Optimization, Computational Geometry, Graphs and High-Dimensional Data

                 Jose Costa
   Center for the Mathematics of Information
       California Institute of Technology

ABSTRACT: Increasingly intricate and rich data sets at the heart of
today's most common applications, from video surveillance to
biomedical information systems, are raising new questions in data
storage, access and especially data exploration. The high-dimensional
nature of such data sets makes them prone to the curse of
dimensionality, with the consequent failure of standard inference
tools. This series of two lectures will address some challenging
problems in analyzing high-dimensional data and information that must
be transmitted, stored and processed accurately using manageable
computational resources. Originally motivated by computational
considerations, we demonstrate how computational efficient and
scalable graph constructions can be used to encode both statistical
and spatial information and address the problems of dimension
reduction and structure discovery in high-dimensional data, with
provable results.

In this first lecture, we will discuss the asymptotic behavior of
graphs such as Minimal Spanning Trees or k-Nearest Neighbor graphs,
and show how this can be used to estimate the intrinsic dimension and
entropy of data sets that span a high-dimensional space but contain
fundamental features concentrated on a low-dimensional manifold.