Related papers: On Mergable Coresets for Polytope Distance
A coreset is a small set that can approximately preserve the structure of the original input data set. Therefore we can run our algorithm on a coreset so as to reduce the total computational complexity. Conventional coreset techniques…
Polygonal finite elements generally do not pass the patch test as a result of quadrature error in the evaluation of weak form integrals. In this work, we examine the consequences of lack of polynomial consistency and show that it can lead…
We prove that a well-distributed subset of R^2 can have a separated distance set only if the distance is induced by a polygon.
Let $\mathcal{P}$ be an $\mathcal{H}$-polytope in $\mathbb{R}^d$ with vertex set $V$. The vertex centroid is defined as the average of the vertices in $V$. We prove that computing the vertex centroid of an $\mathcal{H}$-polytope is #P-hard.…
Motivated by practical generalizations of the classic $k$-median and $k$-means objectives, such as clustering with size constraints, fair clustering, and Wasserstein barycenter, we introduce a meta-theorem for designing coresets for…
Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer…
Several central problems in quantum information theory (such as measurement compatibility and quantum steering) can be rephrased as membership in the minimal matrix convex set corresponding to special polytopes (such as the hypercube or its…
In this paper, we study the mesoscopic fluctuations at edges of orthogonal polynomial ensembles with both continuous and discrete measures. Our main result is a Central limit Theorem (CLT) for linear statistics at mesoscopic scales. We show…
Basis set incompleteness error and finite size error can manifest concurrently in systems for which the two effects are phenomenologically well-separated in length scale. When this is true, we need not necessarily remove the two sources of…
Is perfect error correction always worth the trouble? A framework is presented for the analysis of error detection and correction in multi-level systems of communication that takes into account degrees of freedom attended and ignored by…
We study the existence and structure of $d$-polytopes for which the number $f_1$ of edges is small compared to the number $f_0$ of vertices. Our results are more elegantly expressed in terms of the excess degree of the polytope, defined as…
Clustering in high dimension spaces is a difficult task; the usual distance metrics may no longer be appropriate under the curse of dimensionality. Indeed, the choice of the metric is crucial, and it is highly dependent on the dataset…
We investigate the following question: how close can two disjoint lattice polytopes contained in a fixed hypercube be? This question stems from various contexts where the minimal distance between such polytopes appears in complexity bounds…
We show an interesting PBD-closure result for the set of lengths of constant-composition codes whose distance and size meet certain conditions. A consequence of this PBD-closure result is that the size of optimal constant-composition codes…
Parameterless stopping criteria for recursive polynomial expansions to construct the density matrix in electronic structure calculations are proposed. Based on convergence order estimation the new stopping criteria automatically and…
We consider block codes whose rate converges to the channel capacity with increasing block length at a certain speed and examine the best possible decay of the probability of error. We prove that a moderate deviation principle holds for all…
Uniform sampling is a highly efficient method for data summarization. However, its effectiveness in producing coresets for clustering problems is not yet well understood, primarily because it generally does not yield a strong coreset, which…
In this note, we consider a complete intersection $X=\{x\in \mathbb{R}^n : f_1(x)= \ldots = f_m(x)=0\}, n>m$ and study its Euclidean distance degree in terms of the mixed volume of the Newton polytopes. We show that if the Newton polytopes…
Motivated by the problem of bounding the number of iterations of the Simplex algorithm we investigate the possible lengths of monotone paths followed by the Simplex method inside the oriented graphs of polyhedra (oriented by the objective…
Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between embedded graphs and present a metric to measure the distance between two geometric…