Functional Attention: Reinterpreting Attention as Functional Correspondences for Operator Learning
This paper introduces Functional Attention, a novel attention mechanism for operator learning that replaces standard softmax token-wise affinities with structured linear operators inspired by geometric functional maps. The approach treats attention as a correspondence between adaptive bases rather than discrete tokens, yielding a resolution-invariant, globally-aware representation. Experiments show competitive or state-of-the-art performance on PDE solving, 3D segmentation, and regression tasks, with robustness to varying discretizations.
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Lie-Algebra Attention: tokens as bare matrix Lie group elements with closed-form geometric scores
A new arXiv preprint introduces Lie-Algebra Attention, an attention mechanism where tokens are elements of a matrix Lie group rather than feature vectors, with pairwise attention scores computed as the closed-form algebra norm of the relative pose (log of the group inverse product). The construction achieves equivariance tautologically without representation-theoretic machinery such as irreps, spherical harmonics, or Clebsch-Gordan products, and extends to non-compact affine groups that existing methods cannot handle. Experiments on SE(2), SO(3), and Aff(2) sequence-completion tasks show the closed-form score matches or outperforms learned MLP kernels while using 50–80x fewer score parameters.
Positional vs. Symbolic Attention Heads: Learning Dynamics, RoPE Geometry, and Length Generalization
Researchers train a decoder-only Transformer (GPT-J) on two structurally equivalent multi-hop reasoning tasks to study how attention heads specialize into positional or symbolic roles during learning. They find that successful task learning correlates with the emergence of 'pure' heads—exclusively positional or symbolic—and provide theoretical constructions showing how single-layer RoPE-based attention realizes these functions geometrically. A novel 'discrepancy' metric formalizes the robustness difference between the two head types, with symbolic mechanisms shown to extrapolate more reliably to longer sequences than positional ones. The findings have implications for understanding length generalization failures in RoPE-based models.
ATLAS: Unified Agentic and Latent Visual Reasoning via Functional Tokens
ATLAS proposes a framework where a single discrete 'functional token' serves dual roles as both an agentic operation trigger and a latent visual reasoning unit in multimodal models. This design avoids the computational cost of generating intermediate images while sidestepping the context-switching latency of external tool calls and the generalization limitations of pure latent methods. The framework is compatible with standard SFT and RL training pipelines without architectural changes, and introduces Latent-Anchored GRPO (LA-GRPO) to stabilize reinforcement learning when functional tokens are sparse. Experiments show strong performance on visual reasoning benchmarks with maintained interpretability.
ConSA: Learned FA/SWA allocation for efficient hybrid attention in LLMs
ConSA is a framework that learns optimal assignments between full attention and sliding-window attention layers under a user-specified sparsity target, using L0 regularization and augmented Lagrangian constraints. Evaluated on 0.6B and 1.7B parameter models, learned allocations consistently outperform hand-crafted rule-based baselines, with KV-head-wise granularity outperforming layer-wise. A consistent structural pattern emerges: SWA concentrates in bottom layers while FA clusters in contiguous middle-layer blocks, diverging from the evenly interleaved patterns used in existing hybrid architectures.
Neuronal Stochastic Attention Circuit (NSAC) for Probabilistic Representation Learning
Researchers introduce NSAC, a biologically-inspired continuous-time attention architecture that models attention logits as solutions to an Ornstein-Uhlenbeck stochastic differential equation, drawing on C. elegans Neuronal Circuit Policy wiring to induce Gaussian distributions over attention weights. The architecture enables joint quantification of aleatoric and epistemic uncertainty via a two-term objective combining Gaussian negative log-likelihood with an epistemic-separation regularizer. Empirical evaluation spans irregular time-series function approximation, multivariate regression, long-range forecasting, Industry 4.0 tasks, and autonomous vehicle lane-keeping, showing competitive accuracy with well-calibrated uncertainty estimates.
Express: Efficient causal attention approximation with formal guarantees and FlashAttention 2 speedups
A new tool called Express converts non-causal attention approximations into causal ones with matching theoretical guarantees, achieving log^(3/2)(n)/s approximation error with O(s) memory. Combined with the Thinformer approximation and an I/O-aware Triton implementation, it demonstrates substantial speedups over FlashAttention 2. The work targets four practical bottlenecks: long-context prefill, KV cache compression, and both memory- and compute-constrained long-form decoding.
Nyströmformer: Approximating Self-Attention in Linear Time and Memory via the Nyström Method
This Hugging Face blog post covers Nyströmformer, a transformer variant that approximates standard self-attention using the Nyström method to achieve linear time and memory complexity. The approach addresses the quadratic scaling bottleneck of standard attention, enabling processing of longer sequences at reduced computational cost. The post likely covers the model's integration into the Hugging Face ecosystem and its practical use cases.
HydraHead: Head-level hybridization of full and linear attention for long-context efficiency
Researchers introduce HydraHead, an architecture that hybridizes Full Attention (FA) and Linear Attention (LA) at the head level rather than the conventional layer level, motivated by interpretability findings showing functional heterogeneity among heads within the same layer. An interpretability-driven selection strategy preserves FA only for retrieval-critical heads, achieving a 7:1 LA-to-FA ratio while matching the long-context performance of a 3:1 layer-wise hybrid. Trained on only 15B tokens, HydraHead achieves over 69% improvement over the baseline at 512K context length, approaching Qwen3.5's performance despite that model having a native 256K context window. The work suggests head-level hybridization is a significantly underexplored and high-potential design axis for efficient long-context models.