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We advance a probabilistic approach to the Hadwiger-Nelson problem initially developed by the Polymath16 project, in particular relating the approach to finite unit-distance graphs. We define the numerical \textit{badness} of a given…

We consider the coloring of certain distance graphs on the Euclidean plane. Namely, we ask for the minimal number of colors needed to color all points of the plane in such a way that pairs of points at distance in the interval $[1,b]$ get…

We consider the Hadwiger-Nelson problem on the chromatic number of the plane under conditions of coloring a map containing a finite number of vertices in any bounded region. Woodall (1973) and Townsend (1981) showed that at least 6 colors…

We study colorings of the hyperbolic plane, analogously to the Hadwiger-Nelson problem for the Euclidean plane. The idea is to color points using the minimum number of colors such that no two points at distance exactly $d$ are of the same…

We explore unique considerations involved in fitting ML models to data with very high precision, as is often required for science applications. We empirically compare various function approximation methods and study how they scale with…

The famous Hadwiger-Nelson problem asks for the minimum number of colors needed to color the points of the Euclidean plane so that no two points unit distance apart are assigned the same color. In this note we consider a variant of the…

The Hadwiger--Nelson problem is about determining the chromatic number of the plane (CNP), defined as the minimum number of colours needed to colour the plane so that no two points of distance 1 have the same colour. In this paper we…

One of the most important combinatorial optimization problems is graph coloring. There are several variations of this problem involving additional constraints either on vertices or edges. They constitute models for real applications, such…

Deep Neural Networks and Reinforcement Learning methods have empirically shown great promise in tackling challenging combinatorial problems. In those methods a deep neural network is used as a solution generator which is then trained by…

Deep learning has received considerable empirical successes in recent years. However, while many ad hoc tricks have been discovered by practitioners, until recently, there has been a lack of theoretical understanding for tricks invented in…

Combinatorial optimization problems near algorithmic phase transitions represent a fundamental challenge for both classical algorithms and machine learning approaches. Among them, graph coloring stands as a prototypical constraint…

Deep learning has consistently defied state-of-the-art techniques in many fields over the last decade. However, we are just beginning to understand the capabilities of neural learning in symbolic domains. Deep learning architectures that…

There is a famous problem in geometric graph theory to find the chromatic number of the unit distance graph on Euclidean space; it remains unsolved. A theorem of Erdos and De-Bruijn simplifies this problem to finding the maximum chromatic…

Divide and conquer is an established algorithm design paradigm that has proven itself to solve a variety of problems efficiently. However, it is yet to be fully explored in solving problems with a neural network, particularly the problem of…

Recent success in training deep neural networks have prompted active investigation into the features learned on their intermediate layers. Such research is difficult because it requires making sense of non-linear computations performed by…

Deep neural networks (DNNs) are increasingly proposed as models of human vision, bolstered by their impressive performance on image classification and object recognition tasks. Yet, the extent to which DNNs capture fundamental aspects of…

In this work, we explore the ability of NN (Neural Networks) to serve as a tool for finding eigen-pairs of ordinary differential equations. The question we aime to address is whether, given a self-adjoint operator, we can learn what are the…

Deep networks have shown impressive performance in the image restoration tasks, such as image colorization. However, we find that previous approaches rely on the digital representation from single color model with a specific mapping…

Deep neural networks have achieved impressive supervised classification performance in many tasks including image recognition, speech recognition, and sequence to sequence learning. However, this success has not been translated to…

Drawings of non-planar graphs always result in edge crossings. When there are many edges crossing at small angles, it is often difficult to follow these edges, because of the multiple visual paths resulted from the crossings that slow down…