Related papers: Hyperorthogonal well-folded Hilbert curves
Decision trees (DTs) and their random forest (RF) extensions are workhorses of classification and regression in Euclidean spaces. However, algorithms for learning in non-Euclidean spaces are still limited. We extend DT and RF algorithms to…
Scaling regression to large datasets is a common problem in many application areas. We propose a two step approach to scaling regression to large datasets. Using a regression tree (CART) to segment the large dataset constitutes the first…
An ultrametric topology formalizes the notion of hierarchical structure. An ultrametric embedding, referred to here as ultrametricity, is implied by a natural hierarchical embedding. Such hierarchical structure can be global in the data…
Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. They are also objects of interest in pure mathematics, such as algebraic geometry and combinatorics, due to their discrete geometry.…
We extend decision tree and random forest algorithms to product space manifolds: Cartesian products of Euclidean, hyperspherical, and hyperbolic manifolds. Such spaces have extremely expressive geometries capable of representing many…
Tree structured graphical models are powerful at expressing long range or hierarchical dependency among many variables, and have been widely applied in different areas of computer science and statistics. However, existing methods for…
Let R^d -> A be a query problem over R^d for which there exists a data structure S that can compute P(q) in O(log n) time for any query point q in R^d. Let D be a probability measure over R^d representing a distribution of queries. We…
Net-trees are a general purpose data structure for metric data that have been used to solve a wide range of algorithmic problems. We give a simple randomized algorithm to construct net-trees on doubling metrics using $O(n\log n)$ time in…
This paper introduces a novel tree-based model, Learning Hyperplane Tree (LHT), which outperforms state-of-the-art (SOTA) tree models for classification tasks on several public datasets. The structure of LHT is simple and efficient: it…
As a flexible nonparametric learning tool, the random forests algorithm has been widely applied to various real applications with appealing empirical performance, even in the presence of high-dimensional feature space. Unveiling the…
Nonlinear dimension reduction methods provide a low-dimensional representation of high-dimensional data by applying a Nonlinear transformation. However, the complexity of the transformations and data structures can create wildly different…
The metric space of phylogenetic trees defined by Billera, Holmes, and Vogtmann, which we refer to as BHV space, provides a natural geometric setting for describing collections of trees on the same set of taxa. However, it is sometimes…
The segment tree is an extremely versatile data structure. In this paper, a new heap based implementation of segment trees is proposed. In such an implementation of segment tree, the structural information associated with the tree nodes can…
Natural slender structures, such as plant leaves, petals, and tendrils, often exhibit complex three-dimensional (3D) morphologies-including twisting, helical coiling, and saddle-bending-driven by differential growth. The resulting internal…
The recent developments of basis pursuit and compressed sensing seek to extract information from as few samples as possible. In such applications, since the number of samples is restricted, one should deploy the sampling points wisely. We…
Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion…
High-dimensional motion planning problems can often be solved significantly faster by using multilevel abstractions. While there are various ways to formally capture multilevel abstractions, we formulate them in terms of fiber bundles.…
We consider a variant of treewidth that we call clique-partitioned treewidth in which each bag is partitioned into cliques. This is motivated by the recent development of FPT-algorithms based on similar parameters for various problems. With…
Taking projections of high-dimensional data is a common analytical and visualisation technique in statistics for working with high-dimensional problems. Sectioning, or slicing, through high dimensions is less common, but can be useful for…
Tree embedding has been a fundamental method in algorithm design with wide applications. We focus on the efficiency of building tree embedding in various computational settings under high-dimensional Euclidean $\mathbb{R}^d$. We devise a…