Related papers: Epsilon-approximations and epsilon-nets
Consider a point set D with a measure function w : D -> R. Let A be the set of subsets of D induced by containment in a shape from some geometric family (e.g. axis-aligned rectangles, half planes, balls, k-oriented polygons). We say a range…
Random networks are increasingly used to analyse complex transportation networks, such as airline routes, roads and rail networks. So far, this research has been focused on describing the properties of the networks with the help of random…
We use mathematical methods from the theory of tailored random graphs to study systematically the effects of sampling on topological features of large biological signalling networks. Our aim in doing so is to increase our quantitative…
Bipartite networks manifest as a stream of edges that represent transactions, e.g., purchases by retail customers. Many machine learning applications employ neighborhood-based measures to characterize the similarity among the nodes, such as…
Computing the exact likelihood of data in large Bayesian networks consisting of thousands of vertices is often a difficult task. When these models contain many deterministic conditional probability tables and when the observed values are…
This is basically a review of the field of Quasi-Monte Carlo intended for computational physicists and other potential users of quasi-random numbers. As such, much of the material is not new, but is presented here in a style hopefully more…
Random graphs are more and more used for modeling real world networks such as evolutionary networks of proteins. For this purpose we look at two different models and analyze how properties like connectedness and degree distributions are…
Distance queries are a basic tool in data analysis. They are used for detection and localization of change for the purpose of anomaly detection, monitoring, or planning. Distance queries are particularly useful when data sets such as…
One of the core applications of machine learning to knowledge discovery consists on building a function (a hypothesis) from a given amount of data (for instance a decision tree or a neural network) such that we can use it afterwards to…
In this work we give precise asymptotic expressions on the probability of the existence of fixed-size components at the threshold of connectivity for random geometric graphs.
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
The Krakow-Orsay collaboration has applied methods borrowed from equilibrium statistical mechanics and analytic combinatorics to study the geometry of random networks. Results contained in a series of recent publications and concerning…
We present improved upper bounds for the size of relative (p,Epsilon)-approximation for range spaces with the following property: For any (finite) range space projected onto (that is, restricted to) a ground set of size n and for any…
Specify a randomized algorithm that, given a very large graph or network, extracts a random subgraph. What can we learn about the input graph from a single subsample? We derive laws of large numbers for the sampler output, by relating…
We present effective numerical algorithms for locally recovering unknown governing differential equations from measurement data. We employ a set of standard basis functions, e.g., polynomials, to approximate the governing equation with high…
In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs,…
The amount of large-scale real data around us increase in size very quickly and so does the necessity to reduce its size by obtaining a representative sample. Such sample allows us to use a great variety of analytical methods, whose direct…
The random networks enriched with additional structures as metric and group-symmetry in background metric space are investigated. The important quantities like he clustering coefficient as well as the mean degree of separation in such…
This article presents a general approximation-theoretic framework to analyze measure transport algorithms for probabilistic modeling. A primary motivating application for such algorithms is sampling -- a central task in statistical…
When planning motions in a configuration space that has underlying symmetries (e.g. when manipulating one or multiple symmetric objects), the ideal planning algorithm should take advantage of those symmetries to produce shorter…