For spacetimes that are not asymptotic to anti-de Sitter Space (non AAdS), we adapt the Lewkowycz-Maldacena procedure to find the holographic entanglement entropy. The key observation, which to our knowledge is not very well appreciated, is that asymptotic boundary conditions play an essential role on extending the replica trick to the bulk. For non AAdS, we expect the following three main modifications: (1) the expansion near the special surface has to be compatible with the asymptotic expansion; (2) periodic conditions are imposed to coordinates on the phase space with diagonalized symplectic structure, not to all fields appearing in the action; (3) evaluating the entanglement functional using the boundary term method amounts to evaluating the presymplectic structure at the special surface, where some additional exact form may contribute. An explicit calculation is carried out for three-dimensional warped anti-de Sitter spacetime (WAdS3) in a consistent truncation of string theory, the so-called S-dual dipole theory. It turns out that the generalized gravitational entropy in WAdS3 is captured by the least action of a charged particle in WAdS3 space, or equivalently, by the geodesic length in an auxiliary AdS3. Consequently, the bulk calculation agrees with the CFT results, providing another piece of evidence for the WAdS3/CFT2 correspondence.

We propose a definition of generalized semi-inner products (g.s.i.p.). By relating them to duality mappings from a normed vector space to its dual space, a characterization for all g.s.i.p. satisfying this definition is obtained. We then study the Riesz representation of continuous linear functionals via g.s.i.p. As applications, we establish a representer theorem and characterization equation for the minimizer of a regularized learning from finite or infinite samples in Banach spaces of functions.

Frames in a Banach space B B mathContainer Loading Mathjax were defined as a sequence in its dual space B 68 B 68 mathContainer Loading Mathjax in some recent references. We propose to define them as a collection of elements in B B mathContainer Loading Mathjax by making use of semi-inner products. Classical theory on frames and Riesz bases is generalized under this new perspective. We then aim at establishing the Shannon sampling theorem in Banach spaces. The existence of such expansions in translation invariant reproducing kernel Hilbert and Banach spaces is discussed.

We view regularized learning of a function in a Banach space from its finite samples as an optimization problem. Within the framework of reproducing kernel Banach spaces, we prove the representer theorem for the minimizer of regularized learning schemes with a general loss function and a nondecreasing regularizer. When the loss function and the regularizer are differentiable, a characterization equation for the minimizer is also established.

Motivated by multi-task machine learning with Banach spaces, we propose the notion of vector-valued reproducing kernel Banach spaces (RKBSs). Basic properties of the spaces and the associated reproducing kernels are investigated. We also present feature map constructions and several concrete examples of vector-valued RKBSs. The theory is then applied to multi-task machine learning. Especially, the representer theorem and characterization equations for the minimizer of regularized learning schemes in vector-valued RKBSs are established.