The Pseudo-Pascal Triangle of Maximum Deng Entropy

PPascal triangle (known as Yang Hui Triangle in Chinese) is an important model in mathematics while the entropy has been heavily studied in physics or as uncertainty measure in information science. How to construct the the connection between Pascal triangle and uncertainty measure is an interesting topic. One of the most used entropy, Tasllis entropy, has been modelled with Pascal triangle. But the relationship of the other entropy functions with Pascal triangle is still an open issue. Dempster-Shafer evidence theory takes the advantage to deal with uncertainty than probability theory since the probability distribution is generalized as basic probability assignment, which is more efficient to model and handle uncertain information. Given a basic probability assignment, its corresponding uncertainty measure can be determined by Deng entropy, which is the generalization of Shannon entropy. In this paper, a Pseudo-Pascal triangle based the maximum Deng entropy is constructed. Similar to the Pascal triangle modelling of Tasllis entropy, this work provides the a possible way of Deng entropy in physics and information theory

Unimodality polynomials and generalized Pascal triangles

This paper is an extension of Boros and Moll’s result “A criterion for unimodality”, who proved that the polynomial P(x + 1) is unimodal

Pascal pyramid in the space ( H^{2}) xR

In this article we introduce a new type of Pascal pyramids. A regular squared mosaic in the hyperbolic plane yields an ( h^{2}r)-cube mosaic in space ( H^{2}) xR and the definition of the pyramid is based on this regular mosaic. The levels of the pyramid inherit some properties from the Euclidean and hyperbolic Pascal triangles. We give the growing method from level to level and show some illustrating figures

Hilbert metric, beyond convexity

The Hilbert metric on convex subsets of $\mathbb R^n$ has proven a rich notion and has been extensively studied. We propose here a generalization of this metric to subset of complex projective spaces and give examples of applications to diverse fields. Basic examples include the classical Hilbert metric which coincides with the hyperbolic metric on real hyperbolic spaces as well as the complex hyperbolic metric on complex hyperbolic spaces

Computationally Tractable Riemannian Manifolds for Graph Embeddings

Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic geometry. However, going beyond embedding spaces of constant sectional curvature, while potentially more representationally powerful, proves to be challenging as one can easily lose the appeal of computationally tractable tools such as geodesic distances or Riemannian gradients. Here, we explore computationally efficient matrix manifolds, showcasing how to learn and optimize graph embeddings in these Riemannian spaces. Empirically, we demonstrate consistent improvements over Euclidean geometry while often outperforming hyperbolic and elliptical embeddings based on various metrics that capture different graph properties. Our results serve as new evidence for the benefits of non-Euclidean embeddings in machine learning pipelines.Comment: Submitted to the Thirty-fourth Conference on Neural Information Processing System