The statistical Minkowski distances: Closed-form formula for Gaussian Mixture Models
This work provides a theoretical extension for distance calculations in machine learning, particularly for mixture models, but appears incremental as it builds on existing Minkowski concepts.
The authors tackled the problem of computing distances between probability distributions by proposing statistical Minkowski distances based on Minkowski's inequality, which admit closed-form formulas for Gaussian mixture models and other exponential families with integer exponents.
The traditional Minkowski distances are induced by the corresponding Minkowski norms in real-valued vector spaces. In this work, we propose novel statistical symmetric distances based on the Minkowski's inequality for probability densities belonging to Lebesgue spaces. These statistical Minkowski distances admit closed-form formula for Gaussian mixture models when parameterized by integer exponents. This result extends to arbitrary mixtures of exponential families with natural parameter spaces being cones: This includes the binomial, the multinomial, the zero-centered Laplacian, the Gaussian and the Wishart mixtures, among others. We also derive a Minkowski's diversity index of a normalized weighted set of probability distributions from Minkowski's inequality.