Related papers: Modules in Robinson Spaces
A Robinson space is a dissimilarity space $(X,d)$ on $n$ points for which there exists a compatible order, {\it i.e.} a total order $<$ on $X$ such that $x<y<z$ implies that $d(x,y)\le d(x,z)$ and $d(y,z)\leq d(x,z)$. Recognizing if a…
Robinson spaces are structures equipped with a total order that encodes comparative dissimilarity relationships. We study the problem of representing Robinson dissimilarity spaces into low-dimensional metric spaces. These representations…
In this paper, we present a factor 16 approximation algorithm for the following NP-hard distance fitting problem: given a finite set X and a distance d on X, find a Robinsonian distance dR on X minimizing the l\infty-error ||d - dR||\infty…
In this work, we extend the concept of Robinson spaces to asymmetric dissimilarities, enhancing their applicability in representing and analyzing complex data. Within this generalized framework, we introduce two different problems that…
A Robinson similarity matrix is a symmetric matrix where the entry values on all rows and columns increase toward the diagonal. Decompose the Robinson matrix into the sum of k {0, 1}-matrices, then these k {0, 1}-matrices are the adjacency…
This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We…
$\newcommand{\floor}[1]{\left\lfloor {#1} \right\rfloor} \renewcommand{\Re}{\mathbb{R}}$ Tverberg's theorem states that a set of $n$ points in $\Re^d$ can be partitioned into $\floor{n/(d+1)}$ sets with a common intersection. A point in…
We study the space of all tilings which can be obtained using the Robinson tiles (this is a two-dimensional subshift of finite type). We prove that it has a unique minimal subshift, and describe it by means of a substitution. This…
A \textit{distinguishing partition} of a group $X$ with automorphism group ${aut}(X)$ is a partition of $X$ that is fixed by no nontrivial element of ${aut}(X)$. In the event that $X$ is a complete multipartite graph with its automorphism…
Partitionings (or segmentations) divide a given domain into disjoint connected regions whose union forms again the entire domain. Multi-dimensional partitionings occur, for example, when analyzing parameter spaces of simulation models,…
A metric space $\mathrm{M}=(M,\de)$ is {\em indivisible} if for every colouring $\chi: M\to 2$ there exists $i\in 2$ and a copy $\mathrm{N}=(N, \de)$ of $\mathrm{M}$ in $\mathrm{M}$ so that $\chi(x)=i$ for all $x\in N$. The metric space…
An 'isomorphism' between the 'moduli space' of star products on $\R^2$ and the 'moduli space' of all formal Poisson structures on $\R^2$ is established.
We give a polynomial time algorithm to decode multivariate polynomial codes of degree $d$ up to half their minimum distance, when the evaluation points are an arbitrary product set $S^m$, for every $d < |S|$. Previously known algorithms can…
A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a…
A $k$-regular partition into distinct parts is a partition into distinct parts with no part divisible by $k$. In this paper, we provide a general method to establish the unimodality of $k$-regular partition into distinct parts where the…
To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson…
One fruitful formulation of Deep Networks (DNs) enabling their theoretical study and providing practical guidelines to practitioners relies on Piecewise Affine Splines. In that realm, a DN's input-mapping is expressed as per-region affine…
A Lorentzian manifold is defined here as a smooth pseudo-Riemannian manifold with a metric tensor of signature ((2n +1, 1)). A Robinson manifold is a Lorentzian manifold (M) of dimension (\geqslant 4) with a subbundle (N) of the…
A square symmetric matrix is a Robinson similarity matrix if entries in its rows and columns are non-decreasing when moving towards the diagonal. A Robinson similarity matrix can be viewed as the affinity matrix between objects arranged in…
Although many machine learning algorithms involve learning subspaces with particular characteristics, optimizing a parameter matrix that is constrained to represent a subspace can be challenging. One solution is to use Riemannian…