Transformer Attention

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Transformer attention uses queries, keys, and values to decide which tokens should contribute information to one another, with positional encodings and masks shaping those interactions. Work seeks both to improve this basic mechanism and to reduce its costs, using approaches such as sparse, linear, kernel, and local attention, as well as efficient GPU kernels and compressed key-value caches. Researchers also add structural or domain-specific biases and study attention’s role in learning, retrieval, stability, and interpretability across language, vision, graphs, and time series. A central tension remains: methods that reduce computation or constrain interactions can improve speed and memory use, but may lose accuracy or long-range retrieval quality, and no single design is best across settings.

Latest papers 308

41 papers in the last four weeks, up 86% on the four weeks before. 0.2% of all new papers.

May 9, 2026

Kinetic theory for Transformers and the lost-in-the-middle phenomenon

We study causal self-attention dynamics -- a toy model for decoder Transformers -- which we interpret as a non-exchangeable interacting particle system. Adapting cumulant expansions to the triangular causal dependency structure of the model, and appealing to non-hierarchical methods to estimate correlations using Glauber calculus, we prove a quantitative mean-field limit result and a next-order characterization of correlations. For iid uniformly distributed tokens, the limiting correlation equation can be solved in closed form and we obtain a rigorous explanation of the empirically observed lost-in-the-middle phenomenon: the token retrieval profile, as a function of the source position in the prompt, is

\mathsf{U}

-shaped, with primacy, recency, and a unique interior minimum under an explicit smallness condition.

Mitia Duerinckx, Borjan Geshkovski, Stefano Rossi

Self-Attention Transformer Attention Mean-Field Theory

May 9, 2026

When and Why Grouping Attention Heads Accelerates Muon Optimization

Muon orthogonalizes matrix updates, but multi-head attention naturally operates at the level of heads. This granularity mismatch raises the question of whether Muon should be applied to the full attention projection, to individual heads, or to intermediate head groups. We study this question through a one-step descent comparison between full-matrix Muon and group-wise Muon. Our analysis reveals a trade-off between the group-wise whitening gain from group-wise updates and the grouping-induced norm cost, an additional update-norm cost caused by replacing full-matrix whitening with group-wise whitening. Motivated by this trade-off, we propose Group Muon, which treats head group size and grouping rule as optimizer hyperparameters. On GPT-2 Small trained on FineWeb, appropriate grouping improves validation loss over both full-QKV Muon and fully head-wise MuonSplit.

Hongtao Zhang, Wenjie Zhou, Wei Chen +1

State Key Laboratory of AI Safety, Institute of Computing Technology, Chinese Academy of Sciences · School of Advanced Interdisciplinary Sciences, University of Chinese Academy of Sciences · University of Chinese Academy of Sciences

Attention Head Analysis Muon Optimizer Transformer Attention Multi-Head Attention

May 8, 2026

Sink vs. diagonal patterns as mechanisms for attention switch and oversmoothing prevention

This paper studies the role of sinks and diagonal patterns as attention switch and anti-oversmoothing mechanisms. We analyze geometric conditions under which sinks can be represented, showing a necessary alignment between the embedding of the sink and all other embeddings. Next, we refine the current understanding of the role of sinks in oversmoothing prevention: we specify the conditions under which dense attention provably smooths more than sparse attention, and empirically verify that such conditions are often satisfied in practice. We further prove an equivalence between sinks and hard attention switch, in which the output of the attention is identically 0. Finally, we relax the hard attention switch by allowing token self-communication: we provide a quantitative comparison of the costs of representing sinks vs.\ diagonal patterns, showing why sinks are favored in pretrained transformers. The introduction and analysis of diagonal patterns and the generalization of the attention switch close the gap between what oversmoothing prevention requires and what sinks provide, while also establishing when and why attention layers act like MLPs if token communication is not necessary.

Peter Súkeník, Cristina López Amado, Christoph H. Lampert +1

Institute of Science and Technology (ISTA) Austria

Self-Attention Transformer Attention Attention Sinks Attention Mechanisms

May 8, 2026

Convergent Stochastic Training of Attention and Understanding LoRA

Transformers have revolutionized machine learning and deploying attention layers in the model is increasingly standard across a myriad of applications. Further, for large models, it is common to implement Low Rank Adaptation (LoRA), whereby a factorized parameterization of them is trained, to achieve a surprisingly beneficial accuracy-size trade-off. In this work, via a unified framework we rigorously establish trainability of such models under stochastic methods. We prove that for any mild regularization, the empirical regression loss on a attention layer and LoRA on a shallow neural net, both induce Poincaré inequality for the corresponding Gibbs' measure. Then it follows via invoking recent results that a certain SDE, which mimics the SGD, minimizes the corresponding losses. In both the cases, our first-of-its-kind results of trainability on attention and nets, do not rely on any assumptions on the data or the size of the architecture.

