Related papers: Commutative Algebra of Statistical Ranking
Inspired by the commutator and anticommutator algebras derived from algebras graded by groups, we introduce noncommutatively graded algebras. We generalize various classical graded results to the noncommutatively graded situation concerning…
In the unidimensional unfolding model, given m objects in general position there arise 1+m(m-1)/2 rankings. The set of rankings is called the ranking pattern of the m given objects. By changing these m objects, we can generate various…
Ranking theories according to their strength is a recurring motif in mathematical logic. We introduce a new ranking of arbitrary (not necessarily recursively axiomatized) theories in terms of the encoding power of their $\beta$-models:…
Online learning to rank is a sequential decision-making problem where in each round the learning agent chooses a list of items and receives feedback in the form of clicks from the user. Many sample-efficient algorithms have been proposed…
Using the theory of noncommutative symmetric functions, we introduce the higher order peak algebras, a sequence of graded Hopf algebras which contain the descent algebra and the usual peak algebra as initial cases (N = 1 and N = 2). We…
We model stochastic choices with categorization. The agent preliminarly groups alternatives in homogenous disjoint classes, then randomly chooses one class and randomly picks an item within the selected class. We give a formal definition of…
A simple generative model for rank ordered data with ties is presented. The model is based on ordering geometric latent variables and can be seen as the discrete counterpart of the Plackett-Luce (PL) model, a popular, relatively tractable…
In this paper we build an Orlik-Solomon model for the canonical gradation of the cohomology algebra with integer coefficients of the complement of a toric arrangement. We give some results on the uniqueness of the representation of…
The concept of diagrammatic combinatorial Hopf algebras in the form introduced for describing the Heisenberg-Weyl algebra in~\cite{blasiak2010combinatorial} is extended to the case of so-called rule diagrams that present graph rewriting…
We consider the problem of statistical inference for ranking data, specifically rank aggregation, under the assumption that samples are incomplete in the sense of not comprising all choice alternatives. In contrast to most existing methods,…
We introduce two new partial orders on the standard Young tableaux of a given partition shape, in analogy with the strong and weak Bruhat orders on permutations. Both posets are ranked by the major index statistic offset by a fixed shift.…
A perspective of statistical language models which emphasizes their collocational aspect is advocated. It is suggested that strings be generalized in terms of classes of relationships instead of classes of objects. The single most important…
We propose a topic modeling approach to the prediction of preferences in pairwise comparisons. We develop a new generative model for pairwise comparisons that accounts for multiple shared latent rankings that are prevalent in a population…
This paper considers ranking inference of $n$ items based on the observed data on the top choice among $M$ randomly selected items at each trial. This is a useful modification of the Plackett-Luce model for $M$-way ranking with only the top…
Conjunctive table algebras are introduced and axiomatically characterized. A conjunctive table algebra is a variant of SPJR algebra (a weaker form of relational algebra), which corresponds to conjunctive queries with equality. The table…
This paper presents State Algebra, a novel framework designed to represent and manipulate propositional logic using algebraic methods. The framework is structured as a hierarchy of three representations: Set, Coordinate, and Row…
A new class of general exponential ranking models is introduced which we label angle-based models for ranking data. A consensus score vector is assumed, which assigns scores to a set of items, where the scores reflect a consensus view of…
For massive and heterogeneous modern datasets, it is of fundamental interest to provide guarantees on the accuracy of estimation when computational resources are limited. In the application of learning to rank, we provide a hierarchy of…
We present a new method of analysis of associative algebras. This method bears a certain resemblance to the famous analysis of commutative $C^*$-algebras in which an important role is played by multiplicative functionals over the algebra.…
We propose a novel combinatorial inference framework to conduct general uncertainty quantification in ranking problems. We consider the widely adopted Bradley-Terry-Luce (BTL) model, where each item is assigned a positive preference score…