Beyond scalar quasi-arithmetic means: Quasi-arithmetic averages and quasi-arithmetic mixtures in information geometry

We generalize quasi-arithmetic means beyond scalars by considering the gradient map of a Legendre type real-valued function. The gradient map of a Legendre type function is proven strictly comonotone with a global inverse. It thus yields a generalization of strictly mononotone and differentiable functions generating scalar quasi-arithmetic means. Furthermore, the Legendre transformation gives rise to pairs of dual quasi-arithmetic averages via the convex duality. We study the invariance and equivariance properties under affine transformations of quasi-arithmetic averages via the lens of dually flat spaces of information geometry. We show how these quasi-arithmetic averages are used to express points on dual geodesics and sided barycenters in the dual affine coordinate systems. We then consider quasi-arithmetic mixtures and describe several parametric and non-parametric statistical models which are closed under the quasi-arithmetic mixture operation.Comment: 20 page

Geographic Information Science (GIScience) and Geospatial Approaches for the Analysis of Historical Visual Sources and Cartographic Material

This book focuses on the use of GIScience in conjunction with historical visual sources to resolve past scenarios. The themes, knowledge gained and methodologies conducted might be of interest to a variety of scholars from the social science and humanities disciplines

Circular Average Filtering and Circular Linear Interpolation in Complex Color Spaces

In color spaces where the chromatic term is given in polar coordinates, the shortest distance between colors of the same value is circular. By converting such a space into a complex polar form with a real-valued value axis, a color algebra for combining colors is immediately available. In this work, we introduce two complex space operations utilizing this observation: circular average filtering and circular linear interpolation. These operations produce Archimedean Spirals, thus guaranteeing that they operate along the shortest paths. We demonstrate that these operations provide an intuitive way to work in certain color spaces and that they are particularly useful for obtaining better filtering and interpolation results. We present a set of examples based on the perceptually uniform color space CIELAB or L*a*b* with its polar form CIEHLC. We conclude that representing colors in a complex space with circular operations can provide better visual results by exploitation of the strong algebraic properties of complex space C.Comment: 10 page

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282