Labeled sample compression schemes for complexes of oriented matroids

We show that the topes of a complex of oriented matroids (abbreviated COM) of VC-dimension d admit a proper labeled sample compression scheme of size d. This considerably extends results of Moran and Warmuth and the authors and is a step towards the sample compression conjecture – one of the oldest open in computational learning theory. On the one hand, our approach exploits the rich combinatorial cell structure of COMs via oriented matroid theory. On the other hand viewing tope graphs of COMs as partial cubes creates a fruitful link to metric

[1]  Hans-Jürgen Bandelt,et al.  COMs: Complexes of oriented matroids , 2015, J. Comb. Theory A.

[2]  K. Knauer,et al.  Finitary Affine Oriented Matroids , 2020, Discrete & Computational Geometry.

[3]  V. Itskov,et al.  Hyperplane Neural Codes and the Polar Complex , 2018, Topological Data Analysis.

[4]  Manfred K. Warmuth,et al.  Unlabeled Compression Schemes for Maximum Classes, , 2007, COLT.

[5]  Shay Moran,et al.  Labeled Compression Schemes for Extremal Classes , 2015, ALT.

[6]  David Eppstein,et al.  Isometric Diamond Subgraphs , 2008, GD.

[7]  Victor Reiner,et al.  Whitney Numbers for Poset Cones , 2019, Order.

[8]  Gábor Tardos,et al.  Unlabeled Compression Schemes Exceeding the VC-dimension , 2018, Discret. Appl. Math..

[9]  M. Deza,et al.  On Bouquets of Matroids and Orientations , 1986 .

[10]  Jim Lawrence,et al.  Oriented matroids , 1978, J. Comb. Theory B.

[11]  Corners and simpliciality in oriented matroids and partial cubes , 2020, 2002.11403.

[12]  Béla Bollobás,et al.  Defect Sauer Results , 1995, J. Comb. Theory A.

[13]  Frederic Paulin,et al.  Simplicite de groupes d'automorphismes d'espaces a courbure negative , 1998, math/9812167.

[14]  Andrea Baum,et al.  The axiomatization of affine oriented matroids reassessed , 2016, 1603.03228.

[15]  David Eppstein,et al.  Media theory - interdisciplinary applied mathematics , 2010 .

[16]  Kolja B. Knauer,et al.  Convexity in partial cubes: The hull number , 2013, Discret. Math..

[17]  Paul H. Edelman,et al.  The theory of convex geometries , 1985 .

[18]  Victor Chepoi,et al.  Distance-Preserving Subgraphs of Johnson Graphs , 2015, Comb..

[19]  Vladimir Vapnik,et al.  Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities , 1971 .

[20]  Andreas W. M. Dress,et al.  Gated sets in metric spaces , 1987 .

[21]  Andreas W. M. Dress,et al.  Towards a theory of holistic clustering , 1996, Mathematical Hierarchies and Biology.

[22]  Ilda P. F. da Silva,et al.  Axioms for maximal vectors of an oriented matroid; a combinatorial characterization of the regions determined by an arrangement of pseudohyperplanes , 1995, Eur. J. Comb..

[23]  D. Djoković.  Distance-preserving subgraphs of hypercubes , 1973 .

[24]  Peter L. Bartlett,et al.  Bounding Embeddings of VC Classes into Maximum Classes , 2014, ArXiv.

[25]  Kolja B. Knauer,et al.  On Tope Graphs of Complexes of Oriented Matroids , 2017, Discret. Comput. Geom..

[26]  Hans-Jürgen Bandelt,et al.  Graphs with intrinsic s3 convexities , 1989, J. Graph Theory.

[27]  Shai Ben-David,et al.  Combinatorial Variability of Vapnik-chervonenkis Classes with Applications to Sample Compression Schemes , 1998, Discret. Appl. Math..

[28]  J. Lawrence.  Lopsided sets and orthant-intersection by convex sets , 1983 .

[29]  Benjamin Steinberg,et al.  Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry , 2015, Memoirs of the American Mathematical Society.

[30]  Winfried Hochstättler,et al.  The Varchenko determinant for oriented matroids , 2018, Mathematische Zeitschrift.

[31]  Hans-Jürgen Bandelt,et al.  Combinatorics of lopsided sets , 2006, Eur. J. Comb..

[32]  R. P. Kurshan,et al.  On the addressing problem of loop switching , 1972 .

[33]  David Eppstein,et al.  Journal of Graph Algorithms and Applications Recognizing Partial Cubes in Quadratic Time 270 Eppstein Recognizing Partial Cubes in Quadratic Time , 2022 .

[34]  Sergei Ovchinnikov,et al.  Graphs and Cubes , 2011 .

[35]  Manfred K. Warmuth,et al.  Relating Data Compression and Learnability , 2003 .

[36]  Victor Chepoi,et al.  Two-dimensional partial cubes , 2019, Electron. J. Comb..

[37]  Vo D. Chepoi,et al.  Isometric subgraphs of Hamming graphs and d-convexity , 1988 .

[38]  Manfred K. Warmuth,et al.  Learning Nested Differences of Intersection-Closed Concept Classes , 1989, COLT '89.

[39]  Tilen Marc,et al.  Mirror graphs: Graph theoretical characterization of reflection arrangements and finite Coxeter groups , 2016, Eur. J. Comb..

[40]  Sally Floyd,et al.  Space-bounded learning and the Vapnik-Chervonenkis dimension , 1989, COLT '89.

[41]  Shay Moran,et al.  Sample compression schemes for VC classes , 2015, 2016 Information Theory and Applications Workshop (ITA).

[42]  Victor Chepoi,et al.  Hypercellular graphs: Partial cubes without Q3- as partial cube minor , 2016, Discret. Math..

[43]  Shay Moran,et al.  Unlabeled sample compression schemes and corner peelings for ample and maximum classes , 2018, ICALP.

[44]  Manfred K. Warmuth.  Compressing to VC Dimension Many Points , 2003, COLT.

[45]  Lajos Rónyai,et al.  Shattering-Extremal Set Systems of VC Dimension at most 2 , 2014, Electron. J. Comb..