In this article, we study a class of Kahler manifolds defined on tube domains in C-n, and in particular those which have O(n) x R-n symmetry. For these, we prove a uniqueness result showing that any such manifold which is complete and has non-negative orthogonal bisectional curvature (n >= 3) or non-negative bisectional curvature (n >= 2) is biholomorphically isometric to C-n. We also consider another curvature tensor called the orthogonal anti-bisectional curvature. We find necessary and sufficient conditions for a complete O(n)-symmetric tube domain to have non-negative orthogonal anti-bisectional curvature and provide several examples of complete metrics which satisfy this condition. Finally, we discuss some applications of these spaces within optimal transport. In particular, we study "synthetic" curvature bounds for non-smooth geometries and how they can be applied to the rough geometry induced by the Monge cost c(x, y) = parallel to x-y parallel to.

We revisit the work of Rieffel and van Daele on pairs of subspaces of a real Hilbert space, while relaxing as much as possible the assumption that all the relevant subspaces are in general position with respect to each other. We work out, in detail, how two real projection operators lead to the construction of a complex Hilbert space where the theory of the modular operator is applicable, with emphasis on the relevance of a central extension of the group of split quaternions. Two examples are given for which the subspaces have unequal dimension and therefore are not in generic position.

For a smooth manifold M, the tangent groupoid “glues” the set M × M with TM as two underlying pieces in smooth transition from one to the other. We show that any contrast function defined on M × M naturally leads to a Riemannian metric and a pair of torsion-free conjugate connections (so-called “statistical structure”) that are objects defined for sections of TM. This is achieved through smooth “extension” of the contrast function and its anti-symmetrized version on M × M to, respectively, a quadratic and a cubic function on TM. We recovered the standard formulae [1, 4–6] linking contrast functions to statistical structure through differentiation of the former by two and three vector fields to obtain the metric and the connections, respectively.

The λ-exponential family generalizes the standard exponential family via a generalized convex duality motivated by optimal transport. It is the constant-curvature analogue of the exponential family from the information-geometric point of view, but the development of computational methodologies is still in an early stage. In this paper, we propose a fixed point iteration for maximum likelihood estimation under i.i.d. sampling, and prove using the duality that the likelihood is monotone along the iterations. We illustrate the algorithm with the q-Gaussian distribution and the Dirichlet perturbation.

An affine connection is said to be flat if its curvature tensor vanishes identically. Koszul-Vinberg (KV for abbreviation) cohomology has been invoked to study the deformation theory of flat and torsion-free affine connections on tangent bundle. In this Note, we compute explicitly the differentials of various specific KV cochains, and study their relation to classical objects in information geometry, including deformations associated with projective and dual-projective transformations of a flat and torsion-free affine connection. As an application, we also give a simple yet non-trivial example of a KV algebra of which second cohomology group does not vanish.

We investigate evidence lower bound (ELBO) with generalized/deformed entropy and generalized/deformed divergence, in place of Shannon entropy and KL divergence in the standard framework. Two equivalent forms of deformed ELBO have been proposed, suitable for either Tsallis or Rényi deformation that have been unified in the recent framework of λ -deformation (Wong and Zhang, 2022, IEEE Trans Inform Theory). The decomposition formulae are developed for λ -deformed ELBO, or λ -ELBO in short, now for real-valued λ (with λ = 0 reducing to the standard case). The meaning of the deformation factor λ in the λ -deformed ELBO and its performance for variational autoencoder (VAE) are investigated. Naturally emerging from our formulation is a deformation homotopy probability distribution function that extrapolates encoder distribution and the latent prior. Results show that λ values around 0.5 generally achieve better performance in image reconstruction for generative models.

This paper reviews the role of convex duality in Information Geometry. It clarifies the notion of bi-orthogonal coordinates associated with Legendre duality by treating its two underlying aspects separately: as a dual coordinate system and as a bi-orthogonal frame. It addresses the deformation of exponential families in a way that stills preserves the dually-flat geometry of 1- and (-1)-connections. The deformation involves a metric which generalizes the Fisher–Rao metric controlled by one degree of freedom and a pair of connections controlled by an additional degree of freedom.

