Word embeddings seek to recover a Euclidean metric space by mapping words into vectors, starting from words cooccurrences in a corpus. Word embeddings may underestimate the similarity between nearby words, and overestimate it between distant words in the Euclidean metric space. In this paper, we re-embed pre-trained word embeddings with a stage of manifold learning which retains dimensionality....

We investigate higher-order Voronoi diagrams in the city metric. This metric is induced by quickest paths in the L1 metric in the presence of an accelerating transportation network of axis-parallel line segments. For the structural complexity of k-order city Voronoi diagrams of n point sites, we show an upper bound of O(k(n − k) + kc) and a lower bound of Ω(n+ kc), where c is the complexity of ...

S.S. Chern raised the problem to find necessary and sufficient conditions for a given Riemannian manifold to be realizable on a minimal submanifold of a Euclidean space. The aim of this paper is to provide new necessary conditions. For minimal submanifolds in a Euclidean space we consider the negative of the Ricci tensor as defining a new metric, which is nothing but the third fundamental form,...

Symmetric positive definite (spd) matrices pervade numerous scientific disciplines, including machine learning and optimization. We consider the key task of measuring distances between two spd matrices; a task that is often nontrivial whenever the distance function must respect the non-Euclidean geometry of spd matrices. Typical non-Euclidean distance measures such as the Riemannian metric δR(X...

The discovery of pseudoparticle solutions to the euclidean SU(2) Yang-Mills theory [1 ] has suggested the possibility that analogous solutions might occur in Einstein's theory of gravitation. The existence of such solutions would have a profound effect on the quantum theory of gravitation [2,3]. Since fire Yang-Mills pseudoparticles possess self-dual field strengths, one likely possibility is t...

Symmetric positive definite (spd) matrices pervade numerous scientific disciplines, including machine learning and optimization. We consider the key task of measuring distances between two spd matrices; a task that is often nontrivial whenever the distance function must respect the non-Euclidean geometry of spd matrices. Typical non-Euclidean distance measures such as the Riemannian metric δR(X...

Compact metric spaces that admit intrinsic isometries to the Euclidean d-space are considered. Roughly, the main result states that the class of such spaces coincides with the class of inverse limits of Euclidean d-polyhedra. §

Given a scattering metric on the Euclidean space. We consider the Schrödinger equation corresponding to the metric, and study the propagation of singularities for the solution in terms of the homogeneous wavefront set. We also prove that the notion of the homogeneous wavefront set is essentially equivalent to that of the quadratic scattering wavefront set introduced by J. Wunsch [21]. One of th...

The discovery of self-dual instanton solutions in Euclidean Yang-Mills theory [I] has recently stimulated a great deal of interest in self-dual solutions to Einstein’s theory of gravitation. One would expect that the relevant instanton-like metrics would be those whose gravitational fields are self-dual, localized in Euclidean spacetime and free of singularities. In fact, solutions have been fo...

Many natural language processing still depend on the Euclidean distance function between the two feature vectors, but it has severe defects as to feature weightings and feature correlations. In this paper we propose an optimal metric distance function that can be used as an alternative to the Euclidean distance, accommodating the two problems at the same time. This metric is optimal in the sens...