Zhengkai Sun, Dibyakanti Kumar, Alejandro F Frangi +2

Stochastic Optimization Low-Rank Adaptation Transformer Attention Neural Network Training Dynamics

May 8, 2026

Training-Induced Escape from Token Clustering in a Mean-Field Formulation of Transformers

Transformers perform inference by iteratively transforming token representations across layers. This layerwise computation has been studied empirically, and recent mean-field theories of Transformer dynamics explain how attention can drive token distributions toward clustering. However, existing mean-field analyses largely treat model parameters as prescribed, leaving open how training reshapes this clustering picture. We study this question in a noisy mean-field Transformer in which only a parameter-linear FFN is trained under

L^2

regularization. We find and analyze a training-induced phase in the dynamics: after initially following attention-driven clustering, the token distribution can leave the clustered regime near the final layers. Our mathematical analysis is based on an entropy-regularized interaction energy that captures the clustering bias of attention. More broadly, our results point toward a training-aware mean-field theory of Transformer dynamics, in which training and inference dynamics are treated together.

Noboru Isobe, Daisuke Inoue, Masaaki Imaizumi

1RIKEN Center for Advanced Intelligence Project · 2Imperial College London · 3The University of Tokyo +1

Transformer Clustering Transformer Attention Mean-Field Theory Neural Network Training Dynamics

May 8, 2026

Beyond Linear Attention: Softmax Transformers Implement In-Context Reinforcement Learning

In-context reinforcement learning (ICRL) studies agents that, after pretraining, adapt to new tasks by conditioning on additional context without parameter updates. Existing theoretical analyses of ICRL largely rely on linear attention, which replaces the softmax function in the standard attention with an identity mapping. This paper provides the first theoretical understanding of ICRL without making the unrealistic linear attention simplification. In particular, we consider the standard softmax attention used in practice. We show that, with certain parameters, the layerwise forward pass of a Transformer with such softmax attention is equivalent to iterative updates of a weighted softmax temporal difference (TD) learning algorithm. Here, weighted softmax TD is a new RL algorithm that performs policy evaluation in kernel space and adopts both linear TD and tabular TD as special cases. We also prove that under a certain contraction condition, the policy evaluation error decays as the number of layers grows, with the identified parameters above. Finally, we prove that those parameters are a global minimizer of a pretraining loss, explaining their emergence in our numerical experiments.

Zixuan Xie, Xinyu Liu, Claire Chen +3

University of Virginia · California Institute of Technology · Purdue University

Reinforcement Learning Transformer-Based RL Temporal-Difference Learning Transformer Attention In-Context RL

May 7, 2026

Transformers Efficiently Perform In-Context Logistic Regression via Normalized Gradient Descent

Transformers have demonstrated remarkable in-context learning (ICL) capabilities. The strong ICL performance of transformers is commonly believed to arise from their ability to implicitly execute certain algorithms on the context, thereby enhancing prediction and generation. In this work, we investigate how transformers with softmax attention perform in-context learning on linear classification data. We first construct a class of multi-layer transformers that can perform in-context logistic regression, with each layer exactly performing one step of normalized gradient descent on an in-context loss. Then, we show that our constructed transformer can be obtained through (i) training a single self-attention layer supervised by one-step gradient descent, and (ii) recurrently applying the trained layer to obtain a looped model. Training convergence guarantees of the self-attention layer and out-of-distribution generalization guarantees of the looped model are provided. Our results advance the theoretical understanding of ICL mechanism by showcasing how softmax transformers can effectively act as in-context learners.

Chenyang Zhang, Yuan Cao

School of Computing & Data Science, The University of Hong Kong

Logistic Regression Self-Attention Gradient Descent Transformer Attention In-Context Learning

May 7, 2026

Cubit: Token Mixer with Kernel Ridge Regression

Since its introduction in 2017, the Transformer has become one of the most widely adopted architectures in modern deep learning. Despite extensive efforts to improve positional encoding, attention mechanisms, and feed-forward networks, the core token-mixing mechanism in Transformers remains attention. In this work, we show that the attention module in Transformers can be interpreted as performing Nadaraya-Watson regression, where it computes similarities between tokens and aggregates the corresponding values accordingly. Motivated by this perspective, we propose Cubit, a potential next-generation architecture that leverages Kernel Ridge Regression (KRR), while the vanilla Transformer relies on Nadaraya-Watson regression. Specifically, Cubit modifies the classical attention computation by incorporating the closed-form solution of KRR, combining value aggregation through kernel similarities with normalization via the inverse of the kernel matrix. To improve the training stability, we further propose the Limited-Range Rescale (LRR), which rescales the value layer within a controlled range. We argue that Cubit, as a KRR-based architecture, provides a stronger mathematical foundation than the vanilla Transformer, whose attention mechanism corresponds to Nadaraya-Watson regression. We validate this claim through comprehensive experiments. The experimental results suggest that Cubit may exhibit stronger long-sequence modeling capability. In particular, its performance gain over the Transformer appears to increase as the training sequence length grows.