This paper systematically presents the λ-deformation as the canonical framework of deformation to the dually flat (Hessian) geometry, which has been well established in information geometry. We show that, based on deforming the Legendre duality, all objects in the Hessian case have their correspondence in the λ-deformed case: λ-convexity, λ-conjugation, λ-biorthogonality, λ-logarithmic divergence, λ-exponential and λ-mixture families, etc. In particular, λ-deformation unifies Tsallis and Rényi deformations by relating them to two manifestations of an identical λ-exponential family, under subtractive or divisive probability normalization, respectively. Unlike the different Hessian geometries of the exponential and mixture families, the λ-exponential family, in turn, coincides with the λ-mixture family after a change of random variables. The resulting statistical manifolds, while still carrying a dualistic structure, replace the Hessian metric and a pair of dually flat conjugate affine connections with a conformal Hessian metric and a pair of projectively flat connections carrying constant (nonzero) curvature. Thus, λ-deformation is a canonical framework in generalizing the well-known dually flat Hessian structure of information geometry.

The primary objects of study in information geometry are statistical manifolds, which are parametrized families of probability measures, induced with the Fisher-Rao metric and a pair of torsion-free conjugate connections. In recent work, the authors considered parametrized probability distributions as partially-flat statistical manifolds admitting torsion and showed that there is a complex to symplectic duality on the tangent bundles of such manifolds, based on the dualistic geometry of the underlying manifold. In this paper, we explore this correspondence further in the context of Hessian manifolds, in which case the conjugate connections are both curvature- and torsion-free, and the associated dual pair of spaces are K\"ahler manifolds. We focus on several key examples and their geometric features. In particular, we show that the moduli space of univariate normal distributions gives rise to a correspondence between the Siegel half-space and the Siegel-Jacobi space, which are spaces that appear in the context of automorphic forms.

Information geometry and optimal transport are two distinct geometric frameworks for modeling families of probability measures. During the recent years, there has been a surge of research endeavors that cut across these two areas and explore their links and interactions. This paper is intended to provide an (incomplete) survey of these works, including entropy-regularized transport, divergence functions arising from c-duality, density manifolds and transport information geometry, the para-Kähler and Kähler geometries underlying optimal transport and the regularity theory for its solutions. Some outstanding questions that would be of interest to audience of both these two disciplines are posed. Our piece also serves as an introduction to the Special Issue on Optimal Transport of the journal Information Geometry.

Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation and functional analysis. Existing constructions include the reflexive RKBS via a bilinear form, the semi-inner-product RKBS, the RKBS with ℓ1 norm, the p-norm RKBS via generalized Mercer kernels, etc. The definitions of RKBS and the associated reproducing kernel in those references are dependent on the construction. Moreover, relations among those constructions are unclear. We explore a generic definition of RKBS and the reproducing kernel for RKBS that is independent of construction. Furthermore, we propose a framework of constructing RKBSs that leads to new RKBSs based on Orlicz spaces and unifies existing constructions mentioned above via a continuous bilinear form and a pair of feature maps. Finally, we develop representer theorems for machine learning in RKBSs constructed in our framework, which also unifies representer theorems in existing RKBSs.

