New Algorithm for Density Estimation Using Hellinger Distance Achieves Near-Linear Time Complexity
Researchers have developed a new minimum-distance estimator approach for density estimation that works with Hellinger distance, extending techniques previously limited to total variation distance. The method leverages connections to reverse data processing inequalities and VC dimension theory to enable efficient learning. This advance enables the first near-linear time algorithms for estimating complex probability distributions like mixtures of Gaussians and log-concave densities with near-optimal sample complexity.
A new theoretical computer science paper presents an extension of the minimum-distance estimator framework for density estimation, a fundamental problem in machine learning and statistics. The key innovation is adapting this classical approach to work with Hellinger distance rather than just total variation distance, by drawing connections to recent reverse data processing inequality results. The authors demonstrate that their recipe—which relies on bounding the VC dimension of a related concept class—is flexible enough to accommodate fast algorithms. By modifying prior work by Acharya et al. (2017), they achieve the first near-linear time algorithm for learning important function classes including univariate mixtures of log-concave densities and mixtures of Gaussians with arbitrary variances, while maintaining near-optimal sample complexity bounds.
What's missing
The paper does not discuss practical applications or empirical validation of the proposed algorithms. Additionally, the specific constants hidden in the 'near-linear' and 'near-optimal' complexity bounds are not detailed in the abstract, and the paper does not address how the approach scales to high-dimensional settings beyond univariate cases.
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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