Two Proofs for Shallow Packings

We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let V be a finite set system defined over an n-point set X; we view V as a set of indicator vectors over the n-dimensional unit cube. A δ-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u,v ∈W is greater than δ, where δ > 0 is an integer parameter. The δ-packing number is then defined as the cardinality of the largest δ-separated subcollection of V. Haussler showed an asymptotically tight bound of Θ((n/δ)) on the δ-packing number if V has VC-dimension (or primal shatter dimension) d. We refine this bound for the scenario where, for any subset, X ′ ⊆ X of size m ≤ n and for any parameter 1 ≤ k ≤ m, the number of vectors of length at most k in the restriction of V to X ′ is only O(md1kd−d1), for a fixed integer d > 0 and a real parameter 1 ≤ d1 ≤ d (this generalizes the standard notion of bounded primal shatter dimension when d1 = d). In this case when V is “k-shallow” (all vector lengths are at most k), we show that its δ-packing number is O(nd1kd−d1/δd), matching Haussler’s bound for the special cases where d1 = d or k = n. We present two proofs, the first is an extension of Haussler’s approach, and the second extends the proof of Chazelle, originally presented as a simplification for Haussler’s proof.