Sign rank versus Vapnik-Chervonenkis dimension

Abstract

This work studies the maximum possible sign rank of sign -matrices with a given Vapnik-Chervonenkis dimension . For , this maximum is three. For , this maximum is . For , similar but slightly less accurate statements hold. The lower bounds improve on previous ones by Ben-David et al., and the upper bounds are novel. The lower bounds are obtained by probabilistic constructions, using a theorem of Warren in real algebraic topology. The upper bounds are obtained using a result of Welzl about spanning trees with low stabbing number, and using the moment curve. The upper bound technique is also used to: (i) provide estimates on the number of classes of a given Vapnik-Chervonenkis dimension, and the number of maximum classes of a given Vapnik-Chervonenkis dimension — answering a question of Frankl from 1989, and (ii) design an efficient algorithm that provides an multiplicative approximation for the sign rank. We also observe a general connection between sign rank and spectral gaps which is based on Forster's argument. Consider the adjacency - matrix of a -regular graph with a second eigenvalue of absolute value and . We show that the sign rank of the signed version of this matrix is at least . We use this connection to prove the existence of a maximum class with Vapnik-Chervonenkis dimension and sign rank . This answers a question of Ben-David et al. regarding the sign rank of large Vapnik-Chervonenkis classes. We also describe limitations of this approach, in the spirit of the Alon-Boppana theorem. We further describe connections to communication complexity, geometry, learning theory, and combinatorics. Bibliography: 69 titles.