Related papers: Commutative Algebra of Statistical Ranking
Ranking data arises in a wide variety of application areas but remains difficult to model, learn from, and predict. Datasets often exhibit multimodality, intransitivity, or incomplete rankings---particularly when generated by humans---yet…
We study the problem of collaborative filtering where ranking information is available. Focusing on the core of the collaborative ranking process, the user and their community, we propose new models for representation of the underlying…
We introduce a general covariate-assisted statistical ranking model within the Plackett--Luce framework. Unlike previous studies focusing on individual effects with fixed covariates, our model allows covariates to vary across comparisons.…
In this paper we parameterize non-negative matrices of sum one and rank at most two. More precisely, we give a family of parameterizations using the least possible number of parameters. We also show how these parameterizations relate to a…
We study probability distributions over free algebras of trees. Probability distributions can be seen as particular (formal power) tree series [Berstel et al 82, Esik et al 03], i.e. mappings from trees to a semiring K . A widely studied…
A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…
In many contexts involving ranked preferences, agents submit partial orders over available alternatives. Statistical models often treat these as marginal in the space of total orders, but this approach overlooks information contained in the…
Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the…
Given an undirected graph representing similarities between a set of items and an additive measure evaluating the items, we treat the position of a special subset of items in an ordinal ranking through a collection of combinatorial…
We consider the problem of counting the number of possible sets of rankings (called ranking patterns) generated by unfolding models of codimension one. We express the ranking patterns as slices of the braid arrangement and show that all…
Rankings are central to decision-making in fields ranging from education to online platforms, yet classical deterministic methods such as the Borda count method or Copeland-type pairwise methods ignore uncertainty due to sampling noise or…
We consider sequential or active ranking of a set of n items based on noisy pairwise comparisons. Items are ranked according to the probability that a given item beats a randomly chosen item, and ranking refers to partitioning the items…
In decision theory an act is a function from a set of conditions to the set of real numbers. The set of conditions is a partition in some algebra of events. The expected value of an act can be calculated when a probability measure is given.…
Ranking over sets arise when users choose between groups of items. For example, a group may be of those movies deemed $5$ stars to them, or a customized tour package. It turns out, to model this data type properly, we need to investigate…
We propose a new statistical model for computational linguistics. Rather than trying to estimate directly the probability distribution of a random sentence of the language, we define a Markov chain on finite sets of sentences with many…
We classify (possibly non commutative) algebras of low rank over a domain R. We first review results for algebras of rank 2 and for finite-dimensional division algebras over the real numbers. These results motivate us to consider which…
This paper addresses the general problem of modelling and learning rank data with ties. We propose a probabilistic generative model, that models the process as permutations over partitions. This results in super-exponential combinatorial…
We introduce a family of toric algebras defined by maximal chains of a finite distributive lattice. Applying results on stable set polytopes we conclude that every such algebra is normal and Cohen-Macaulay, and give an interpretation of its…
Ranking metrics are a family of metrics largely used to evaluate recommender systems. However they typically suffer from the fact the reward is affected by the order in which recommended items are displayed to the user. A classical way to…
An algebraic linear ordering is a component of the initial solution of a first-order recursion scheme over the continuous categorical algebra of countable linear orderings equipped with the sum operation and the constant 1. Due to a general…