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Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously.…
We consider the problem of approximating a two-dimensional shape contour (or curve segment) using discrete assembly systems, which allow to build geometric structures based on limited sets of node and edge types subject to edge length and…
We study a class of geometric covering and packing problems for bounded regions on the plane. We are given a set of axis-parallel line segments that induces a planar subdivision with a set of bounded (rectilinear) faces. We are interested…
Distance metric learning is a branch of machine learning that aims to learn distances from the data, which enhances the performance of similarity-based algorithms. This tutorial provides a theoretical background and foundations on this…
We present and study approximate notions of dimensional and margin complexity, which correspond to the minimal dimension or norm of an embedding required to approximate, rather then exactly represent, a given hypothesis class. We show that…
We consider the number of configurations of a surface in two dimensions that has a prescribed length and encloses a prescribed perimeter with respect to a baseline. An approximate analytical treatment in a semi--continuum compares…
The generation of curves and surfaces from given data is a well-known problem in Computer-Aided Design that can be approached using subdivision schemes. They are powerful tools that allow obtaining new data from the initial one by means of…
In this work, we revisit the problem of uniformity testing of discrete probability distributions. A fundamental problem in distribution testing, testing uniformity over a known domain has been addressed over a significant line of works, and…
With the recent advances in complex networks theory, graph-based techniques for image segmentation has attracted great attention recently. In order to segment the image into meaningful connected components, this paper proposes an image…
We study the problem of estimating the Earth Mover's Distance (EMD) between probability distributions when given access only to samples. We give closeness testers and additive-error estimators over domains in $[0, \Delta]^d$, with sample…
Dimension reduction algorithms are a crucial part of many data science pipelines, including data exploration, feature creation and selection, and denoising. Despite their wide utilization, many non-linear dimension reduction algorithms are…
Automating algorithm configuration is growing increasingly necessary as algorithms come with more and more tunable parameters. It is common to tune parameters using machine learning, optimizing performance metrics such as runtime and…
Finding the k-medianin a network involves identifying a subset of k vertices that minimize the total distance to all other vertices in a graph. This problem has been extensively studied in computer science, graph theory, operations…
Gradient-based dimension reduction decreases the cost of Bayesian inference and probabilistic modeling by identifying maximally informative (and informed) low-dimensional projections of the data and parameters, allowing high-dimensional…
Starting with a similarity function between objects, it is possible to define a distance metric on pairs of objects, and more generally on probability distributions over them. These distance metrics have a deep basis in functional analysis,…
Error bounds and complexity bounds in numerical analysis and information-based complexity are often proved for functions that are defined on very simple domains, such as a cube, a torus, or a sphere. We study optimal error bounds for the…
We show how complexity theory can be introduced in machine learning to help bring together apparently disparate areas of current research. We show that this new approach requires less training data and is more generalizable as it shows…
We present a fast algorithm for global rigid symmetry detection with approximation guarantees. The algorithm is guaranteed to find the best approximate symmetry of a given shape, to within a user-specified threshold, with very high…
Fully supervised deep neural networks for segmentation usually require a massive amount of pixel-level labels which are manually expensive to create. In this work, we develop a multi-task learning method to relax this constraint. We regard…
The subdiffusion model that involves a Caputo fractional derivative in time is widely used to describe anomalously slow diffusion processes. In this work we aim at recovering the locations of small conductivity inclusions in the model from…