Chuanyang Zheng, Jiankai Sun, Yihang Gao +6

Kernel Ridge Regression Transformer Attention Long-Context Modeling

May 7, 2026

FedFrozen: Two-Stage Federated Optimization via Attention Kernel Freezing

Federated learning with heterogeneous clients remains a significant challenge for deep learning, primarily due to client drift arising from inconsistent local updates. Existing federated optimization methods typically address this issue through objective-level regularization or update-correction mechanisms. Recent studies, however, suggest that Transformer-based architectures may be inherently more robust than conventional models under heterogeneous federated training. Motivated by this observation, we investigate how different parameter components within the attention mechanism influence federated optimization. Specifically, we decompose the attention module into a query/key block, which determines the attention kernel, and a value block, which performs semantic transformation under the induced kernel. Based on this perspective, we propose FedFrozen, a two-stage federated optimization framework that first performs full-model warm-up training and then freezes the query/key block while continuing to optimize the value block. Under a linear-attention formulation, we show that the warm-up stage can be interpreted as an inexact descent procedure on a regularized kernel-profile objective, while the frozen stage reduces to a restricted value-block optimization problem under a fixed attention kernel. Our analysis further reveals an explicit trade-off that governs the choice of warm-up length. Simulations validate the predicted bias-drift behavior, and real-data experiments demonstrate that FedFrozen improves both the stability and effectiveness of Transformer models in heterogeneous federated learning.

Junye Du, Zhenghao Li, Yushi Feng +1

The University of Hong Kong

Transformer Attention Federated Learning Heterogeneous Federated Learning

May 6, 2026

Understanding In-Context Learning for Nonlinear Regression with Transformers: Attention as Featurizer

Pre-trained transformers are able to learn from examples provided as part of the prompt without any weight updates, a remarkable ability known as in-context learning (ICL). Despite its demonstrated efficacy across various domains, the theoretical understanding of ICL is still developing. Whereas most existing theory has focused on linear models, we study ICL in the nonlinear regression setting. Through the interaction mechanism in attention, we explicitly construct transformer networks to realize nonlinear features, such as polynomial or spline bases, which span a wide class of functions. Based on this construction, we establish a framework to analyze end-to-end in-context nonlinear regression with the constructed features. Our theory provides finite-sample generalization error bounds in terms of context length and training set size. We numerically validate the theory on synthetic regression tasks.

Alexander Hsu, Zhaiming Shen, Wenjing Liao +1

Department of Mathematics, Purdue University · School of Mathematics, Georgia Institute of Technology

Transformer Attention In-Context Learning Statistical Learning Theory

May 6, 2026

CuBridge: An LLM-Based Framework for Understanding and Reconstructing High-Performance Attention Kernels

Efficient CUDA implementations of attention mechanisms are critical to modern deep learning systems, yet supporting diverse and evolving attention variants remains challenging. Existing frameworks and compilers trade performance for flexibility, while expert-written kernels achieve high efficiency but are difficult to adapt. Recent work explores large language models (LLMs) for GPU kernel generation, but prior studies report unstable correctness and significant performance gaps for complex operators such as attention. We present CuBridge, an LLM-based framework that adapts expert-written attention kernels through a structured lift-transfer-lower workflow. CuBridge starts from expert-written CUDA attention kernels and lifts them into an executable intermediate representation that makes execution orchestration explicit while abstracting low-level CUDA syntax. Given a user-provided PyTorch specification, CuBridge generates and verifies a target IR program, then reconstructs optimized CUDA code via reference-guided lowering. Across diverse attention variants and GPU platforms, CuBridge consistently produces correct kernels and substantially outperforms general frameworks, compiler-based approaches, and prior LLM-based methods.

Xing Ma, Yangjie Zhou, Wu Sun +6

Shanghai Jiao Tong University · National University of Singapore · Shanghai Qi Zhi Institute

GPU Kernel Optimization GPU Acceleration Self-Attention Large Language Model-Guided Optimization LLM-Based Program Synthesis Transformer Attention+1

May 6, 2026

HEXST: Hexagonal Shifted-Window Transformer for Spatial Transcriptomics Gene Expression Prediction

Spatial transcriptomics offers spatially resolved gene expression profiling within tissue sections, but its cost and limited throughput hinder large-scale deployment. To extend this capability to routine practice, recent computational methods aim to infer spatial gene expression directly from ubiquitous hematoxylin and eosin-stained histology slides. However, most existing models assume Cartesian or geometry-agnostic locality, despite the hexagonal sampling of widely used spot-array platforms, and point-wise regression objectives often yield over-smoothed gene expression profiles, obscuring gene-specific spatial heterogeneity. To address these, we propose HEXST, a geometry-aligned Transformer for spatial gene expression prediction from histology. HEXST operates directly on hexagonal spot coordinates to enable efficient local-to-global contextual modeling via tailored shifted-window attention mechanism and hexagonal rotary positional encoding. To enhance gene-wise spatial contrast, HEXST complements point-wise regression with a contrast-sensitive differential objective and transcriptomic priors from a pretrained single-cell foundation model during training. Across seven spatial transcriptomics datasets, HEXST consistently outperforms state-of-the-art models, providing accurate and robust spatial gene expression predictions while preserving gene-wise contrast and spatial heterogeneity.

