Shattering News

 We explore the concepts of shattered and order-shattered sets. In particular, for every family ℱ of subsets of {1,2,…,m} there are exactly |ℱ| subsets of {1,2,…,m} order-shattered by ℱ. This provides proofs and strengthenings of the result of Sauer, Perles and Shelah, Vapnik and Chervonenkis (sometimes known as Sauer's lemma) and a new approach to the reverse Sauer Inequality of Bollobás and Radcliffe. We characterize those sets which can be order-shattered by a uniform family and those sets which can be order-shattered by an antichain. We also give an algebraic interpretation of order shattering using Gr?bner bases. This results in sharpening of a theorem of Frankl and Pach. It also points out an interesting and promising connection between combinatorial and algebraic objects. Received: May 9, 2000 Final version received: July 3, 2001     

On the complexity of constrained VC-classes

Sauer's lemma is extended to classes H_N of binary-valued functions h on [n]={1,…,n} which have a margin less than or equal to N on all x∈[n] with h(x)=1, where the margin μ_h(x) of h at x∈[n] is defined as the largest non-negative integer a such that h is constant on the interval I_a(x)=[x-a,x+a]⊆[n]. Estimates are obtained for the cardinality of classes of binary-valued functions with a margin of at least N on a positive sample S⊆[n].     

On the transversal number and VC-dimension of families of positive homothets of a convex body

Let F be a family of positive homothets (or translates) of a given convex body K in Rⁿ. We investigate two approaches to measuring the complexity of F. First, we find an upper bound on the transversal number τ(F) of F in terms of n and the independence number ν(F). This question is motivated by a problem of Grünbaum [L. Danzer, B. Grünbaum, V. Klee, Helly’s theorem and its relatives, in: Proc. Sympos. Pure Math., vol. VII, Amer. Math. Soc., Providence, RI, 1963, pp. 101-180]. Our bound is exponential in n, an improvement from the previously known bound of Kim, Nakprasit, Pelsmajer and Skokan [S.-J. Kim, K. Nakprasit, M.J. Pelsmajer, J. Skokan, Transversal numbers of translates of a convex body, Discrete Math. 306 (18) (2006) 2166-2173], which was of order nⁿ. By a lower bound, we show that the right order of magnitude is exponential in n.Next, we consider another measure of complexity, the Vapnik-?ervonenkis dimension of F. We prove that vcdim(F)≤3 if n=2 and is infinite for some F if n≥3. This settles a conjecture of Grünbaum [B. Grünbaum, Venn diagrams and independent families of sets, Math. Mag. 48 (1975) 12-23]: Show that the maximum dual VC-dimension of a family of positive homothets of a given convex body K in Rⁿ is n+1. This conjecture was disproved by Naiman and Wynn [D.Q. Naiman, H.P. Wynn, Independent collections of translates of boxes and a conjecture due to Grünbaum, Discrete Comput. Geom. 9 (1) (1993) 101-105] who constructed a counterexample of dual VC-dimension . Our result implies that no upper bound exists.     

On the VC-dimension of uniform hypergraphs

Let be a k-uniform hypergraph on [n] where k−1 is a power of some prime p and n≥ n ₀(k). Our main result says that if , then there exists E ₀∊ such that {E∩ E ₀: E∊ } contains all subsets of E ₀. This improves a longstanding bound of due to Frankl and Pach [7].Research supported in part by NSF grants DMS-0400812 and an Alfred P. Sloan Research Fellowship.Research supported in part by NSA grant H98230-05-1-0079. Part of this research was done while working at University of Illinois at Chicago.     

Pseudo-dimension and entropy of manifolds formed by affine-invariant dictionary

We consider the manifolds H _n(φ) formed by all possible linear combinations of n functions from the set {φ(A⋅+b)}, where x→Ax+b is arbitrary affine mapping in the space ℝ^d. For example, neural networks and radial basis functions are the manifolds of type H _n(φ). We obtain estimates for pseudo-dimension of the manifold H _n(φ) for wide collection of the generator function φ. The estimates have the order O(d ² n) in degree scale, that is the order is proportional to number of parameters of the manifold H _n(φ). Moreover the estimates for ɛ-entropy of the manifold H _n(φ) are obtained. Mathematics subject classifications (2000) 41A46, 41A50, 42A61, 42C10 V. Maiorov: Supported by the Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel.     

Forbidden configurations and repeated induction

For a given k×? matrix F, we say a matrix A has no configurationF if no k×? submatrix of A is a row and column permutation of F. We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. We define as the maximum number of columns in an m-rowed simple matrix which has no configuration F. A fundamental result of Sauer, Perles and Shelah, and Vapnik and Chervonenkis determines exactly, where K_k denotes the k×2^k simple matrix. We extend this in several ways. For two matrices G,H on the same number of rows, let [G∣H] denote the concatenation of G and H. Our first two sets of results are exact bounds that find some matrices B,C where and . Our final result provides asymptotic boundary cases; namely matrices F for which is O(m^p) yet for any choice of column α not in F, we have is Ω(m^p+1). This is evidence for a conjecture of Anstee and Sali. The proof techniques in this paper are dominated by repeated use of the standard induction employed in forbidden configurations. Analysis of base cases tends to dominate the arguments. For a k-rowed (0,1)-matrix F, we also consider a function which is the minimum number of columns in an m-rowed simple matrix for which each k-set of rows contains F as a configuration.