  Many learning-to-rank (LtR) algorithms focus on query-independent model, in
which query and document do not lie in the same feature space, and the rankers
rely on the feature ensemble about query-document pair instead of the
similarity between query instance and documents. However, existing algorithms
do not consider local structures in query-document feature space, and are
fragile to irrelevant noise features. In this paper, we propose a novel
Riemannian metric learning algorithm to capture the local structures and
develop a robust LtR algorithm. First, we design a concept called \textit{ideal
candidate document} to introduce metric learning algorithm to query-independent
model. Previous metric learning algorithms aiming to find an optimal metric
space are only suitable for query-dependent model, in which query instance and
documents belong to the same feature space and the similarity is directly
computed from the metric space. Then we extend the new and extremely fast
global Geometric Mean Metric Learning (GMML) algorithm to develop a localized
GMML, namely L-GMML. Based on the combination of local learned metrics, we
employ the popular Normalized Discounted Cumulative Gain~(NDCG) scorer and
Weighted Approximate Rank Pairwise (WARP) loss to optimize the \textit{ideal
candidate document} for each query candidate set. Finally, we can quickly
evaluate all candidates via the similarity between the \textit{ideal candidate
document} and other candidates. By leveraging the ability of metric learning
algorithms to describe the complex structural information, our approach gives
us a principled and efficient way to perform LtR tasks. The experiments on
real-world datasets demonstrate that our proposed L-GMML algorithm outperforms
the state-of-the-art metric learning to rank methods and the stylish
query-independent LtR algorithms regarding accuracy and computational
efficiency.
