2 research outputs found
Ranking a set of objects: a graph based least-square approach
We consider the problem of ranking objects starting from a set of noisy
pairwise comparisons provided by a crowd of equal workers. We assume that
objects are endowed with intrinsic qualities and that the probability with
which an object is preferred to another depends only on the difference between
the qualities of the two competitors. We propose a class of non-adaptive
ranking algorithms that rely on a least-squares optimization criterion for the
estimation of qualities. Such algorithms are shown to be asymptotically optimal
(i.e., they require comparisons
to be -PAC). Numerical results show that our schemes are
very efficient also in many non-asymptotic scenarios exhibiting a performance
similar to the maximum-likelihood algorithm. Moreover, we show how they can be
extended to adaptive schemes and test them on real-world datasets
Least Squares Ranking on Graphs
Given a set of alternatives to be ranked, and some pairwise comparison data,
ranking is a least squares computation on a graph. The vertices are the
alternatives, and the edge values comprise the comparison data. The basic idea
is very simple and old: come up with values on vertices such that their
differences match the given edge data. Since an exact match will usually be
impossible, one settles for matching in a least squares sense. This formulation
was first described by Leake in 1976 for rankingfootball teams and appears as
an example in Professor Gilbert Strang's classic linear algebra textbook. If
one is willing to look into the residual a little further, then the problem
really comes alive, as shown effectively by the remarkable recent paper of
Jiang et al. With or without this twist, the humble least squares problem on
graphs has far-reaching connections with many current areas ofresearch. These
connections are to theoretical computer science (spectral graph theory, and
multilevel methods for graph Laplacian systems); numerical analysis (algebraic
multigrid, and finite element exterior calculus); other mathematics (Hodge
decomposition, and random clique complexes); and applications (arbitrage, and
ranking of sports teams). Not all of these connections are explored in this
paper, but many are. The underlying ideas are easy to explain, requiring only
the four fundamental subspaces from elementary linear algebra. One of our aims
is to explain these basic ideas and connections, to get researchers in many
fields interested in this topic. Another aim is to use our numerical
experiments for guidance on selecting methods and exposing the need for further
development.Comment: Added missing references, comparison of linear solvers overhauled,
conclusion section added, some new figures adde