Training neural network language models over large vocabularies is computationally costly compared to count-based models such as Kneser-Ney. We present a systematic comparison of neural strategies to represent and train large vocabularies, including softmax, hierarchical softmax, target sampling, noise contrastive estimation and self normalization. We extend self normalization to be a proper es...

We derive a new lower bound for the bandwidth of a graph that is based on a new lower bound for the minimum cut problem. Our new semidefinite programming relaxation of the minimum cut problem is obtained by strengthening the known semidefinite programming relaxation for the quadratic assignment problem (or for the graph partition problem) by fixing two vertices in the graph; one on each side of...

Let V be a vector space of dimension v over a field of order q. The q-Kneser graph has the k-dimensional subspaces of V as its vertices, where two subspaces α and β are adjacent if and only if α ∩ β is the zero subspace. This paper is motivated by the problem of determining the chromatic numbers of these graphs. This problem is trivial when k = 1 (and the graphs are complete) or when v < 2k (an...

We give a simple and elementary proof of Kř́ıž’s lower bound on the chromatic number of the Kneser r-hypergraph of a set system S.

We prove that the propositional translations of the KneserLovász theorem have polynomial size extended Frege proofs and quasipolynomial size Frege proofs. We present a new counting-based combinatorial proof of the Kneser-Lovász theorem that avoids the topological arguments of prior proofs. We introduce a miniaturization of the octahedral Tucker lemma, called the truncated Tucker lemma. The prop...