4 papers
Variational formulations of ODE-Net as a mean-field optimal control problem and existence results
Noboru Isobe, Mizuho Okumura
This paper presents a mathematical analysis of ODE-Net, a continuum model of deep neural networks (DNNs). In recent years, Machine Learning researchers have introduced ideas of rep…
Profile decomposition in Sobolev spaces and decomposition of integral functionals II: homogeneous case
Mizuho Okumura
The present paper is devoted to a theory of profile decomposition for bounded sequences in \emph{homogeneous} Sobolev spaces, and it enables us to analyze the lack of compactness o…
Profile decomposition in Sobolev spaces and decomposition of integral functionals I: inhomogeneous case
Mizuho Okumura
The present paper is devoted to analysis of the lack of compactness of bounded sequences in \emph{inhomogeneous} Sobolev spaces, where bounded sequences might fail to be compact du…
Generalization of the Ehrling inequality and universal characterization of completely continuous operators
Mizuho Okumura
The present work is devoted to an extension of the well-known Ehrling inequalities, which quantitatively characterize compact embeddings of function spaces, to more general operato…