Two Proofs for Shallow Packings

Kunal Dutta, Esther Ezra, Arijit Ghosh

Abstract: We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let \(\mathcal {V}\) be a finite set system defined over an n-point set X; we view \(\mathcal {V}\) as a set of indicator vectors over the n-dimensional unit cube. A \(\delta \)-separated set of \(\mathcal {V}\) is a subcollection \(\mathcal {W}\), s.t. the Hamming distance between each pair \(\mathbf{u}, \mathbf{v}\in \mathcal {W}\) is greater than \(\delta \), where \(\delta > 0\) is an integer parameter. The \(\delta \)-packing number is then defined as the cardinality of a largest \(\delta \)-separated subcollection of \(\mathcal {V}\). Haussler showed an asymptotically tight bound of \(\Theta ((n/\delta )^d)\) on the \(\delta \)-packing number if \(\mathcal {V}\) has VC-dimension (or primal shatter dimension) d. We refine this bound for the scenario where, for any subset, \(X' \subseteq X\) of size \(m \le n\) and for any parameter \(1 \le k \le m\), the number of vectors of length at most k in the restriction of \(\mathcal {V}\) to \(X'\) is only \(O(m^{d_1} k^{d-d_1})\), for a fixed integer \(d > 0\) and a real parameter \(1 \le d_1 \le d\) (this generalizes the standard notion of bounded primal shatter dimension when \(d_1 = d\)). In this case when \(\mathcal {V}\) is “k-shallow” (all vector lengths are at most k), we show that its \(\delta \)-packing number is \(O(n^{d_1} k^{d-d_1}/\delta ^d)\), 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.