Giuseppe Bruno, Shi Chen, Zhengjiang Lin, Yury Polyanskiy, Philippe Rigollet
We study the long-context limit of softmax self-attention with a fixed query and a random context of $n$ i.i.d. keys on the sphere with the intrinsic dimension of the key distribution held fixed, viewing the inverse temperature $β_n$ as the scaling parameter that decides whether attention degenerates into uniform averaging or collapses onto the single closest key. We show that the critical scale at which selectivity emerges is determined by the local exponent of the distance-to-query distribution near zero rather than by global features of the context, and scales like $β_n^\ast \asymp n^{2/(d-1)}$ for uniform keys on $\mathbb{S}^{d-1}$. Furthermore, we characterize the limiting laws of the ordered attention weights and of the attention output across all regimes of $β_n$: a subcritical regime in which the output reduces to a local average around $q$ with explicit deterministic bias and Gaussian fluctuations; a critical regime in which a finite collection of nearest keys retains macroscopic mass without single-key collapse; and a supercritical regime in which all mass concentrates on the closest key. Of notable interest is the subcritical case with identity value matrix where the attention map approximately implements a backward heat equation.
Krishnakumar Balasubramanian, Sayan Banerjee, Philippe Rigollet
We study stationary solutions of McKean-Vlasov equations on the circle. We establish an exact equivalence between solutions of the stationary McKean-Vlasov equation and an infinite-dimensional quadratic system of equations over Fourier coefficients, allowing explicit characterization of stationary states in a sequence space rather than a function space. This framework provides a transparent description of local bifurcations, their periodicity and resonance structures, while accommodating singular potentials. We derive analytic expressions for the emergence, form and shape (supercritical, critical, subcritical or transcritical) of bifurcations involving possibly multiple Fourier modes and connect them with discontinuous phase transitions. We also characterize, under suitable assumptions, the detailed structure of the stationary bifurcating solutions through leading small-amplitude coefficients for arbitrarily many fixed Fourier modes. At the global level, we establish regularity and concavity properties of the free energy landscape, proving existence, compactness, and coexistence of globally minimizing stationary measures, and identifying discontinuous phase transitions with points of non-differentiability of the minimum free energy map. We apply the theory to the Noisy Mean-Field Transformer model, where we show how changing the inverse temperature $β$ affects the geometry of the infinitely many bifurcations from the uniform measure. We also discuss how increasing $β$ motivates the study of approximate multi-mode stationary profiles as candidates for a separate metastability analysis; persistence times require additional dynamical estimates. Moreover, we determine the exact threshold $β_0\simeq2.447$ separating continuous and discontinuous phase transitions in this model. The same threshold marks the change of the first bifurcation from supercritical to subcritical.
Learning developmental dynamics from lineage context
Gokul Gowri, Nic Fishman, William N. Colgan, Minming Wang, Yusuke Kijima, Sachin A. Kumar, Aleksandr Zimin, Luke W. Koblan 等 13 位作者
AI 速览：作者提出“谱系修补”（lineage inpainting）生成建模方法，利用树结构Transformer与随机插值从谱系上下文预测细胞状态动态。
A central goal of biology is to predict the dynamics of multicellular systems. Lineage context (the states of a cell's sisters, cousins, etc.) could inform these predictions, but how best to leverage it remains unclear. Here we introduce lineage inpainting, a generative modeling approach that uses lineage context with tree-structured transformers and stochastic interpolants to model cell state dynamics across lineage trees. Applied to a lineage-traced transcriptional atlas of mouse embryogenesis, it recovers dynamics not resolved by gene expression alone, including multiple transcriptional convergence events where distinct lineages are drawn to a common state. Despite being fit on unperturbed development, lineage inpainting can generate hypotheses about perturbation responses by simulating counterfactuals. These simulations agree in many cases with observed developmental phenotypes, including a single-cell measurement of the Tbx5 knockout response. Our results show that lineage information provides a rich source of context for learning multicellular dynamics.