Geometry is Everywhere, Part XLVII: Metrics, Nets, Dimensions, Measures

Ken Clarkson

IBM Almaden Research

Geometers are Everywhere

ACM Symp. Advances in GIS	ACM Symp. on Parallelism in Algorithms and Architectures (SPAA)	ACM Symp. on Principles of Distributed Computing (PODC),	ACM Transactions on Algorithms	ALENEX
ALGOSENSORS	Advances in Mathematics	Air Traffic Control Quarterly	Algorithmic Aspects of Large and Complex Networks	Algorithms in Bioinformatics
Annual IEEE Intl. Conf. Very Large Databases	Applied Cryptography and Network Security Conference	BioInformatics	Biochemistry	Bioinformatics
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Computer Graphics Forum	Computer Graphics, and Massive Data Exploration	Computers and Games	Conference on Mobile Data Management	DCOSS
Discrete Applied Mathematics	Discrete and Computational Geometry	Ecology Letters	Eurographics Symposium on Point-Based Graphics	Eurographics/ACM SIGGRAPH symposium
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Geombinatorics	Graphs and Combinatorics	ICCAD	IEEE Conference on Computer Vision and Pattern Recognition	IEEE Conference on Pervasive Computing and Communications Workshops
IEEE PERVASIVE COMPUTING	IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS	IEEE Trans. Computer-Aided Design	IEEE Trans. on Knowledge and Data Engineering	IEEE Transactions on Wireless Communications
IEEE Vehicular Technology Conference	IEEE Visualization	IEEE Wireless Communications and Networking Conferences	IFIP WG Working Conference on Data and Application Secuirity (DBSec)	IJCAC

Geometers are Even More Everywhere

INFOCOM	IPSN	Indian Statistical Institute Platinum Jubilee Commemorative Volume--Architecture and Algorithms	Information Processing in Medical Imaging	International Conference on Computational Science
International Conference on Computer Graphics and Interactive Techniques	International Conference on Sensor Technologies and Applications	International Symposium on Graph Drawing	International Workshop on the Algorithmic Foundations of Robotics	International symposium on Information processing in sensor networks
Intl. Conf. on Database Theory	Intl. Symp. Spatial Data Handling	Israel J. Math	J. Comput. Biology (JCB)	J. Intell. Inf. Syst. (JIIS)
J. London Math. Soc	J. Proteome Res	J. of Graph Algorithms and Applications	Journal of Algorithms	Journal of Combinatorial Theory
Journal of Computational Biology	Journal of Differential Geometry	Journal of Graph Theory	Journal of Intelligent Information Systems	Journal of Symbolic Computation,
Journal of the ACM	Középiskolai Matematikai Lapok	Mobile Networks and Applications	Multimedia Information Retrieval	PROTEINS: Structure, Function, and Bioinformatics
PSB	Pacific Symposium on Biocomputing	Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing	Proc. International Symposium on Spatial Data Handling	Proc. Intl. Conf. Computational Systems Bioinformatic
Proc. Latin American Symposium on Theoretical Informatics (LATIN)	Proc. Symbolic-Numeric Workshop	Proc. of International Workshop on Ubiquitous Access Control	Proc.~Intl. Symp. Symbolic and Algebraic Comp. (ISSAC'06)	Proceedings of Information Processing in Medical Imaging
Proceedings of Robotics: Science and Systems	Revue Roumaine de Mathématiques Pures et Appliquées	SIAM Data Mining	SIAM J. Discrete Math,	Symposium on Geometry Processing
The Computer Journal	The International Journal of Robotics Research	Theory of Computing Systems	Trans. Comput. Biology Bioinform. (TCBB)	Workshop on Internet and Network Economics
Electronic Journal of Combinatorics

Overview

Let `U` be a set of `n` points in the plane in general position
`U` + distance measure `D` == metric space `(U,D)`
"Real" title: A Light Survey of Some Basic Constructions Related To Metric Spaces

Metric transformations
`epsilon`-nets, the greedy algorithm, and applications
A non-greedy algorithm and a few non-`epsilon`-nets
Box dimension and coping with finiteness

Definitions of dimension that make sense for finite sets
Estimation of dimension using finite samples

Neighbors
Interpolation

Point clouds == point sets in high-d Euclidean space, often samples of a manifold

Metric Transformations

Sometimes metrics appear unexpectedly

For integers `a` and `b`, `D(a,b) := |~ log_2 quad 1 + |a-b| ~|` is a metric

That is, the number of bits needed to delta-encode `b`, given that you know `a`
In general `log(1+D(x,y))` is a metric if `D(x,y)` is

If `f` and `g` are functions on a random variable `x`, then `D(f,g) := Prob{f(x) != g(x)}` is a metric

If `x_1, x_2,...` are `n` independent instances of `x`, then

`hat D(f,g) := Prob{exists x_i, f(x_i) != g(x_i)} = 1 - (1 - D(f,g))^n`

is a metric

For any metric `D`, `1 - exp(-nD(x,y)) approx 1 - (1 - D(f,g))^n` is a metric

Metric Transformations: Scaling by Distance

If `A subset U`, then the distance `D` scaled by distance to `A`
`hat D(x,y) := {:2 D(x,y):} / {: D(x,y) + min_{a in A} D(x,a) + D(a,y) :}`
is a metric (Steinhaus/biotope) [C06]
Similarly: given a nonnegative `1`-Lipschitz function `F`, `{:D(x,y):} / {: D(x,y) + F(x) + F(y) :}` is a metric

`epsilon`-nets and the Greedy Algorithm