Sign rank versus Vapnik-Chervonenkis dimension

This work studies the maximum possible sign rank of sign (N ×N)-matrices with a given Vapnik-Chervonenkis dimension d. For d = 1, this maximum is three. For d = 2, this maximum is Θ̃(N). For d > 2, 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 O(N/ log(N)) 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 (N×N)matrix of a ∆-regular graph with a second eigenvalue of absolute value λ and ∆ 6 N/2. We show that the sign rank of the signed version of this matrix is at least ∆/λ. We use this connection to prove the existence of a maximum class C ⊆ {±1} with Vapnik-Chervonenkis dimension 2 and sign rank Θ̃(N). 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.