In every dimension d≥2 there exists a unique β_*^{(d)}>0 such that the uniform density on the sphere is the unique global minimizer of the USA free energy up to the linear-stability threshold K_# for β≤β_*, yielding a continuous transition, while for β>β_* the uniform density is not globally minimiz
On the Structure of Stationary Solutions to
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Defines the L² over Wasserstein space to equip random probability measures with inherited Riemannian geometry, enabling statistical convergence results and Bayesian posterior consistency in the Wasserstein topology.
Transformers converge pathwise to a stochastic particle system and SPDE in the scaling limit, exhibiting synchronization by noise and exponential energy dissipation when common noise is coercive relative to self-attention drift.
For 1/(n+1)-periodic interactions with Fourier decay, the phase transition in mean-field free energies on the circle is continuous at the linear stability threshold of the uniform state.
Models multi-head transformer data flow as time-dependent Wasserstein gradient flows of an attention-capturing interaction energy, with proofs on omega-limit stationary points and stability under weight and input perturbations.
In the low-temperature regime, the token distribution in mean-field transformers concentrates onto the push-forward under a key-query-value projection with Wasserstein distance scaling as √(log(β+1)/β) exp(Ct) + exp(-ct).
citing papers explorer
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Phase transitions for the noisy transformer model in arbitrary dimension
In every dimension d≥2 there exists a unique β_*^{(d)}>0 such that the uniform density on the sphere is the unique global minimizer of the USA free energy up to the linear-stability threshold K_# for β≤β_*, yielding a continuous transition, while for β>β_* the uniform density is not globally minimiz
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$L^2$ over Wasserstein: Statistical Analysis for Optimal Transport
Defines the L² over Wasserstein space to equip random probability measures with inherited Riemannian geometry, enabling statistical convergence results and Bayesian posterior consistency in the Wasserstein topology.
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Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models
Transformers converge pathwise to a stochastic particle system and SPDE in the scaling limit, exhibiting synchronization by noise and exponential energy dissipation when common noise is coercive relative to self-attention drift.
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Phase transitions in Doi-Onsager, Noisy Transformer, and other multimodal models
For 1/(n+1)-periodic interactions with Fourier decay, the phase transition in mean-field free energies on the circle is continuous at the linear stability threshold of the uniform state.
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Multi-Headed Transformer Architectures as Time-dependent Wasserstein Gradient Flows
Models multi-head transformer data flow as time-dependent Wasserstein gradient flows of an attention-capturing interaction energy, with proofs on omega-limit stationary points and stability under weight and input perturbations.
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Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime
In the low-temperature regime, the token distribution in mean-field transformers concentrates onto the push-forward under a key-query-value projection with Wasserstein distance scaling as √(log(β+1)/β) exp(Ct) + exp(-ct).