This work studies the maximum possible sign rank of N×N sign
matrices with a given VC dimension d. For d=1, this maximum is {three}. For
d=2, this maximum is Θ~(N1/2). For d>2, similar but
slightly less accurate statements hold. {The lower bounds improve over 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 VC dimension, and the number of maximum classes of
a given VC dimension -- answering a question of Frankl from '89, 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 N×N adjacency
matrix of a Δ regular graph with a second eigenvalue of absolute value
λ and Δ≤N/2. We show that the sign rank of the signed
version of this matrix is at least Δ/λ. We use this connection to
prove the existence of a maximum class C⊆{±1}N with VC
dimension 2 and sign rank Θ~(N1/2). This answers a question
of Ben-David et al.~regarding the sign rank of large VC 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.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC
dimension". Additional results in this version: (i) Estimates on the number
of maximum VC classes (answering a question of Frankl from '89). (ii)
Estimates on the sign rank of large VC classes (answering a question of
Ben-David et al. from '03). (iii) A discussion on the computational
complexity of computing the sign-ran