We are concerned with divergences on the Cartan–Hadamard Riemannian manifold of symmetric cones, self-dual homogeneous cones in Euclidean spaces, and related optimization problems. We introduce a parameterized version of fidelity on symmetric cones, namely sandwiched quasi-relative entropies, and construct a one-parameter family of divergences based on these entropies. We consider the median minimization problem of finite points over these divergences and establish existence and uniqueness of minimizer. The global linear rate convergence of a gradient projection algorithm for solving the median minimization problem is analyzed based on the derived upper bound of the condition number of the Hessian function.
The work of the first author was supported by Basic Science Research Program through NRF Grant No. NRF-2017R1A2B1002008. The work of the second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) Nos. NRF-2015R1A3A2031159 and 2016R1A5A1008055. The work of the third author was supported by the National Research Foundation of Korea (NRF) Nos. NRF-2016R1A5A1008055 and NRF-2019R1F1A1057051.
"Divergences on Symmetric Cones and Medians." Taiwanese J. Math. Advance Publication 1 - 20, 2022. https://doi.org/10.11650/tjm/220106