A Size-Sensitive Discrepancy Bound for Set Systems of Bounded Primal Shatter Dimension

A Size-Sensitive Discrepancy Bound for Set Systems of Bounded Primal Shatter Dimension

Let (X, S) be a set system on an n-point set X . The discrepancy of S is defined as the minimum of the largest deviation from an even split, over all subsets of S ∈ S and two-coloringsχ on X . We consider the scenario where, for any subset X ′ ⊆ X of size m ≤ n and for any parameter 1 ≤ k ≤ m, the number of restrictions of the sets of S to X ′ of size at most k is only O(m1k1), for fixed intege...

Size sensitive packing number for Hamming cube and its consequences

We prove a size-sensitive version of Haussler’s Packing lemma [7] for set-systems with bounded primal shatter dimension, which have an additional size-sensitive property. This answers a question asked by Ezra [9]. We also partially address another point raised by Ezra regarding overcounting of sets in her chaining procedure. As a consequence of these improvements, we get an improvement on the s...

On discrepancy bounds via dual shatter function∗

Recently, it has been shown that tight or almost tight upper bounds for the discrepancy of many geometrically defined set systems can be derived from simple combinatorial parameters of these set systems. Namely, if the primal shatter function of a set system R on an n-point set X is bounded by const.m, then disc(R) = O(n1/2−1/2d) (which is known to be tight), and if the dual shatter function is...

Tight bounds on the maximum size of a set of permutations with bounded VC-dimension

The VC-dimension of a family P of n-permutations is the largest integer k such that the set of restrictions of the permutations in P on some k-tuple of positions is the set of all k! permutation patterns. Let rk(n) be the maximum size of a set of n-permutations with VCdimension k. Raz showed that r2(n) grows exponentially in n. We show that r3(n) = 2 Θ(n logα(n)) and for every t ≥ 1, we have r2...

Lecture 11 . Discrepancy of Set Systems