• URL: http://arxiv.org/abs/2603.25204v1
• Date: Thu, 26 Mar 2026 09:09:00 GMT
• Title: A CDF-First Framework for Free-Form Density Estimation
• Abstract summary: A core challenge is free-form density estimation, capturing distributions that exhibit multimodality, asymmetry, or topological complexity without restrictive assumptions.<n>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 probability density function (PDF) via differentiation of the learned smooth CDF.<n>Our framework guarantees valid PDFs by construction, enables tractable approximate likelihood training, and preserves complex distributional shapes.
• Score: 8.204788282365657
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 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.

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