Intrinsic dimension identification via graph-theoretic methods

Intrinsic dimension identification via graph-theoretic methods

Three graph theoretical statistics are considered for the problem of estimating the intrinsic dimension of a data set. The first is the ‘‘reach’’ statistic, r j,k, proposed in Brito et al. (2002) [4] for the problem of identification of Euclidean dimension. The second,Mn, is the sample average of squared degrees in the minimum spanning tree of the data, while the third statistic, Uk n , is base...

Multiscale Geometric Methods for Estimating Intrinsic Dimension

We present a novel approach for estimating the intrinsic dimension of certain point clouds: we assume that the points are sampled from a manifold M of dimension k, with k << D, and corrupted by D-dimensional noise. When M is linear, one may analyze this situation by PCA: with no noise one would obtain a rank k matrix, and noise may be treated as a perturbation of the covariance matrix. WhenM is...

Estimation of Intrinsic Dimension via Clustering

The problem of estimating the intrinsic dimension of a data set from pairwise distances is a critical issue for a wide range of disciplines, including genomics, finance, and networking. Current estimation techniques are agnostic to the structure of the data, resulting in techniques that may be computationally intractable for large data sets. In this paper, we present a methodology that exploits...

Graph Theoretic Methods in Coding Theory

This paper is a tutorial on the application of graph theoretic techniques in classical coding theory. A fundamental problem in coding theory is to determine the maximum size of a code satisfying a given minimum Hamming distance. This problem is thought to be extremely hard and still not completely solved. In addition to a number of closed form expressions for special cases and some numerical re...

Graph Theoretic Methods in Multiagent Networks