Bounding Embeddings of VC Classes into Maximum Classes

One of the earliest conjectures in computational learning theory—the Sample Compression conjecture—asserts that concept classes (equivalently set systems) admit compression schemes of size linear in their VC dimension. To-date this statement is known to be true for maximum classes—those that possess maximum cardinality for their VC dimension. The most promising approach to positively resolving the conjecture is by embedding general VC classes into maximum classes without super-linear increase to their VC dimensions, as such embeddings would extend the known compression schemes to all VC classes. We show that maximum classes can be characterised by a local-connectivity property of the graph obtained by viewing the class as a cubical complex. This geometric characterisation of maximum VC classes is applied to prove a negative embedding result which demonstrates VC-d classes that cannot be embedded in any maximum class of VC dimension lower than 2d. On the other hand, we show that every VC-d class C embeds in a VC-(d +D) maximum class where D is the deficiency of C, i.e., the difference between the cardinalities of a maximum VC-d class and of C. For VC-2 classes in binary n-cubes for 4≤ n≤ 6, we give best possible results on embedding into maximum classes. For some special classes of Boolean functions, relationships with maximum classes are investigated. Finally we give a general recursive procedure for embedding VC-d classes into VC-(d + k) maximum classes for smallest k. J. Hyam Rubinstein Department of Mathematics & Statistics, The University of Melbourne, Australia e-mail: [email protected] Benjamin I. P. Rubinstein Department of Computing & Information Systems, The University of Melbourne, Australia e-mail: [email protected] Peter L. Bartlett Depts. Electrical Engineering & Computer Sciences and Statistics, UC Berkeley, USA Faculty of Science and Engineering, Queensland University of Technology, Australia e-mail: [email protected] 1 ar X iv :1 40 1. 73 88 v1 [ cs .L G ] 2 9 Ja n 20 14 2 Rubinstein, Rubinstein, Bartlett