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Given a simple graph $G$ with $n$ vertices and a natural number $i \leq n$, let $L_G(i)$ be the maximum number of leaves that can be realized by an induced subtree $T$ of $G$ with $i$ vertices. We introduce a problem that we call the…
The pure-exploration problem in stochastic multi-armed bandits aims to find one or more arms with the largest (or near largest) means. Examples include finding an {\epsilon}-good arm, best-arm identification, top-k arm identification, and…
We show that the algorithm to extract diverse M -solutions from a Conditional Random Field (called divMbest [1]) takes exactly the form of a Herding procedure [2], i.e. a deterministic dynamical system that produces a sequence of hypotheses…
We study the problem of cutting a length-$n$ string of positive real numbers into $k$ pieces so that every piece has sum at least $b$. The problem can also be phrased as transforming such a string into a new one by merging adjacent numbers.…
Let $T$ be an $n$-node tree of maximum degree 4, and let $P$ be a set of $n$ points in the plane with no two points on the same horizontal or vertical line. It is an open question whether $T$ always has a planar drawing on $P$ such that…
Due to their long-standing reputation as excellent off-the-shelf predictors, random forests continue remain a go-to model of choice for applied statisticians and data scientists. Despite their widespread use, however, until recently, little…
A graph $\mathcal G = (\mathcal V, \mathcal E)$ is said to satisfy the Neighbour Sum Property if there exists some $f:\mathcal V\to\mathbb R$ such that $f\not\equiv 0$ and it maps every vertex to the sum of the values taken by its…
We consider a pruning of the inhomogeneous continuum random trees, as well as the cut trees that encode the genealogies of the fragmentations that come with the pruning. We propose a new approach to the reconstruction problem, which has…
We propose random hinge forests, a simple, efficient, and novel variant of decision forests. Importantly, random hinge forests can be readily incorporated as a general component within arbitrary computation graphs that are optimized…
In this article, we study a simple random walk on a decorated Galton-Watson tree, obtained from a Galton-Watson tree by replacing each vertex of degree $n$ with an independent copy of a graph $G_n$ and gluing the inserted graphs along the…
We consider the task of learning the maximum-likelihood polytree from data. Our first result is a performance guarantee establishing that the optimal branching (or Chow-Liu tree), which can be computed very easily, constitutes a good…
The Hamming oracle returns the Hamming distance between an unknown binary $n$-vector $x$ and a binary query $n$-vector y. The objective is to determine $x$ uniquely using a sequence of $m$ queries. What are the minimum number of queries…
A bipartite graph $G=(L,R;E)$ with at least one edge is said to be identifiable if for every vertex $v\in L$, the subgraph induced by its non-neighbors has a matching of cardinality $|L|-1$. An $\ell$-subgraph of $G$ is an induced subgraph…
Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of a graph a random number is assigned and a path through the graph is called accessible if all numbers along the path are in…
Hunters and Rabbit game is played on a graph $G$ where the Hunter player shoots at $k$ vertices in every round while the Rabbit player occupies an unknown vertex and, if not shot, must move to a neighbouring vertex after each round. The…
Given a set S of n \geq d points in general position in R^d, a random hyperplane split is obtained by sampling d points uniformly at random without replacement from S and splitting based on their affine hull. A random hyperplane search tree…
Let $\mathcal{A}$ be a set of positive numbers. A graph $G$ is called an $\mathcal{A}$-embeddable graph in $\mathbb{R}^d$ if the vertices of $G$ can be positioned in $\mathbb{R}^d$ so that the distance between endpoints of any edge is an…
We investigate a graph parameter called the total vertex irregularity strength ($tvs(G)$), i.e. the minimal $s$ such that there is a labeling $w: E(G)\cup V(G)\rightarrow \{1,2,..,s\}$ of the edges and vertices of $G$ giving distinct…
Counting objects in images is a pattern recognition problem that focuses on identifying an element to determine its incidence and is approached in the literature as Visual Object Counting (VOC). In this work, we propose a methodology to…
Imagine an invisible rabbit that starts at some unknown integer point $A$ on the number line. At each time step, it hops by a fixed but unknown integer stride $B$. Both $A$ and $B$ are fixed integers, but their values are unknown. Suppose…