We consider the problem of similarity search within a set of top-k lists under the Kendall's Tau distance function. This distance describes how related two rankings are in terms of concordantly and discordantly ordered items. As top-k lists are usually very short compared to the global domain of possible items to be ranked, creating an inverted index to look up overlapping lists is possible but does not capture tight enough the similarity measure. In this work, we investigate locality sensitive hashing schemes for the Kendall's Tau distance and evaluate the proposed methods using two real-world datasets.
There is an ample work on computing relatedness between ranked lists of items, such as to mine correlations or anti-correlations between lists ranked by different attributes. Arguably, the two most prominent similarity measures are Kendall's tau and Spearman's Footrule. @cite_1 study comparing incomplete top- @math lists, that is, lists capturing a subset of a global set of items, rendering the lists incomplete in nature. In the scenarios motivating our work, like similarity search favorite preference rankings, lists are naturally incomplete, capturing, e.g., only the top-10 movies of all times. In this work, we focus on the computation of Kendall's Tau distance.
@cite_1
2160266360 1966835268 1980940798 1707848225
Spearman's footrule and Kendall's tau are two well established distances between rankings. They, however, fail to take into account concepts crucial to evaluating a result set in information retrieval: element relevance and positional information. That is, changing the rank of a highly-relevant document should result in a higher penalty than changing the rank of an irrelevant document; a similar logic holds for the top versus the bottom of the result ordering. In this work, we extend both of these metrics to those with position and element weights, and show that a variant of the Diaconis-Graham inequality still holds - the generalized two measures remain within a constant factor of each other for all permutations. We continue by extending the element weights into a distance metric between elements. For example, in search evaluation, swapping the order of two nearly duplicate results should result in little penalty, even if these two are highly relevant and appear at the top of the list. We extend the distance measures to this more general case and show that they remain within a constant factor of each other. We conclude by conducting simple experiments on web search data with the proposed measures. Our experiments show that the weighted generalizations are more robust and consistent with each other than their unweighted counter-parts. In the field of information retrieval, one is often faced with the problem of computing the correlation between two ranked lists. The most commonly used statistic that quantifies this correlation is Kendall's Τ. Often times, in the information retrieval community, discrepancies among those items having high rankings are more important than those among items having low rankings. The Kendall's Τ statistic, however, does not make such distinctions and equally penalizes errors both at high and low rankings. In this paper, we propose a new rank correlation coefficient, AP correlation (Τap), that is based on average precision and has a probabilistic interpretation. We show that the proposed statistic gives more weight to the errors at high rankings and has nice mathematical properties which make it easy to interpret. We further validate the applicability of the statistic using experimental data. We propose a new family of distance measures on rankings, derived through an axiomatic approach, that consider the nonuniform relevance of the top and bottom of ordered lists and similarities between candidates. The proposed distance functions include specialized weighted versions of the Kendall τ distance and the Cayley distance, and are suitable for comparing rankings in a number of applications, including information retrieval and rank aggregation. In addition to proposing the distance measures and providing the theoretical underpinnings for their applications, we also analyze algorithmic and computational aspects of weighted distance-based rank aggregation. We present an aggregation method based on approximating weighted distance measures by a generalized version of Spearman's footrule distance as well as a Markov chain method inspired by PageRank, where transition probabilities of the Markov chain reflect the chosen weighted distances. Understanding the correlation between two different scores for the same set of items is a common problem in graph analysis and information retrieval. The most commonly used statistics that quantifies this correlation is Kendall's tau; however, the standard definition fails to capture that discordances between items with high rank are more important than those between items with low rank. Recently, a new measure of correlation based on average precision has been proposed to solve this problem, but like many alternative proposals in the literature it assumes that there are no ties in the scores. This is a major deficiency in a number of contexts, and in particular when comparing centrality scores on large graphs, as the obvious baseline, indegree, has a very large number of ties in social networks and web graphs. We propose to extend Kendall's definition in a natural way to take into account weights in the presence of ties. We prove a number of interesting mathematical properties of our generalization and describe an O(n n) algorithm for its computation. We also validate the usefulness of our weighted measure of correlation using experimental data on social networks and web graphs.