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Divergence functions are the non-symmetric “distance” on the manifold, Μθ, of parametric probability density functions over a measure space, ( Χ,μ). Classical information geometry prescribes, on Μθ: (i) a Riemannian metric given by the Fisher information; (ii) a pair of dual connections (giving rise to the family of α-connections) that preserve the metric under parallel transport by their joint actions; and (iii) a family of divergence functions ( α-divergence) defined on Μθ x？ Μθ, which induce the metric and the dual connections. Here, we construct an extension of this differential geometric structure from Μθ (that of parametric probability density functions) to the manifold, Μ, of non-parametric functions on X, removing the positivity and normalization constraints. The generalized Fisher information and α-connections on M are induced by an α-parameterized family of divergence functions, reflecting the fundamental convex inequality associated with any smooth and strictly convex function. The infinite-dimensional manifold, M, has zero curvature for all these α-connections; hence, the generally non-zero curvature of M can be interpreted as arising from an embedding of Μθ into Μ. Furthermore, when a parametric model (after a monotonic scaling) forms an affine submanifold, its natural and expectation parameters form biorthogonal coordinates, and such a submanifold is dually flat for α？= ± 1, generalizing the results of Amari’s α-embedding. The present analysis illuminates two different types of duality in information geometry, one concerning the referential status of a point (measurable function) expressed in the divergence function (“referential duality”) and the other concerning its representation under an arbitrary monotone scaling (“representational duality”).

A paper was published (Harsha and Subrahamanian Moosath, 2014) in which the authors claimed to have discovered an extension to Amari's \(\alpha\)-geometry through a general monotone embedding function. It will be pointed out here that this so-called \((F, G)\)-geometry (which includes \(F\)-geometry as a special case) is identical to Zhang's (2004) extension to the \(\alpha\)-geometry, where the name of the pair of monotone embedding functions \(\rho\) and \(\tau\) were used instead of \(F\) and \(H\) used in Harsha and Subrahamanian Moosath (2014). Their weighting function \(G\) for the Riemannian metric appears cosmetically due to a rewrite of the score function in log-representation as opposed to \((\rho, \tau)\)-representation in Zhang (2004). It is further shown here that the resulting metric and \(\alpha\)-connections obtained by Zhang (2004) through arbitrary monotone embeddings is a unique extension of the \(\alpha\)-geometric structure. As a special case, Naudts' (2004) \(\phi\)-logarithm embedding (using the so-called \(\log_\phi\) function) is recovered with the identification \(\rho=\phi, \, \tau=\log_\phi\), with \(\phi\)-exponential \(\exp_\phi\) given by the associated convex function linking the two representations.

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.

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.

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 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.