Bayesian inference has been used in the past to model visual perception (Kersten, Mamassian, Yuille, 2004), accounting for the Helmholtz principle of perception as "unconscious inference" that is constrained by bottom-up sensory evidence (likelihood) while subject to top-down expectation, priming, or other contextual influences (prior bias); here "unconsciousness" merely relates to the "directness" of perception in the sense of Gibson. Here, we adopt the same Bayesian framework to model emotion process in accordance with Schachter-Singer's Two-Factor theory, which argues that emotion is the outcome of cognitive labeling or attribution of a diffuse pattern of autonomic arousal (Schachter Singer, 1962). In analogous to visual perception, we conceptualize the emotion process as an instance of Bayesian inference, combining the contextual information with a person's physiological arousal patterns. Drift-diffusion models were constructed to simulate emotional processes, where the decision boundaries correspond to the emotional state experienced by the participants, and boundary-crossing constitutes "labeling" in Schachter-Singer's sense. Our model is tested against experimental data from the Schachter Singer's study (1962) and the Ross et al. study (1969). Two model scenarios are investigated, in which arousal pattern as one factor is pitted against contextual interaction with an confederate (in Schachter-Singer case) or explicitly instructed mis-attribution (in Ross et al. case) as another factor, mapping onto the Bayesian prior (initial position of the drift) and the likelihood function (evidence accumulation or drift rate). We find that the first scenario (arousal as the prior and context as the likelihood) has a better fit with Schachter Singer (1962) whereas the second scenario (context as the prior and arousal as the likelihood) has a better fit with Ross et al. (1969).

Tsallis and Rényi entropies, which are monotone transformations of each other, are deformations of the celebrated Shannon entropy. Maximization of these deformed entropies, under suitable constraints, leads to the $q$ -exponential family which has applications in non-extensive statistical physics, information theory and statistics. In previous information-geometric studies, the $q$ -exponential family was analyzed using classical convex duality and Bregman divergence. In this paper, we show that a generalized $\lambda $ -duality, where $\lambda = 1 - q$ is to be interpreted as the constant information-geometric curvature, leads to a generalized exponential family which is essentially equivalent to the $q$ -exponential family and has deep connections with Rényi entropy and optimal transport. Using this generalized convex duality and its associated logarithmic divergence, we show that our $\lambda $ -exponential family satisfies properties that parallel and generalize those of the exponential family. Under our framework, the Rényi entropy and divergence arise naturally, and we give a new proof of the Tsallis/Rényi entropy maximizing property of the $q$ -exponential family. We also introduce a $\lambda $ -mixture family which may be regarded as the dual of the $\lambda $ -exponential family, and connect it with other mixture-type families. Finally, we discuss a duality between the $\lambda $ -exponential family and the $\lambda $ -logarithmic divergence, and study its statistical consequences.

To investigate mechanisms of rational representation, I consider (1) construction of an ordered continuum of psychophysical scale of magnitude of sensation; (2) counting mechanism leading to an approximate numerosity scale for integers; and (3) conjoint measurement structure pitting the denominator against the numerator in tradeoff positions. Number sense of resulting rationals is neither intuitive nor expedient in their manipulation.

We investigate integrability conditions for an almost (para-)complex structure L, $$L^2 = \pm \text{ id }$$ , on manifolds that admit affine connections carrying torsion in general. The affine connections $$\nabla $$ under consideration are not assumed to be (para-)complex $$\nabla L \ne 0$$ . Two kinds of torsion matching conditions leading to integrability of L are analyzed, which are generalizations of the first and second canonical connections along the Gauduchon line. We also discuss (para-)holomorphicity of $$\nabla $$ encoded by the second condition.

This chapter investigates deformations to the exponential family and the mixture family of probability density functions, under both subtractive and divisive normalizations. We study the two normalizations under the so-called "lambda-deformation" which captures the Tsallis deformation and Renyi deformation as two sides of the same coin; the resulting deformed exponential families only differ by a reparameterization. Our lambda-deformation is essentially a parametric deformation of the exponential function by exp(lambda)(t) = (1+lambda t)(1/lambda) coupled with a lambda-deformed Legendre duality characterized by kappa(lambda)(t) = 1/lambda log (1+lambda t) = log (exp(lambda)(t)). This lambda-deformed Legendre duality and the regular Legendre duality induce the same conjugate variable once the notion of lambda-gradient is introduced. The deformed lambda-exponential and lambda-mixture families turn out to be nearly identical apart from a transformation of the underlying random variables. We remark that the induced information geometry has a pair of constant-curvature dual connections and a conformal Hessian metric, where the lambda parameter encodes the curvature value.

