相关论文: Random trees between two walls: Exact partition fu…

Consider a density $f$ on $[0,1]$ that must be estimated from an i.i.d. sample $X_1,...,X_n$ drawn from $f$. In this note, we study binary-tree-based histogram estimates that use recursive splitting of intervals. If the decision to split an…

Learning a regression function using censored or interval-valued output data is an important problem in fields such as genomics and medicine. The goal is to learn a real-valued prediction function, and the training output labels indicate an…

This work presents novel algorithms for learning Bayesian network structures with bounded treewidth. Both exact and approximate methods are developed. The exact method combines mixed-integer linear programming formulations for structure…

We shift the perspective on the interval fragmentation problem from division points to division spacings. This leads to a proof that is both simpler and stronger, establishing limiting distributions for partition points and spacings and,…

The phase transition in the number partitioning problem (NPP), i.e., the transition from a region in the space of control parameters in which almost all instances have many solutions to a region in which almost all instances have no…

We develop a finite-sample, design-based theory for random forests in which each tree is a randomized conditional predictor acting on fixed covariates and the forest is their Monte Carlo average. An exact variance identity separates Monte…

We revisit binary decision trees from the perspective of partitions of the data. We introduce the notion of partitioning function, and we relate it to the growth function and to the VC dimension. We consider three types of features:…

Consider the d-dimensional lattice Z^d where each vertex is ``open'' or ``closed'' with probability p or 1-p, respectively. An open vertex v is connected by an edge to the closest open vertex w such that the dth co-ordinates of v and w…

The exactly solvable four-vertex model with the fixed boundary conditions in the presence of inhomogeneous linearly growing external field is considered. The partition function of the model is calculated and represented in the determinantal…

Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfying approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that…

The recursive and hierarchical structure of full rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is…

Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Renyi's parking problem, alternatively called blocking…

We derive the exact generating function for planar maps (genus zero fatgraphs) with vertices of arbitrary even valence and with two marked points at a fixed geodesic distance. This is done in a purely combinatorial way based on a bijection…

Optimal transport (\OT) theory defines a powerful set of tools to compare probability distributions. \OT~suffers however from a few drawbacks, computational and statistical, which have encouraged the proposal of several regularized variants…

We study random unrooted plane trees with $n$ vertices sampled according to the weights corresponding to the vertex-degrees. Our main result shows that if the generating series of the weights has positive radius of convergence, then this…

We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with $N$ nodes. First, we consider complete graphs. Second, we study Erd\H{o}s-R\'{e}nyi (ER) random graphs with edge probability $p=c/N$…

In the planted partition problem, the $n$ vertices of a random graph are partitioned into $k$ "clusters," and edges between vertices in the same cluster and different clusters are included with constant probability $p$ and $q$, respectively…

We study fragmentation of a random recursive tree into a forest by repeated removal of nodes. The initial tree consists of N nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly-selected…

In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erd\H{o}s-R\'enyi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every…

We study the problem of estimating the density $f(\boldsymbol x)$ of a random vector ${\boldsymbol X}$ in $\mathbb R^d$. For a spanning tree $T$ defined on the vertex set $\{1,\dots ,d\}$, the tree density $f_{T}$ is a product of bivariate…