Related papers: Small One-Dimensional Euclidean Preference Profile…
Every graph G can be embedded in a Euclidean space as a two-distance set. The Euclidean representation number of G is the smallest dimension in which G is representable by such an embedding. We consider spherical and J-spherical…
We define the lower and upper mutual dimensions $mdim(x:y)$ and $Mdim(x:y)$ between any two points $x$ and $y$ in Euclidean space. Intuitively these are the lower and upper densities of the algorithmic information shared by $x$ and $y$. We…
Let $B_n$ be the $n$-dimensional unit ball given by the inequality $\|x\|\leq 1$, where $\|x\|$ is the standard Euclid norm in ${\mathbb R}^n$. For an $n$-dimensional nondegenerate simplex $S$, we denote by $E$ the ellipsoid of minimum…
Learning high-quality feature embeddings efficiently and effectively is critical for the performance of web-scale machine learning systems. A typical model ingests hundreds of features with vocabularies on the order of millions to billions…
There are just 10 closed flat 3-manifolds, following [1], we call them platycosms. The aim of this paper is to classify types of n-coverings over amphicosms, i.e. some kinds of platycosms, and enumerate the numbers of them. Key words:…
Beyond accuracy, there are a variety of aspects to the quality of recommender systems, such as diversity, fairness, and robustness. We argue that many of the prevalent problems in recommender systems are partly due to low-dimensionality of…
A graph is called (generically) rigid in R^d if, for any choice of sufficiently generic edge lengths, it can be embedded in R^d in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining…
Inspired by the spread of discontent as in the 2016 presidential election, we consider a voter model in which 0's are ordinary voters and 1's are zealots. Thinking of a social network, but desiring the simplicity of an infinite object that…
Multidimensional unfolding methods are widely used for visualizing item response data. Such methods project respondents and items simultaneously onto a low-dimensional Euclidian space, in which respondents and items are represented by ideal…
Every isometry of a finite dimensional euclidean space is a product of reflections and the minimum length of a reflection factorization defines a metric on its full isometry group. In this article we identify the structure of intervals in…
In this paper we study number fields which are Euclidean with respect to a function different from the absolute value of the norm. We also show that the Euclidean minimum with respect to weighted norms may be irrational and not isolated.
A preference matrix $M$ has an entry for each pair of candidates in an election whose value $p_{ij}$ represents the proportion of voters that prefer candidate $i$ over candidate $j$. The matrix is rationalizable if it is consistent with a…
Given five points in a three-dimensional euclidean space, one can consider five tetrahedra, using those points as vertices. We present a pentagon-like formula containing the product of three volumes of those tetrahedra in its l.h.s. and the…
Quantitative estimates related to the classical Borsuk problem of splitting set in Euclidean space into subsets of smaller diameter are considered. For a given $k$ there is a minimal diameter of subsets at which there exists a covering with…
We study the problem of representing all distances between $n$ points in $\mathbb R^d$, with arbitrarily small distortion, using as few bits as possible. We give asymptotically tight bounds for this problem, for Euclidean metrics, for…
The winner determination problems of many attractive multi-winner voting rules are NP-complete. However, they often admit polynomial-time algorithms when restricting inputs to be single-peaked. Commonly, such algorithms employ dynamic…
We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the Euclidean plane is either very sparse or has all but an…
In this work we study the coupled dynamics of social balance and opinion formation. We propose a model where agents form opinions under bounded confidence, but only considering the opinions of their friends. The signs of social ties…
Level-1 Consensus is a property of a preference-profile. Intuitively, it means that there exists a preference relation which induces an ordering of all other preferences such that frequent preferences are those that are more similar to it.…
Our main result states that whenever we have a non-Euclidean norm $\|\cdot\|$ on a two-dimensional vector space $X$, there exists some $x\neq 0$ such that for every $\lambda\neq 1, \lambda>0$, there exist $y, z\in X$ verifying that…