Keunho Byeon, Jin Tae Kwak

School of Electrical Engineering, Korea University, Seoul

Computational Pathology Spatial Transcriptomics Transformer Attention Spatial Gene Expression Prediction

May 6, 2026

FLUID: Continuous-Time Hyperconnected Sparse Transformer for Sink-Free Learning

Continuous-time (CT) Transformers improve irregular and long-range modeling over CT-RNNs by exploiting inputs or outputs embeddings with continuous dynamics. However, the core scaled-dot-product-attention (SDPA) mechanism remains inherently discrete. We propose FLUID (Flexible Unified Information Dynamics), a CT Transformer that incorporates continuous dynamics directly into the attention computation by replacing it with Liquid Attention Network (LAN). LAN reinterprets attention logits as continuous dynamical system and reformulates them as the solution to a linear ODE modulated by input-dependent nonlinear recurrent gates. Theoretically, we establish stability guarantees for LAN dynamics and show that it serves as an interpolating middle ground between SDPA and CT-RNNs, recovering each as special case under well-defined parameterization of its gating functions. LAN also introduces an explicit attention-sink gate to eliminate disproportionate attention mass on uninformative nodes. FLUID replaces standard residual connections with input-dependent Liquid Hyper-Connections to adaptively regulate interlayer information flow. Empirically, we evaluate FLUID on a broad set of learning tasks, including (i) irregular time-series, (ii) long-range modeling, (iii) lane-keeping control of autonomous vehicles, and (iv) learning physical dynamics under a scarce data regime. Across all the tasks, FLUID consistently matches or outperforms CT baselines, achieving improvements of up to 47% in certain scenarios and enhancing generalization under distributional shifts. Additionally, FLUID demonstrates superior noise robustness and a self-correcting inductive bias in autonomous vehicle control. We also provide a detailed analysis of key hyperparameters to guide tuning and show that FLUID occupies an intermediate position among competing approaches in terms of runtime and memory efficiency.

Waleed Razzaq, Yun-Bo Zhao

School of Automation University of Science and Technology China Hefei, Anhui

Transformer Transformer Attention Attention Mechanisms Time Series Transformers

May 5, 2026

Jordan-RoPE: Non-Semisimple Relative Positional Encoding via Complex Jordan Blocks

Relative positional encodings determine which functions of query-key lag can enter the primitive attention logit. RoPE supplies a rotary phase, while ALiBi supplies an additive distance bias. Motivated by group-theoretic views of linear translation-invariant positional encodings, we study a non-semisimple case in which a complex rotary eigenvalue and a nilpotent response live in the same defective Jordan block. The resulting relative operator generates oscillatory-polynomial features such as

e^{-γd}\cos(ωd)

,

e^{-γd}\sin(ωd)

,

d e^{-γd}\cos(ωd)

, and

d e^{-γd}\sin(ωd)

, for causal lag

d=i-j\geq 0

. Thus the construction realizes a distance-modulated phase basis

d e^{iωd}

, rather than merely adding a separate distance channel to RoPE. We formulate Exact Jordan-RoPE as a non-semisimple one-parameter representation, give its real block form, and specify the contragredient query action required by non-orthogonal positional maps. We also distinguish this exact representation from stabilized variants whose bounded shear improves numerical behavior but breaks the exact group law. Kernel-level diagnostics and a Jordan-friendly synthetic language-model task show that the coupled Jordan basis is useful when the target contains distance-modulated phase interactions. On a small WikiText-103 byte language model, a scaled-exact variant improves over RoPE and direct-sum baselines within the Jordan family, while RoPE+ALiBi remains strongest overall. The evidence is structural rather than a broad performance claim.

Yaobo Zhang

School of Physics, Ningxia University Yinchuan, 750021, China

Rotary Positional Embeddings Self-Attention Transformer Attention Positional Encoding

May 5, 2026

Transformers with Selective Access to Early Representations

Several recent Transformer architectures expose later layers to representations computed in the earliest layers, motivated by the observation that low-level features can become harder to recover as the residual stream is repeatedly transformed through depth. The cheapest among these methods add static value residuals: learned mixing coefficients that expose the first-layer value projection V_1 uniformly across tokens and heads. More expressive dense or dynamic alternatives recover finer-grained access, but at higher memory cost and lower throughput. The usefulness of V_1 is unlikely to be constant across tokens, heads, and contexts; different positions plausibly require different amounts of access to early lexical or semantic information. We therefore treat early-representation reuse as a retrieval problem rather than a connectivity problem, and introduce Selective Access Transformer (SATFormer), which preserves the first-layer value pathway while controlling access with a context-dependent gate. Across models from 130M to 1.3B parameters, SATFormer consistently improves validation loss and zero-shot accuracy over the static value-residual and Transformer baselines. Its strongest gains appear on retrieval-intensive benchmarks, where it improves over static value residuals by approximately 1.5 average points, while maintaining throughput and memory usage close to the baseline Transformer. Gate analyses suggest sparse, depth-dependent, head-specific, and category-sensitive access patterns, supporting the interpretation that SATFormer learns selective reuse of early representations rather than uniform residual copying. Our code is available at https://github.com/SkyeGunasekaran/SATFormer.

Skye Gunasekaran, Téa Wright, Rui-Jie Zhu +1

UC Santa Cruz · UC Berkeley

Transformer Residual Learning Transformer Attention

May 4, 2026

Cascade Token Selection for Transformer Attention Acceleration

A method is presented for reducing the cost of representative token selection in transformer attention layers by exploiting the coherence of the representative set across depth. Activation Decorrelation Attention (ADA) selects

r \ll T

representative tokens at each layer via a Gram threshold and computes attention on the compressed

r \times r

problem, but the selection requires a

T \times T

Gram matrix at every layer. The cascade mechanism introduced here inherits the representative set from layer

l

to layer

l+1

, validates it via a

(T - r) \times r

cross-Gram computation, and updates it with a small number of additions and removals. The cost of the selection step drops from

O(T^2 d)

to

O(T r d)

per layer. Validation on three model families (GPT-2 124M, GPT-J 6B, OPT 6.7B) on AMD MI300X demonstrates Gram operation savings of

22\%

to

63\%

with mean Jaccard overlap of

0.83

to

0.94

between consecutive layers. The cascade reveals that the set of informative tokens is a structural property of the input that propagates coherently through the depth of the network: the same tokens carry the non-redundant information at layer

l

and at layer

l+1

.

