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1. Ordinal Hyperplane Loss

   https://doi.org/10.1109/BigData.2018.8622079  

   Vanderheyden, Bob; Xie, Ying (December 2018, 2018 IEEE International Conference on Big Data (Big Data))

   The problem of ordinal classification occurs in a large and growing number of areas. Some of the most common source and applications of ordinal data include rating scales, medical classification scales, socio-economic scales, meaningful groupings of continuous data, facial emotional intensity, facial age estimation, etc. The problem of predicting ordinal classes is typically addressed by either performing n-1 binary classification for n ordinal classes or treating ordinal classes as continuous values for regression. However, the first strategy doesn't fully utilize the ordering information of classes and the second strategy imposes a strong continuous assumption to ordinal classes. In this paper, we propose a novel loss function called Ordinal Hyperplane Loss (OHPL) that is particularly designed for data with ordinal classes. The proposal of OHPL is a significant advancement in predicting ordinal class data, since it enables deep learning techniques to be applied to the ordinal classification problem on both structured and unstructured data. By minimizing OHPL, a deep neural network learns to map data to an optimal space where the distance between points and their class centroids are minimized while a nontrivial ordinal relationship among classes are maintained. Experimental results show that deep neural network with OHPL not only outperforms the state-of-the-art alternatives on classification accuracy but also scales well to large ordinal classification problems. 

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2. Ordinal Causal Discovery

   Ni, Yang; Mallick, Bani (January 2022, Proceedings of the 38th Conference on Uncertainty in Artificial Intelligence)
3. Learning Lines with Ordinal Constraints

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.45  

   Fan, Bohan; Centurion, Diego I.; Mohammadi, Neshat; Sgherzi, Francesco; Sidiropoulos, Anastasios; Valizadeh, Mina. (January 2020, APPROX)
4. Strongly Compact Cardinals and Ordinal Definability

   https://doi.org/10.1142/S0219061322500106  

   Goldberg, Gabriel (May 2023, Journal of Mathematical Logic)

   This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of Woodin’s HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable omega Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD. 

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5. Ordinal Maximin Share Approximation for Goods

   https://doi.org/10.1613/jair.1.13317  

   Hosseini, Hadi; Searns, Andrew; Segal-Halevi, Erel (May 2022, Journal of Artificial Intelligence Research)

   In fair division of indivisible goods,  ℓ-out-of-d maximin share (MMS) is the value that an agent can guarantee by partitioning the goods into d bundles and choosing the ℓ least preferred bundles. Most existing works aim to guarantee to all agents a constant fraction of their 1-out-of-n MMS. But this guarantee is sensitive to small perturbation in agents' cardinal valuations. We consider a more robust approximation notion, which depends only on the agents' ordinal rankings of bundles. We prove the existence of ℓ-out-of-⌊(ℓ + 1/2)n⌋ MMS allocations of goods for any integer ℓ ≥ 1, and present a polynomial-time algorithm that finds a 1-out-of-⌈3n/2⌉ MMS allocation when ℓ=1. We further develop an algorithm that provides a weaker ordinal approximation to MMS for any ℓ > 1. 

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