2023/05/09 by Ken’ichiro Tanaka, Tanaka, Ken'ichiro
Medicine · #49M37 #62E17 #65K10 #Advanced Neuroimaging Techniques and Applications #Bone and Joint Diseases #FOS: Mathematics #Medical Imaging Techniques and Applications #Numerical Analysis (math.NA) #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2305.05127
openalex publication_date 2023/05/09 · openalex created_date 2023/05/12 · openalex updated_date 2026/07/28
We consider problems of minimizing functionals F of probability measures on the Euclidean space. To propose an accelerated gradient descent algorithm for such problems, we consider gradient flow of transport maps that give push-forward measures of an initial measure. Then we propose a deterministic accelerated algorithm by extending Nesterov's acceleration technique with momentum. This algorithm do not based on the Wasserstein geometry. Furthermore, to estimate the convergence rate of the accelerated algorithm, we introduce new convexity and smoothness for F based on transport maps. As a result, we can show that the accelerated algorithm converges faster than a normal gradient descent algorithm. Numerical experiments support this theoretical result.