4 research outputs found
Mallows Ranking Models: Maximum Likelihood Estimate and Regeneration
This paper is concerned with various Mallows ranking models. We study the
statistical properties of the MLE of Mallows' model. We also make
connections of various Mallows ranking models, encompassing recent progress in
mathematics. Motivated by the infinite top- ranking model, we propose an
algorithm to select the model size automatically. The key idea relies on
the renewal property of such an infinite random permutation. Our algorithm
shows good performance on several data sets.Comment: 10 pages, 2 figures, 5 tables. This paper is published by
http://proceedings.mlr.press/v97/tang19a.htm
Learning Mixtures of Ranking Models
This work concerns learning probabilistic models for ranking data in a
heterogeneous population. The specific problem we study is learning the
parameters of a Mallows Mixture Model. Despite being widely studied, current
heuristics for this problem do not have theoretical guarantees and can get
stuck in bad local optima. We present the first polynomial time algorithm which
provably learns the parameters of a mixture of two Mallows models. A key
component of our algorithm is a novel use of tensor decomposition techniques to
learn the top-k prefix in both the rankings. Before this work, even the
question of identifiability in the case of a mixture of two Mallows models was
unresolved
Multiresolution Analysis of Incomplete Rankings
Incomplete rankings on a set of items are orderings of
the form , with and . Though they arise in many modern
applications, only a few methods have been introduced to manipulate them, most
of them consisting in representing any incomplete ranking by the set of all its
possible linear extensions on . It is the major purpose
of this paper to introduce a completely novel approach, which allows to treat
incomplete rankings directly, representing them as injective words over . Unexpectedly, operations on incomplete rankings have very
simple equivalents in this setting and the topological structure of the complex
of injective words can be interpretated in a simple fashion from the
perspective of ranking. We exploit this connection here and use recent results
from algebraic topology to construct a multiresolution analysis and develop a
wavelet framework for incomplete rankings. Though purely combinatorial, this
construction relies on the same ideas underlying multiresolution analysis on a
Euclidean space, and permits to localize the information related to rankings on
each subset of items. It can be viewed as a crucial step toward nonlinear
approximation of distributions of incomplete rankings and paves the way for
many statistical applications, including preference data analysis and the
design of recommender systems
Tractable Search for Learning Exponential Models of Rankings
We consider the problem of learning the Generalized Mallows (GM) model of [Fligner and Verducci, 1986], which represents a probability distribution over all possible permutations (or rankings) of a given set of objects. The training data consists of a set of permutations. This problem generalizes the well known rank aggregation problem. Maximum Likelihood estimation of the GM model is NP-hard. An exact but inefficient searchbased method was recently proposed for this problem. Here we introduce the first nontrivial heuristic function for this search. We justify it theoretically, and show why it is admissible in practice. We experimentally demonstrate its effectiveness, and show that it is superior to existing techniques for learning the GM model. We also show good performance of a family of faster approximate methods of search.