New Algorithm for Density Estimation Using Hellinger Distance Achieves Near-Linear Time Complexity
A new preprint on arXiv presents a method for extending the classical minimum-distance estimator approach in density estimation from total variation distance to Hellinger distance. The work draws on reverse data processing inequalities to construct an analogous recipe requiring only bounds on the VC dimension of a related concept class. The result yields the first near-linear time algorithm with near-optimal sample complexity for learning univariate mixtures of log-concave densities and mixtures of Gaussians with arbitrary variances.
Researchers have submitted a preprint to arXiv introducing a framework that generalizes the minimum-distance estimator (MDE) approach—a standard tool in density estimation—to work under Hellinger distance, not just total variation distance. The classical MDE technique derives both algorithms and guarantees by bounding the VC dimension of a concept class known as the Yatracos class, but its sharp guarantees had previously been confined largely to total variation distance. By connecting the MDE framework to recent results on reverse data processing inequalities, the authors construct an analogous recipe for Hellinger distance that similarly requires only VC dimension bounds on a related concept class. Adapting the approach of Acharya et al. (2017), the paper claims the first near-linear time algorithm for density estimation of univariate mixtures of log-concave densities and mixtures of Gaussians with arbitrary variances, achieving near-optimal sample complexity. The framework is described as flexible enough to incorporate fast algorithms originally designed for total variation distance. The work sits at the intersection of theoretical computer science, machine learning, and mathematical statistics. As a preprint, it has not yet undergone formal peer review.
What's missing
As a preprint, this work has not been peer-reviewed, and independent verification of the claimed algorithmic guarantees and sample complexity bounds is pending. Practical empirical performance relative to existing estimators is not discussed.
What different sources said
- arXiv cs.LGCenter
Density estimation for Hellinger via minimum-distance estimators: mixtures of Gaussians, log-concave, and more
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