Stephen J. Thomas

Department of Computer Science and Engineering, Lehigh University, Bethlehem, PA 18015

Self-Attention Transformer Attention Efficient Attention

May 4, 2026

Projection-Free Transformers via Gaussian Kernel Attention

Self-attention in Transformers is typically implemented as

\mathrm{softmax}(QK^\top/\sqrt{d})V

, where

Q=XW_Q

,

K=XW_K

, and

V=XW_V

are learned linear projections of the input

X

. We ask whether these learned projections are necessary, or whether they can be replaced by a simpler similarity-based diffusion operator. We introduce Gaussian Kernel Attention (GKA), a drop-in replacement for dot-product attention that computes token affinities directly using a Gaussian radial basis function (RBF) kernel applied to per-head token features. Each head learns only a bandwidth parameter

σ_h

, while a single output projection

W_O

preserves compatibility with the standard Transformer interface. GKA can be interpreted as normalized kernel regression over tokens, linking modern Transformer architectures to classical non-local filtering and kernel smoothing methods. We evaluate GKA in both vision and language modeling settings. For autoregressive language modeling within the nanochat framework, we implement causal masking and sliding-window constraints by masking and renormalizing the Gaussian kernel. At depth 20, a GKA model with

0.42\times

the parameters and

0.49\times

the total training FLOPs of a standard attention baseline trains stably, exhibits a near-zero train-validation gap, and demonstrates competitive behavior on standard benchmarks, albeit with higher bits-per-byte (BPB) at this compute scale. Overall, GKA provides a minimal, interpretable attention mechanism with an explicit locality scale, offering a dimension in the accuracy-efficiency trade-off for Transformer design.

Debarshi Kundu, Archisman Ghosh, Swaroop Ghosh +1

The Pennsylvania State University, University Park, USA

Transformer Self-Attention Transformer Attention Attention Mechanisms

May 3, 2026

Linear-Time Global Visual Modeling without Explicit Attention

Existing research largely attributes the global sequence modeling capability of Transformers to the explicit computation of attention weights, a process that inherently incurs quadratic computational complexity. In this work, we offer a novel perspective: we demonstrate that attention can be mathematically reframed as a Multi-Layer Perceptron (MLP) equipped with dynamically predicted parameters. Through this lens, we explain attention's global modeling power not as explicit token-wise aggregation, but as an implicit process where dynamically generated parameters act as a compressed representation of the global context. Inspired by this insight, we investigate a fundamental question: can we achieve Transformer-level sequence global modeling entirely through dynamic parameterization while maintaining linear complexity, effectively replacing explicit attention? To explore this, we design various dynamic parameter prediction strategies and integrate them into standard network layers. Extensive empirical studies on vision models demonstrate that dynamic parameterization can indeed serve as a highly effective, linear-complexity alternative to explicit attention, opening new pathways for efficient sequence modeling. Code is available at https://github.com/LeapLabTHU/WeightFormer.

Ruize He, Dongchen Han, Gao Huang

Tsinghua University

Transformer Transformer Attention Efficient ViTs Conditional Computation

May 2, 2026

Focus and Dilution: The Multi-stage Learning Process of Attention

Transformer-based models have achieved remarkable success across a wide range of domains, yet our understanding of their training dynamics remains limited. In this work, we identify a recurrent focus-dilution cycle in attention learning and provide a rigorous explanation in a one-layer Transformer setting for Markovian data via gradient-flow analysis. Using stage-wise linearization around critical points, we show that a single focus-dilution cycle can be decomposed into a sequence of distinct stages. First, embedding and projection rapidly condense to a rank-one structure, while attention parameters remain effectively frozen. Then, the attention parameters begin to increase, inducing a frequency-driven focus toward high-frequency tokens. As attention continues to evolve, it generates next-order perturbations in embeddings, leading to a mass-redistribution mechanism that progressively dilutes this focus. Finally, small asymmetries among low-frequency tokens lift a degenerate critical point, opening new embedding directions and initiating the next cycle. Experiments on synthetic Markovian data as well as WikiText and TinyStories corroborate the predicted stages and cyclical dynamics.

