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 a largest δ -separated subcollection of 𝒱 . Haussler showed an asymptotically tight bound of Θ ((n/δ )^d) 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(m^d_1 k^d-d_1) , for a fixed integer d > 0 and a real parameter 1 ≤ d_1 ≤ d (this generalizes the standard notion of bounded primal shatter dimension when d_1 = 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(n^d_1 k^d-d_1/δ ^d) , matching Haussler’s bound for the special cases where d_1=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.