Limitation of Learning Rankings from Partial Information

1.1. Notations. Let n be the number of elements and Sn be set of all possible n! permutations or rankings of these of n elements. Our interest is in learning non-negative valued functions f defined on Sn, i.e. f : Sn → R+, where R+ = {x ∈ R : x ≥ 0}. The support of f is defined as supp (f) = {σ ∈ Sn : f(σ) 6= 0} . The cardinality of support, | supp (f) | is called the sparsity of f and denoted by K. We also call it the l0 norm of f , denoted by |f |0. In this paper, our goal is to learn f from partial information. In order to formally define the partial information we consider, we need some notations. To this end, consider a partition of n, i.e. an ordered tuple λ = (λ1, λ2, . . . , λr), such that λ1 ≥ λ2 ≥ . . . ≥ λr ≥ 1, and n = λ1+λ2+. . .+λr. For example, λ = (n − 1, 1) is a partition of n. Now consider a partition of the n elements, {1, . . . , n}, as per the λ partition, i.e. divide n elements into r bins with ith bin having λi elements. It is easy to see that n elements can be divided as per the λ partition in Dλ distinct ways, with