Negative-Type Diversities, a Multi-dimensional Analogue of Negative-Type Metrics

Diversities are a generalization of metric spaces in which a non-negative value is assigned to all finite subsets of a set, rather than just to pairs of points. Here we provide an analogue of the theory of negative type metrics for diversities. We introduce negative type diversities, and show that, as in the metric space case, they are a generalization of $L_1$-embeddable diversities. We provide a number of characterizations of negative type diversities, including a geometric characterisation. Much of the recent interest in negative type metrics stems from the connections between metric embeddings and approximation algorithms. We extend some of this work into the diversity setting, showing that lower bounds for embeddings of negative type metrics into $L_1$ can be extended to diversities by using recently established extremal results on hypergraphs.

[1]  David Ellis,et al.  An Approximate Vertex-Isoperimetric Inequality for r-sets , 2013, Electron. J. Comb..

[2]  Assaf Naor,et al.  The integrality gap of the Goemans-Linial SDP relaxation for sparsest cut is at least a constant multiple of √log n , 2017, STOC.

[3]  Anthony Weston,et al.  Strict p-negative type of a metric space , 2009, 0901.0695.

[4]  Michel Deza,et al.  Geometry of cuts and metrics , 2009, Algorithms and combinatorics.

[5]  Y. Rabani,et al.  Improved lower bounds for embeddings into L 1 , 2006, SODA 2006.

[6]  Nisheeth K. Vishnoi,et al.  The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into l/sub 1/ , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[7]  Paul F. Tupper,et al.  Diversities and the Geometry of Hypergraphs , 2013, Discret. Math. Theor. Comput. Sci..

[8]  P. Tupper,et al.  Open Problem Statement: Minimal Distortion Embeddings of Diversities in $\ell_1$ , 2017, 1712.01960.

[9]  A. Hora,et al.  Distance-Regular Graphs , 2007 .

[10]  James R. Lee,et al.  Euclidean distortion and the sparsest cut , 2005, STOC '05.

[11]  B. Piatek.  On the gluing of hyperconvex metrics and diversities , 2014 .

[12]  Yuval Rabani,et al.  Improved lower bounds for embeddings into L1 , 2006, SODA '06.

[13]  Additive combination spaces , 2013, 1308.3293.

[14]  An approximate isoperimetric inequality for r-sets , 2012, 1203.3699.

[15]  P. Tupper,et al.  Constant Distortion Embeddings of Symmetric Diversities , 2016, 1604.01863.

[16]  Robert Krauthgamer,et al.  Measured Descent: A New Embedding Method for Finite Metrics , 2004, FOCS.

[17]  James R. Lee,et al.  Extending Lipschitz functions via random metric partitions , 2005 .

[18]  B. Piatek,et al.  Diversities, hyperconvexity and fixed points , 2013, 1303.7146.

[19]  James R. Lee,et al.  Metric structures in L1: dimension, snowflakes, and average distortion , 2005, Eur. J. Comb..

[20]  Nisheeth K. Vishnoi,et al.  The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into l1 , 2005, FOCS.

[21]  I. J. Schoenberg,et al.  Metric spaces and positive definite functions , 1938 .

[22]  P. Tupper,et al.  Hyperconvexity and tight-span theory for diversities , 2010, 1006.1095.

[23]  Andrew Poelstra.  On the Topological and Uniform Structure of Diversities , 2013 .

[24]  James R. Lee,et al.  Metric Structures in L1: Dimension, Snowflakes, and Average Distortion , 2004, LATIN.

[25]  I. J. Schoenberg.  Remarks to Maurice Frechet's Article ``Sur La Definition Axiomatique D'Une Classe D'Espace Distances Vectoriellement Applicable Sur L'Espace De Hilbert , 1935 .

[26]  J. Bourgain.  On lipschitz embedding of finite metric spaces in Hilbert space , 1985 .

[27]  Diversities and Conformities , 2013, 1307.1897.

[28]  A. Nies,et al.  A Universal Separable Diversity , 2015, 1509.07173.