5 research outputs found
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
The Information Geometry of Sparse Goodness-of-Fit Testing
This paper takes an information-geometric approach to the challenging issue of goodness-of-fit testing in the high dimensional, low sample size context where—potentially—boundary effects dominate. The main contributions of this paper are threefold: first, we present and prove two new theorems on the behaviour of commonly used test statistics in this context; second, we investigate—in the novel environment of the extended multinomial model—the links between information geometry-based divergences and standard goodness-of-fit statistics, allowing us to formalise relationships which have been missing in the literature; finally, we use simulation studies to validate and illustrate our theoretical results and to explore currently open research questions about the way that discretisation effects can dominate sampling distributions near the boundary. Novelly accommodating these discretisation effects contrasts sharply with the essentially continuous approach of skewness and other corrections flowing from standard higher-order asymptotic analysis
A numerical approximation method for the Fisher-Rao distance between multivariate normal distributions
We present a simple method to approximate Rao's distance between multivariate
normal distributions based on discretizing curves joining normal distributions
and approximating Rao's distances between successive nearby normal
distributions on the curves by the square root of Jeffreys divergence, the
symmetrized Kullback-Leibler divergence. We consider experimentally the linear
interpolation curves in the ordinary, natural and expectation parameterizations
of the normal distributions, and compare these curves with a curve derived from
the Calvo and Oller's isometric embedding of the Fisher-Rao -variate normal
manifold into the cone of symmetric positive-definite
matrices [Journal of multivariate analysis 35.2 (1990): 223-242]. We report on
our experiments and assess the quality of our approximation technique by
comparing the numerical approximations with both lower and upper bounds.
Finally, we present several information-geometric properties of the Calvo and
Oller's isometric embedding.Comment: 46 pages, 19 figures, 3 table
The Hidden Geometry of Particle Collisions
We establish that many fundamental concepts and techniques in quantum field
theory and collider physics can be naturally understood and unified through a
simple new geometric language. The idea is to equip the space of collider
events with a metric, from which other geometric objects can be rigorously
defined. Our analysis is based on the energy mover's distance, which quantifies
the "work" required to rearrange one event into another. This metric, which
operates purely at the level of observable energy flow information, allows for
a clarified definition of infrared and collinear safety and related concepts. A
number of well-known collider observables can be exactly cast as the minimum
distance between an event and various manifolds in this space. Jet definitions,
such as exclusive cone and sequential recombination algorithms, can be directly
derived by finding the closest few-particle approximation to the event. Several
area- and constituent-based pileup mitigation strategies are naturally
expressed in this formalism as well. Finally, we lift our reasoning to develop
a precise distance between theories, which are treated as collections of events
weighted by cross sections. In all of these various cases, a better
understanding of existing methods in our geometric language suggests
interesting new ideas and generalizations.Comment: 56 pages, 11 figures, 5 tables; v2: minor changes and updated
references; v3: updated to match JHEP versio