The simplest nearest-neighbor spin system on regular graphs:Time dynamics of the mean coverage function

Laws of large numbers and nearest neighbor distances

We consider the sum of power weighted nearest neighbor distances in a sample of size n from a multivariate density f of possibly unbounded support. We give various criteria guaranteeing that this sum satisfies a law of large numbers for large n, correcting some inaccuracies in the literature on the way. Motivation comes partly from the problem of consistent estimation of certain entropies of f.

Random nearest neighbor graphs: the translation invariant case

If $(ω(e))$ is a family of random variables (weights) assigned to the edges of $\mathbb{Z}^d$, the nearest neighbor graph is the directed graph induced by all edges $\langle x,y \rangle$ such that $ω(\{x,y\})$ is minimal among all neighbors $y$ of $x$. That is, each vertex points to its closest neighbor, if the weights are viewed as edge-lengths. Nanda-Newman introduced nearest neighbor graphs when the weights are i.i.d. and continuously distributed and proved that a.s., all components of the undirected version of the graph are finite. We study the case of translation invariant, distinct weigh…

The limit distribution of the maximum probability nearest neighbor ball

Let $X_1, \ldots, X_n$ be independent random points drawn from an absolutely continuous probability measure with density $f$ in $\mathbb{R}^d$. Under mild conditions on $f$, we derive a Poisson limit theorem for the number of large probability nearest neighbor balls. Denoting by $P_n$ the maximum probability measure of nearest neighbor balls, this limit theorem implies a Gumbel extreme value distribution for $nP_n - \ln n$ as $n \to \infty$. Moreover, we derive a tight upper bound on the upper tail of the distribution of $nP_n - \ln n$, which does not depend on $f$.

Orthogonality and probability: beyond nearest neighbor transitions

In this article, we will explore why Karlin-McGregor method of using orthogonal polynomials in the study of Markov processes was so successful for one dimensional nearest neighbor processes, but failed beyond nearest neighbor transitions. We will proceed by suggesting and testing possible fixtures.

共享最近邻图上的图拉普拉斯算子与具有相同极限的$k$-最近邻图上的图拉普拉斯算子

Graph Laplacians on Shared Nearest Neighbor graphs and graph Laplacians on $k$-Nearest Neighbor graphs having the same limit

针对高维数据中广泛使用但缺乏理论支撑的共享最近邻（SNN）图，本文首次证明其图拉普拉斯算子的大规模渐近性质达到与$k$-NN图相同的连续统极限，并给出了高概率下的线性逐点收敛速率。

Triangles Formed via Poisson Nearest Neighbors

We start with certain joint densities (for sides and for angles) corresponding to pinned Poissonian triangles in the plane, then discuss analogous results for staked and anchored triangles.

Graph-based Nearest Neighbor Search: From Practice to Theory

Graph-based approaches are empirically shown to be very successful for the nearest neighbor search (NNS). However, there has been very little research on their theoretical guarantees. We fill this gap and rigorously analyze the performance of graph-based NNS algorithms, specifically focusing on the low-dimensional (d << \log n) regime. In addition to the basic greedy algorithm on nearest neighbor graphs, we also analyze the most successful heuristics commonly used in practice: speeding up via adding shortcut edges and improving accuracy via maintaining a dynamic list of candidates. We believe …

Large and moderate deviations for record numbers in some non-nearest neighbor random walks

The deviation principles of record numbers in random walk models have not been completely investigated, especially for the non-nearest neighbor cases. In this paper, we derive the asymptotic probabilities of large and moderate deviations for the number of "weak records"(or "ladder points") in two kinds of one-dimensional non-nearest neighbor random walks. The proofs depend only on the direct analysis of random walks. We illustrate that the traditional method of analyzing the local time of Brownian motions, which is often adopted for the simple random walks, would lead to wrong conjectures for …

A Nearest Neighbor Characterization of Lebesgue Points in Metric Measure Spaces

The property of almost every point being a Lebesgue point has proven to be crucial for the consistency of several classification algorithms based on nearest neighbors. We characterize Lebesgue points in terms of a 1-Nearest Neighbor regression algorithm for pointwise estimation, fleshing out the role played by tie-breaking rules in the corresponding convergence problem. We then give an application of our results, proving the convergence of the risk of a large class of 1-Nearest Neighbor classification algorithms in general metric spaces where almost every point is a Lebesgue point.