Two Proofs for Shallow Packings

Abstract

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 d1 k d−d1), 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 d1 k d−d1 /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.

When are epsilon-nets small?

“…More specifically, our bound (4) (given the fact the Alexander's capacity is bounded in this case [23]) gives the result that scales as O(log 1 ǫ ), which is significantly better than O( d ǫ ) claimed in [3]. We know (14) that boundedness of the star number is a necessary and sufficient condition for boundedness of τ (ǫ) for all distributions, and it also implies the existence of finite hitting sets for corresponding range spaces. Our results answer the following natural question.…”

“…Lemma H (Shallow-Packing Lemma [14,29]). Let X be an n-element set equipped with the uniform measure P. Assume that a range space (X , R) of at most k-element sets has VC dimension d, shallow-cell complexity ϕ.…”

“…Recent Computational Geometry papers (see e.g. [14,29]) still use the argument of the original proof of Haussler. Apart from a simplified proof, our result may also be used to provide sharper constant factors and will be used in Section 5.…”

Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning

“…Recent efforts have been devoted to finding generalizations of the packing lemma to these finer classifications of set systems. For integers k and δ, call (X, R) a k-shallow δ-packing if R is a δ-packing, and |S| ≤ k for all S ∈ R. After an earlier bound [Ezr14], the following shallow packing lemma has been recently established in [DEG15,Mus16].…”

A Simple Proof of the Shallow Packing Lemma

“…Given a set system (X, P) and an integer δ > 0, we say that (X, P) is δ-separated if |∆(R, S)| ≥ δ for every R, S ∈ P. For any set Y ⊆ X, define the projection of P onto Y as the set system P| Y = S ∩Y | S ∈ P . After a series of partial bounds [8,4,7], the following statement has been recently established in [3], via two different proofs (one building on Haussler's original proof while the other extends Chazelle's proof):…”