Zheng-An Chen, Pengxiao Lin, Zhi-Qin John Xu +1

School of Mathematical Sciences, Shanghai Jiao Tong University. · Institute of Natural Sciences, Shanghai Jiao Tong University. · MOE-LSC, Shanghai Jiao Tong University. +2

Self-Attention Transformer Attention Neural Network Training Dynamics

May 1, 2026

Spiking Sequence Machines and Transformers

Sequence learning reduces to similarity-based retrieval over a temporally indexed representation space, a constraint on any sequence model, not a property of a specific architecture. We show that a spiking Sparse Distributed Memory sequence machine (2007) and the transformer (2017) independently instantiate the same five functional operations (encoding, context maintenance, associative retrieval, storage, and decoding), with cosine similarity as the shared retrieval primitive in both. We formalise a Phase-Latency Isomorphism showing that sinusoidal positional phase and spike timing are linearly related, and prove that dot product attention is invariant to this mapping up to a global scale factor on the positional component (Lemma 1). Empirically, frequency-compressed positional encoding fails to converge on a positionally demanding copy task, while a learned rank-based embedding matches or exceeds sinusoidal encoding, indicating that the critical property for positional representation is distance discriminability under dot-product similarity, not sinusoidal form. Time, phase, and rank are three instantiations of the same computational primitive, an ordered index whose structure survives similarity-based retrieval.

Joy Bose

Independent Researcher Bengaluru, India

Transformer Attention Spiking Neural Networks Positional Encoding Sequence Modeling

Apr 29, 2026

Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models

We prove pathwise convergence of the layerwise evolution of tokens in a finite-depth, finite-width transformer model with MultiLayer Perceptron (MLP) blocks to a continuous-time stochastic interacting particle system. We also identify the stochastic partial differential equation describing the evolution of the tokens' distribution in this limit and prove propagation of chaos when the number of such tokens is large. The bounds we establish are quantitative and the limits we consider commute. We further prove that the limiting stochastic model displays synchronization by noise and establish exponential dissipation of the interaction energy on average, provided that the common noise is sufficiently coercive relative to the deterministic self-attention drift. We finally characterize the activation functions satisfying the former condition.

Andrea Agazzi, Giuseppe Bruno, Eloy Mosig García +2

Department of Mathematics and Statistics, University of Bern (CH) · Department of Mathematics, University of Pisa (IT)

Dynamical Systems Transformer Transformer Attention

Apr 28, 2026

Block-Wise Differentiable Sinkhorn Attention: Tail-Refinement Gradients with a Gap-Aware Dustbin Bridge

We study long-context balanced entropic optimal transport (OT) attention on TPU hardware through a stopped-base, fixed-depth tail-refinement surrogate. After a stopped

T

-step Sinkhorn solve, we unroll a short refinement tail and differentiate that surrogate exactly. For the reported

R=2

TPU path, the backward pass contains four staircase plan factors. We prove an exact one-reference-tile schedule: the

R=2

score cotangent is a single reference plan tile times an explicit modifier field built from vector cotangents and dual differences. This yields block-wise cost

O((T+R)LW)

,

O(Ld)

input storage, and

O(L)

additional HBM usage for fixed head dimension

d

and band width

W

on the balanced fixed-support path. We also formalize the current dustbin\_block path as the same unit-target surrogate on an augmented support, so the adjoint schedule lifts to the single-active-dustbin path used in our TPU runs; this bridge is algebraic and does not claim a general KL-unbalanced or arbitrary-capacity gap model. We provide a local surrogate-bias bound, an a posteriori bias certificate, and a projective contraction certificate for strictly positive active blocks. On synthetic masked problems, the optimized kernel matches exact autodiff of the same centered surrogate to within

10^{-5}

--

10^{-10}

. On TPU v6e-8, a four-configuration Pfam screen completes end-to-end, and a promoted balanced

R=2

run sustains roughly

8.5

examples per second through a three-hour budget, reaching step

1437

. Held-out Pfam test shards improve reconstruction from

5.57

to

2.05

and sparse CE from

5.53

to

5.30

relative to step

0

, with CE logged diagnostically rather than optimized directly; target-barycenter alignment metrics do not materially improve, and a deterministic diagonal reference remains stronger on those metrics.

Dylan Forde

Independent Researcher

Automatic Differentiation Transformer Attention Differentiable Programming Algorithm Unrolling Attention Mechanisms Optimal Transport

Apr 27, 2026

Kwai Summary Attention Technical Report

Long-context ability, has become one of the most important iteration direction of next-generation Large Language Models, particularly in semantic understanding/reasoning, code agentic intelligence and recommendation system. However, the standard softmax attention exhibits quadratic time complexity with respect to sequence length. As the sequence length increases, this incurs substantial overhead in long-context settings, leading the training and inference costs of extremely long sequences deteriorate rapidly. Existing solutions mitigate this issue through two technique routings: i) Reducing the KV cache per layer, such as from the head-level compression GQA, and the embedding dimension-level compression MLA, but the KV cache remains linearly dependent on the sequence length at a 1:1 ratio. ii) Interleaving with KV Cache friendly architecture, such as local attention SWA, linear kernel GDN, but often involve trade-offs among KV Cache and long-context modeling effectiveness. Besides the two technique routings, we argue that there exists an intermediate path not well explored: {Maintaining a linear relationship between the KV cache and sequence length, but performing semantic-level compression through a specific ratio

k

}. This

O(n/k)

path does not pursue a ``minimum KV cache'', but rather trades acceptable memory costs for complete, referential, and interpretable retention of long distant dependency. Motivated by this, we propose Kwai Summary Attention (KSA), a novel attention mechanism that reduces sequence modeling cost by compressing historical contexts into learnable summary tokens.

