Teaching dimension, VC dimension, and critical sets in Latin squares

Abstract

A critical set in an n × n Latin square is a minimal set of entries that uniquely identifies it among all Latin squares of the same size. It is conjectured by Nelder in 1979, and later independently by Mahmoodian, and Bate and van Rees that the size of the smallest critical set is ⌊n 2 /4⌋. We prove a lower-bound of n 2 /10 4 for sufficiently large n, and thus confirm the quadratic order predicted by the conjecture. This improves a recent lower-bound of Ω(n 3/2 ) due to Cavenagh and Ramadurai.From the point of view of computational learning theory, the size of the smallest critical set corresponds to the minimum teaching dimension of the set of Latin squares. We study two related notions of dimension from learning theory. We prove a lower-bound of n 2 −(e+o(1))n 5/3for both of the VC-dimension and the recursive teaching dimension.

On the Distances between Latin Squares and the Smallest Defining Set Size

“…Using the Latin trades constructed in Section of this paper, we show that scs. Although this does not improve the result in , we include it as an interesting application of the theory used to prove Theorem . Theorem The size of the smallest defining set of any Latin square of order n has size .…”

“…Until quite recently, the best known lower bound for large n was scs , which in turn improved results given in and . However, very recently this has been improved by Hatami and Qian () who have shown that scs for sufficiently large n .…”

“…Using the Latin trades constructed in Section 2 of this paper, we show that scs(n) = (n 3/2 ). Although this does not improve the result in [11], we include it as an interesting application of the theory used to prove Theorem 1.3. Theorem 1.4.…”

A Sauer-Shelah-Perles Lemma for Sumsets

“…Vapnik-Chervonenkis dimension [31,32], or VC dimension for short, is a combinatorial parameter of major importance in discrete and computational geometry [9,17,19], statistical learning theory [7,32], and other areas [2,12,15,20]. The VC dimension of a family of binary vectors F ⊆ {0, 1} n is the largest cardinality of a set shattered by the family, that is, a set S ⊆ {1, .…”

Parallel Machine Learning Algorithm Using Fine-Grained-Mode Spark on a Mesos Big Data Cloud Computing Software Framework for Mobile Robotic Intelligent Fault Recognition

“…SUPPORT VECTOR MACHINE LEARNING FOR ROLLER BEARING FAULT DIAGNOSIS PROCESSA. SUPPORT VECTOR MACHIMECorinna Cortes and Vapnik developed the currently used SVM technique to minimize the VC dimension[27][28][29][30][31]. The hyperplane, which separates data (…”