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We consider the chance-constrained program (CCP) with random right-hand side under a finite discrete distribution. It is known that the standard mixed integer linear programming (MILP) reformulation of the CCP is generally difficult to…
The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths),…
This work provides data-processing and majorization inequalities for $f$-divergences, and it considers some of their applications to coding problems. This work also provides tight bounds on the R\'{e}nyi entropy of a function of a discrete…
The in-order traversal provides a natural correspondence between binary trees with a decreasing vertex labeling and endofunctions on a finite set. By suitably restricting the vertex labeling we arrive at a class of trees that we call…
We consider multi-class classification where the predictor has a hierarchical structure that allows for a very large number of labels both at train and test time. The predictive power of such models can heavily depend on the structure of…
Experimental design techniques such as active search and Bayesian optimization are widely used in the natural sciences for data collection and discovery. However, existing techniques tend to favor exploitation over exploration of the search…
Multivariate extreme value distributions are a common choice for modelling multivariate extremes. In high dimensions, however, the construction of flexible and parsimonious models is challenging. We propose to combine bivariate max-stable…
Different types of two- and three-dimensional representations of a finite metric space are studied that focus on the accurate representation of the linear order among the distances rather than their actual values. Lower and upper bounds for…
We develop a theoretical framework for the analysis of oblique decision trees, where the splits at each decision node occur at linear combinations of the covariates (as opposed to conventional tree constructions that force axis-aligned…
Search trees are fundamental data structures in computer science. We study functionals on random search trees that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal…
We study maximal clades in random phylogenetic trees with the Yule-Harding model or, equivalently, in binary search trees. We use probabilistic methods to reprove and extend earlier results on moment asymptotics and asymptotic normality. In…
We consider the problem of learning the structure of undirected graphical models with bounded treewidth, within the maximum likelihood framework. This is an NP-hard problem and most approaches consider local search techniques. In this…
Partitioning trees are efficient data structures for $k$-nearest neighbor search. Machine learning libraries commonly use a special type of partitioning trees called $k$d-trees to perform $k$-nn search. Unfortunately, $k$d-trees can be…
We study the average height of random trees generated by leaf-centric binary tree sources as introduced by Zhang, Yang and Kieffer. A leaf-centric binary tree source induces for every $n \geq 2$ a probability distribution on the set of…
Because biological processes can make different loci have different evolutionary histories, species tree estimation requires multiple loci from across the genome. While many processes can result in discord between gene trees and species…
The Binary Search Tree (BST) is average in computer science which supports a compact data structure in memory and oneself even conducts a row of quick algorithms, by which people often apply it in dynamical circumstance. Besides these…
The ratio of two densities provides a direct characterization of their differences. We consider the two-sample comparison problem by estimating this ratio given i.i.d. observations from two distributions. To this end, we propose additive…
We study some essential arithmetic properties of a new tree-based number representation, {\em hereditarily binary numbers}, defined by applying recursively run-length encoding of bijective base-2 digits. Our representation expresses giant…
Many discrete mathematics problems in phylogenetics are defined in terms of the relative labeling of pairs of leaf-labeled trees. These relative labelings are naturally formalized as tanglegrams, which have previously been an object of…
Tackling simulation optimization problems with non-convex objective functions remains a fundamental challenge in operations research. In this paper, we propose a class of random search algorithms, called Regular Tree Search, which…