It is well known that topological spaces are axiomatically characterized by the topological closure operator satisfying the Kuratowski Closure Axioms. Equivalently, they can be axiomatized by other set operators encoding primitive semantics of topology, such as interior operator, exterior operator, boundary operator, or derived-set operator (or dually, co-derived-set operator). It is also known that a topological closure operator (and dually, a topological interior operator) can be weakened into generalized closure (interior) systems. What about boundary operator, exterior operator, and derived-set (and co-derived-set) operator in the weakened systems? Our paper completely answers this question by showing that the above six set operators can all be weakened (from their topological counterparts) in an appropriate way such that their inter-relationships remain essentially the same as in topological systems. Moreover, we show that the semantics of an interior point, an exterior point, a boundary point, an accumulation point, a co-accumulation point, an isolated point, a repelling point, etc. with respect to a given set, can be extended to an arbitrary subset system simply by treating the subset system as a base of a generalized interior system (and hence its dual, a generalized closure system). This allows us to extend topological semantics, namely the characterization of points with respect to an arbitrary set, in terms of both its spatial relations (interior, exterior, or boundary) and its dynamic convergence of any sequence (accumulation, co-accumulation, and isolation), to much weakened systems and hence with wider applicability. Examples from the theory of matroid and of Knowledge/Learning Spaces are used as an illustration.

We survey some recent research related to the regularity theory of optimal transport and its associated geometry. We discuss recent progress and pose some open questions and a conjecture related to the MTW tensor, which provides a local obstruction to the smoothness of the Monge transport maps. In this paper we survey some recent progress on the Monge problem of optimal transport and its associated geometry. The main goal is to discuss some connections between the MTW tensor and the curvature of pseudo-Riemannian and complex manifolds .

Attempts at generalizing Shannon entropy and Kullback-Leibler divergence (relative entropy) led to a plenthora of deformation models in theoretical physics, including q-model, kappa-model, etc. Naudts and Zhang (Inf. Geom., 1 (2018) 79) established that these models can be unified under two notions: deformed phi-exponential family (Naudts, J., J. Inequal. Pure Appl. Math., 5 (2004) 102) and conjugate (rho, tau)-embedding (Zhang J., Neural Comput., 16 (2004) 159) of probability functions. Conjugate (rho, tau)-embedding has a gauge freedom which, upon its fixing, subsumes the U-model of Eguchi (Sugaku Expositions, 19 (2006) 197) proposed in a statistical machine learning context. The generalization by (rho, tau)-entropy, (rho, tau)-cross-entropy, (rho, tau)-divergence, when applied to the f-exponential family, yields either a Hessian structure or a conformal Hessian structure under different gauge selections -this "splitting" is the hallmark when deforming the exponential family with its dually flat (Hessian) geometry. This letter provides a unified information geometric perspective of deformation of the exponential model, with calculations for Tsallis q-model. Copyright (C) 2021 EPLA

In this article, we study a class of Kahler manifolds defined on tube domains in $\mathbb{C}^n$, and in particular those which have $O(n) \times \mathbb{R}^n$ symmetry. For these, we prove a uniqueness result showing that any such manifold which is complete and has bisectional curvature ($n \geq 3$) or bisectional curvature ($n \geq 2$) is biholomorphically isometric to $\mathbb{C}^n$. We also consider another curvature tensor called the orthogonal curvature and find necessary and sufficient conditions for a complete $O(n)$-symmetric tube domain to have non-negative anti-bisectional curvature. We provide several examples of complete metrics which satisfy this condition. These examples are also of interest to optimal transport, as they can be used to generate new examples of cost functions which only depend on the Euclidean distance between points and satisfy the weak MTW condition. Finally, we discuss how the interplay between optimal transport and complex geometry can be used to define a synthetic version of curvature bounds for Kahler manifolds whose associated potential is merely $C^3$.