We introduce kernel density machines (KDM), an agnostic kernel-based framework for learning the Radon-Nikodym derivative (density) between probability measures under minimal assumptions. KDM applies to general measurable spaces and avoids the structural requirements common in classical nonparametric density estimators. We construct a sample estimator and prove its consistency and a functional central limit theorem. To enable scalability, we develop Nystrom-type low-rank approximations and derive optimal error rates, filling a gap in the literature where such guarantees for density learning have been missing. We demonstrate the versatility of KDM through applications to kernel-based two-sample testing and conditional distribution estimation, the latter enjoying dimension-free guarantees beyond those of locally smoothed methods. Experiments on simulated and real data show that KDM is accurate, scalable, and competitive across a range of tasks.

翻译：我们提出核密度机器（KDM），一种基于核的不可知框架，用于在最小假设下学习概率测度间的Radon-Nikodym导数（密度）。KDM适用于一般可测空间，避免了经典非参数密度估计器中常见的结构要求。我们构造了一个样本估计量，并证明了其一致性和函数中心极限定理。为提升可扩展性，我们开发了Nyström型低秩近似，并推导出最优误差率，填补了密度学习领域中此类保证缺失的文献空白。通过核基双样本检验和条件分布估计的应用，我们展示了KDM的多功能性——后者享有超越局部平滑方法的无维度保证。在模拟和真实数据上的实验表明，KDM在一系列任务中准确、可扩展且具有竞争力。