Zhaonan Qu, Alfred Galichon, Wenzhi Gao, Johan Ugander
arXiv 30 Sep 2023 · Mathematics — Optimization · publishedOperations Research (2025) · 1 citations (OpenAlex)
arXiv:2310.00260 · PDF · DOI · OpenAlex · Extracted main text
For a broad class of models widely used in practice for choice and ranking data based on Luce's choice axiom, including the Bradley--Terry--Luce and Plackett--Luce models, we show that the associated maximum likelihood estimation problems are equivalent to a classic matrix balancing problem with target row and column sums. This perspective opens doors between two seemingly unrelated research areas, and allows us to unify existing algorithms in the choice modeling literature as special instances or analogs of Sinkhorn's celebrated algorithm for matrix balancing. We draw inspirations from these connections and resolve some open problems on the study of Sinkhorn's algorithm. We establish the global linear convergence of Sinkhorn's algorithm for non-negative matrices whenever finite scaling matrices exist, and characterize its linear convergence rate in terms of the algebraic connectivity of a weighted bipartite graph. We further derive the sharp asymptotic rate of linear convergence, which generalizes a classic result of Knight (2008). To our knowledge, these are the first quantitative linear convergence results for Sinkhorn's algorithm for general non-negative matrices and positive marginals. Our results highlight the importance of connectivity and orthogonality structures in matrix balancing and Sinkhorn's algorithm, which could be of independent interest. More broadly, the connections we establish in this paper between matrix balancing and choice modeling could also help motivate further transmission of ideas and lead to interesting results in both disciplines.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Maystre L, Grossglauser M (2017) Choicerank: identifying preferences from node traffic in networks | 1.000 | 21 | 6 | 100% |
| 2 | Hunter DR (2004) Mm algorithms for generalized bradley-terry models | 1.000 | 18 | 7 | 100% |
| 3 | Knight PA (2008) The sinkhorn–knopp algorithm: convergence and applications | 1.000 | 17 | 4 | 100% |
| 4 | Ford LR (1957) Solution of a ranking problem from binary comparisons | 1.000 | 12 | 6 | 100% |
| 5 | Léger F (2021) A gradient descent perspective on sinkhorn | 1.000 | 12 | 4 | 100% |
| 6 | Altschuler J, Niles-Weed J, Rigollet P (2017) Near-linear time approximation algorithms for optimal transport via sinkhorn iteration | 1.000 | 11 | 5 | 100% |
| 7 | Zermelo E (1929) Die berechnung der turnier-ergebnisse als ein maximumproblem der wahrscheinlichkeitsrechnung | 1.000 | 11 | 5 | 100% |
| 8 | Luo ZQ, Tseng P (1992) On the convergence of the coordinate descent method for convex differentiable minimization | 1.000 | 9 | 3 | 100% |
| 9 | Agarwal A, Patil P, Agarwal S (2018) Accelerated spectral ranking | 1.000 | 8 | 4 | 100% |
| 10 | Chakrabarty D, Khanna S (2021) Better and simpler error analysis of the sinkhorn–knopp algorithm for matrix scaling | 1.000 | 8 | 4 | 100% |
Showing the top 10 of 167 scored citations.