Dimension d = O(m^{-2} log n) nearly achieves the optimal margin m^rd(+∞, A) for retrieval embeddings, with matching lower bounds showing d = O(k log(n/k)) suffices and is necessary for m = Θ(k^{-1/2}) on k-sparse query matrices.
Proceedings of the Glasgow Mathematical Association , author=
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Survey presenting the quantitative uniformity theorem for the Mordell conjecture proved by Yu--Yuan--Zhou, building on Vojta, Dimitrov--Habegger--Gao and Kuhne.
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Is Dimensionality a Barrier for Retrieval Models?
Dimension d = O(m^{-2} log n) nearly achieves the optimal margin m^rd(+∞, A) for retrieval embeddings, with matching lower bounds showing d = O(k log(n/k)) suffices and is necessary for m = Θ(k^{-1/2}) on k-sparse query matrices.
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Quantitativity in the Mordell Conjecture
Survey presenting the quantitative uniformity theorem for the Mordell conjecture proved by Yu--Yuan--Zhou, building on Vojta, Dimitrov--Habegger--Gao and Kuhne.