For uniform keys on the d-dimensional sphere, softmax attention becomes selective at inverse temperature scaling β_n* ≍ n^{2/(d-1)}, with explicit limiting laws for attention weights and outputs in each regime.
Mathematical Models and Methods in Applied Sciences , volume=
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Establishes exponential convergence in Wasserstein distance for the mean-field limit and finite-particle approximation of a consensus-based method solving nonconvex bi-level optimization problems.
Consensus-based optimization with structure preservation and cross-dimensional interactions can accurately approximate quantum entanglement.
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For uniform keys on the d-dimensional sphere, softmax attention becomes selective at inverse temperature scaling β_n* ≍ n^{2/(d-1)}, with explicit limiting laws for attention weights and outputs in each regime.
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Establishes exponential convergence in Wasserstein distance for the mean-field limit and finite-particle approximation of a consensus-based method solving nonconvex bi-level optimization problems.
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Consensus-based optimization with structure preservation and cross-dimensional interactions can accurately approximate quantum entanglement.