Chenglong Chu, Guorui Zhou, Guowang Zhang +35

Kuaishou

Self-Attention KV Caching Long-Context Language Model Inference Long-Context Language Modeling Transformer Attention Attention Mechanisms+1

Apr 26, 2026

Transformer as an Euler Discretization of Score-based Variational Flow

Despite the Transformer's dominance across machine learning, its architecture remains largely heuristic and lacks a unified theoretical foundation. We introduce Score-based Variational Flow (SVFlow), a continuous-time dynamical system for representation learning in which the state evolves according to a variational posterior-weighted average of conditional log-likelihood scores, and provide a principled basis for regularization through variational consistency. We show that forward Euler discretization of spherical SVFlow exactly recovers the Transformer architecture. Multi-head attention approximates SVFlow vector field via a vMF kernel-smoothed posterior, while MoE/FFN approximates it in a relaxed network-based way, and the residual-normalization block implements a relaxed retraction that maintains spherical geometry. This unification explains why attention trains stably without explicit regularization while MoE requires auxiliary balancing losses. Experiments on pre-trained language models with prefix shuffling show that SVFlow-induced metrics correlate with task performance, reveal depth-dependent sensitivity, and reflect the intrinsic dynamics of attention.

Huadong Liao

Representation Learning Transformer Self-Attention Decoder-Only Language Models Transformer Attention Attention Mechanisms

Apr 26, 2026

Rank, Head-Channel Non-Identifiability, and Symmetry Breaking: A Precise Analysis of Representational Collapse in Transformers

A widely cited result by Dong et al. (2021) showed that Transformers built from self-attention alone, without skip connections or feed-forward layers, suffer from rapid rank collapse: all token representations converge to a single direction. The proposed remedy was the MLP. We show that this picture, while correct in the regime studied by Dong, is incomplete in ways that matter for architectural understanding. Three results are established. First, layer normalisation is precisely affine-rank-neutral: it preserves the affine rank of the token representation set exactly. The widespread claim that LN "plays no role" is imprecise; the correct statement is sharper. Second, residual connections generically obstruct rank collapse in real Transformers such as BERT-base, in a measure-theoretic sense, without contribution from the MLP. The MLP's irreplaceable function is different: generating feature directions outside the linear span of the original token embeddings, which no stack of attention layers can produce. Third, a phenomenon distinct from rank collapse is identified: head-channel non-identifiability. After multi-head attention sums per-head outputs through the output projection, individual contributions cannot be canonically attributed to a specific head; n(H-1)d_k degrees of freedom per layer remain ambiguous when recovering a single head from the mixed signal. The MLP cannot remedy this because it acts on the post-summation signal. A constructive partial remedy is proposed: a position-gated output projection (PG-OP) at parameter overhead below 1.6% of the standard output projection. The four collapse phenomena identified in the literature -- rank collapse in depth, in width, head-channel non-identifiability, and entropy collapse -- are unified under a symmetry-breaking framework, each corresponding to a distinct symmetry of the Transformer's forward pass.

Giansalvo Cirrincione

aLaboratoire LTI, Université de Picardie Jules Verne, Chemin du Thil, 80025 Amiens, France

Transformer Interpretability Representation Collapse Parameter Identifiability Self-Attention Symmetry Breaking Transformer Attention+2

Apr 26, 2026

A Synergistic CNN-Transformer Network with Pooling Attention Fusion for Hyperspectral Image Classification

In the hyperspectral image (HSI) classification task, each pixel is categorized into a specific land-cover category or material. Convolutional neural networks (CNNs) and transformers have been widely used to extract local and non-local features in HSI classification. Recent works have utilized a multi-scale vision transformer (ViT) to enhance spectral feature capture and yield promising results. However, most existing methods still face challenges in the effective joint use of spatial-spectral information and in preserving information across layers during the propagation process. To address these issues, we propose a synergistic CNN-Transformer network with pooling attention fusion for HSI classification, which collaboratively utilizes CNNs and ViT to process spatial and spectral features separately. Specifically, we propose a Twin-Branch Feature Extraction (TBFE) module, which employs 3D and 2D convolution in parallel to comprehensively extract spectral and spatial features from HSI. A hybrid pooling attention (HPA) module is designed to aggregate spatial attention. Moreover, a cascade transformer encoder is employed for global spectral feature extraction, and a simple yet efficient cross-layer feature fusion (CFF) module is designed to reduce the loss of crucial information in the previous network layers. Extensive experiments are conducted on several representative datasets to demonstrate the superior performance of our proposed method compared to the state-of-the-art works. Code is available at https://github.com/chenpeng052/SCT-Net.git.

Peng Chen, Wenxuan He, Feng Qian +2

College of Engineering, Shantou University, Shantou, 515063, Guangdong, China · School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, China · Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun, 130033, Jilin, China +2

Hybrid CNN-Transformer Architectures Transformer Attention Hyperspectral Image Classification

Apr 24, 2026

Adaptive Head Budgeting for Efficient Multi-Head Attention

Multi-head attention enables Transformers to capture diverse representations, but all attention heads are typically activated for every input, regardless of task complexity. For coarse-grained tasks such as text classification, where relevant information is often global, this fixed allocation can introduce unnecessary computation. We propose BudgetFormer, a Transformer architecture that dynamically allocates attention heads on a per-input basis. The model learns both a head budget and a relevance distribution to select the most informative heads. To support effective head selection, we introduce a training strategy that balances exploration and exploitation. Experiments on text classification tasks show that BudgetFormer reduces FLOPs and memory usage while matching or surpassing the performance of standard multi-head attention. These results highlight adaptive head allocation as an effective approach to improving Transformer efficiency and performance.

