Can a machine or algorithm discover or learn the elliptical orbit of Mars from astronomical sightings alone? Johannes Kepler required two paradigm shifts to discover his First Law regarding the elliptical orbit of Mars. Firstly, a shift from the geocentric to the heliocentric frame of reference. Secondly, the reduction of the orbit of Mars from a three- to a two-dimensional space. We extend AI Feynman, a physics-inspired tool for symbolic regression, to discover the heliocentricity and planarity of Mars' orbit and emulate his discovery of Kepler's first law.

Exemplar based accounts are often considered to be in direct opposition to pure linguistic abstraction in explaining language learners' ability to generalize to novel expressions. However, the recent success of neural network language models on linguistically sensitive tasks suggests that perhaps abstractions can arise via the encoding of exemplars. We provide empirical evidence for this claim by adapting an existing experiment that studies how an LM (BERT) generalizes the usage of novel tokens that belong to lexical categories such as Noun/Verb/Adjective/Adverb from exposure to only a single instance of their usage. We analyze the representational behavior of the novel tokens in these experiments, and find that BERT's capacity to generalize to unseen expressions involving the use of these novel tokens constitutes the movement of novel token representations towards regions of known category exemplars in two-dimensional space. Our results suggest that learners' encoding of exemplars can indeed give rise to abstraction like behavior.

The Figure 3 below shows that GAN comprises of two main parts: generator and discriminator. As the name suggests, generator is responsible to generate new (fake) samples while discriminator attempts to distinguish the real and fake ones. The main objective of training a GAN is to make the generator able to generate new samples such that those samples are indistinguishable by the discriminator. Once this happens, it means that our generator is now able to create samples which the quality is already as good as the originals. As I've mentioned earlier, we are going to work on two-dimensional data since it is a lot simpler as compared to the MNIST dataset we saw earlier.

We consider a swarm of mobile robots evolving in a bidimensional Euclidean space. We study a variant of the crash-tolerant gathering problem: if no robot crashes, robots have to meet at the same arbitrary location, not known beforehand, in finite time; if one or several robots crash at the same location, the remaining correct robots gather at the crash location to rescue them. Motivated by impossibility results in the semi-synchronous setting, we present the first solution to the problem for the fully synchronous setting that operates in the vanilla Look-Compute-Move model with no additional hypotheses: robots are oblivious, disoriented, have no multiplicity detection capacity, and may start from arbitrary positions (including those with multiplicity points). We furthermore show that robots gather in a time that is proportional to the initial maximum distance between robots.

K-means clustering is one of the simplest and popular unsupervised machine learning algorithms. Typically, unsupervised algorithms make inferences from datasets using only input vectors without referring to known, or labelled, outcomes. AndreyBu, who has more than 5 years of machine learning experience and currently teaches people his skills, says that "the objective of K-means is simple: group similar data points together and discover underlying patterns. To achieve this objective, K-means looks for a fixed number (k) of clusters in a dataset." A cluster refers to a collection of data points aggregated together because of certain similarities. You'll define a target number k, which refers to the number of centroids you need in the dataset.

A team of mathematicians has finally finished off Keller's conjecture, but not by working it out themselves. Instead, they taught a fleet of computers to do it for them. Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research develop ments and trends in mathe matics and the physical and life sciences. Keller's conjecture, posed 90 years ago by Ott-Heinrich Keller, is a problem about covering spaces with identical tiles. It asserts that if you cover a two-dimensional space with two-dimensional square tiles, at least two of the tiles must share an edge.

Today, I'll try to give some insights about TDA (for Topological Data Analysis), a mathematical field quickly evolving, that will certainly soon be completely integrated into machine-/deep- learning frameworks. Some use-cases will be presented in the wake of this article, in order to illustrate the power of that theory! Topological Data Analysis, also abbreviated TDA, is a recent field that emerged from various works in applied topology and computational geometry. It aims at providing well-founded mathematical, statistical and algorithmic methods to exploit the topological and underlying geometric structures in data. You will generally find it suitable for three-dimensional data, but experience shows that TDA reveals also to be useful in other cases, such as time-series.

K-means clustering is one of the simplest and popular unsupervised machine learning algorithms. Typically, unsupervised algorithms make inferences from datasets using only input vectors without referring to known, or labelled, outcomes. AndreyBu, who has more than 5 years of machine learning experience and currently teaches people his skills, says that "the objective of K-means is simple: group similar data points together and discover underlying patterns. To achieve this objective, K-means looks for a fixed number (k) of clusters in a dataset." A cluster refers to a collection of data points aggregated together because of certain similarities. You'll define a target number k, which refers to the number of centroids you need in the dataset.