Bilal Faye, Abdoulaye Mbaye, Hanane Azzag +1

Transformer Transformer Attention

Apr 24, 2026

HubRouter: A Pluggable Sub-Quadratic Routing Primitive for Hybrid Sequence Models

We introduce HubRouter, a pluggable module that replaces O(n^2) attention layers with O(nM) hub-mediated routing, where M << n is a small number of learned hub tokens. We demonstrate it in two from-scratch architectures: a Jamba-style hybrid and a 12-layer Transformer; retrofit into pretrained models is a tested negative case. HubRouter implements an encode-decode-score-council pipeline: M learned hubs cross-attend to all tokens, tokens project against hubs for routing fingerprints, a score head selects top-k tokens, and a sparse council attends only to the selected subset. We validate HubRouter in three settings. (1) Hub-Jamba yields a nominal 4.2% PPL improvement (200.2 vs 209.0, single seed; possibly within seed noise) and up to ~90x training throughput at sequence length 1024 in matched PyTorch-native baselines; an optimised baseline would narrow this to ~10-15x. (2) Graduated replacement of 25% of Transformer attention layers gives the best perplexity in our matched-budget sweep (268.0 vs 282.4 pure Transformer). (3) Hub-GPT provides strictly causal routing, achieving PPL 211.5 +/- 0.4 over 3 seeds (post council-causal fix); approximately 3 PPL worse than Jamba's 208.5 +/- 0.7, a measurable quality cost for avoiding O(n^2) computation. Post-fix, chunk size C has little effect; the pre-fix chunk-size benefit was an artifact of a bidirectional-council leak we found in adversarial review. A multi-seed hub-count sweep (~105 runs across M=1-32) reveals M=8-14 as the reliably-converging sub-band (4-5/5 seeds); M=6 is rescued to 5/5 by orthogonal regularization, while M>=20 shows increasing seed sensitivity. Companion paper arXiv:2603.20997 (Basu, 2026) defines the routing diagnostic task. Code and scripts will be released.

Abhinaba Basu

National Institute of Electronics and Information Technology (NIELIT)

Transformer Attention Language Modeling Sparse Attention Attention Mechanisms Sequence Modeling

Apr 24, 2026

Dissociating Decodability and Causal Use in Bracket-Sequence Transformers

When trained on tasks requiring an understanding of hierarchical structure, transformers have been found to represent this hierarchy in distinct ways: in the geometry of the residual stream, and in stack-like attention patterns maintaining a last-in, first-out ordering. However, it remains unclear whether these representations are causally used or merely decodable. We examine this gap in transformers trained on the Dyck language (a formal language of balanced bracket sequences), where the hierarchical ground truth is explicit. By probing and intervening on the residual stream and attention patterns, we find that depth, distance, and top-of-stack signals are all decodable, yet their causal roles diverge. Specifically, masking attention to the true top-of-stack position causes a sharp drop in long-distance accuracy, while ablating low-dimensional residual stream subspaces has comparatively little effect. These results, which extend to a templated natural language setting, suggest that even in a controlled setting where the relevant hierarchical variables are known, decodability alone does not imply causal use.

Aryan Sharma, Cutter Dawes, Shivam Raval

1Yale University · 2Independent · 3Harvard University

Transformer Interpretability Self-Attention Transformer Attention Attention Mechanisms Causal Interpretability Causal Interventions in Language Models

Apr 23, 2026

LayerBoost: Layer-Aware Attention Reduction for Efficient LLMs

Transformers are mostly relying on softmax attention, which introduces quadratic complexity with respect to sequence length and remains a major bottleneck for efficient inference. Prior work on linear or hybrid attention typically replaces softmax attention uniformly across all layers, often leading to significant performance degradation or requiring extensive retraining to recover model quality. This work proposes LayerBoost, a layer-aware attention reduction method that selectively modifies the attention mechanism based on the sensitivity of individual transformer layers. It first performs a systematic sensitivity analysis on a pretrained model to identify layers that are critical for maintaining performance. Guided by this analysis, three distinct strategies can be applied: retaining standard softmax attention in highly sensitive layers, replacing it with linear sliding window attention in moderately sensitive layers, and removing attention entirely in layers that exhibit low sensitivity. To recover performance after these architectural modifications, we introduce a lightweight distillation-based healing phase requiring only 10M additional training tokens. LayerBoost reduces inference latency and improves throughput by up to 68% at high concurrency, while maintaining competitive model quality. It matches base model performance on several benchmarks, exhibits only minor degradations on others, and significantly outperforms state-of-the-art attention linearization methods. These efficiency gains make our method particularly well-suited for high-concurrency serving and hardware-constrained deployment scenarios, where inference cost and memory footprint are critical bottlenecks.

Mohamed Ali Souibgui, Jan Fostier, Rodrigo Abadía-Heredia +3

Openchip & Software Technologies, S.L.

Transformer Attention LLM Inference Acceleration Efficient Attention Linear Attention