Self-Attention

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9 papers in the last four weeks, up 50% on the four weeks before. 0.1% of all new papers.

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Latest papers 195

Oct 6, 2026cs.LG

Tucker Bottleneck Attention for Multi-Dimensional Sequence Modeling

The quadratic cost of self-attention limits scalability to long sequences from multidimensional data. We introduce Tucker bottleneck attention (TuBA), which exploits low-rank tensor structure for efficient global token mixing. TuBA projects hidden tensors into compact Tucker cores, performs multi-head self-attention and linear projections on the cores, and writes updates back to the ambient space, enabling subquadratic computation. Its autoregressive extension combines bidirectional interactions within cores with causal attention across cores. On video prediction and global weather forecasting, TuBA achieves favorable accuracy-efficiency trade-offs over standard and efficient attention and task-specific models. Compared to standard self-attention, TuBA reduces error and computation by up to 24.7% and 66.6% for video prediction and 37.1% and 85.1% for autoregressive weather forecasting, with speedups up to 4.27 times. Low-rank Tucker cores and multi-frame generation also outperform full-rank attention and frame-by-frame generation, respectively.

Ryan Solgi, Parsa Madinei, Zheng Zhang

University of California-Santa Barbara Santa Barbara, CA, USA

Attention Mechanisms Low-Rank Attention Self-Attention Weather Forecasting

Oct 4, 2026cs.LG

An equality condition for the Dobrushin bound on attention rollout and how often it holds in trained transformers

The Dobrushin coefficient of each attention-rollout factor satisfies

κ(\frac12(I+A))\le\frac12(1+κ(A))

, and multiplying these inequalities over layers bounds the coefficient of the whole rollout. We characterise exactly when the layerwise bound is tight: equality holds if and only if some token pair attaining

κ(A)

is mutually self-dominant - each of the two attends to itself at least as strongly as the other attends to it. The condition is far from automatic: uniformly random stochastic matrices satisfy it only 24-30% of the time. When tested on the head-averaged attention of each individual input and restricted to content tokens - image patches, words or tabular features, excluding cls, register and separator tokens - the condition holds for essentially every input at every layer of DINOv2 (three model sizes), RoBERTa and DistilBERT. In the supervised models DeiT-B and ViT-B/16 it holds for 91% and 64% of input-layer pairs respectively, with all failures occurring late in depth. In FT-Transformer trained on two standard tabular benchmarks it holds for only 11-44% of input-layer pairs. The special tokens account for almost all failures in DINOv2 and the language models: when they are included, the condition holds for only 82-97% of input-layer pairs.

Przemysław Rola

Department of Mathematics, Kraków University of Economics

Transformer Interpretability Self-Attention

Oct 1, 2026cs.LG

Universal interpolation for deep residual self-attention networks

Universal approximation is a necessary qualitative property of learning architectures to benefit from scaling laws. While it is generically verified on a variety of neural architectures and random feature models, it typically involves infinite width limits. In this work, we focus on deep self-attention models and consider instead the `dual' regime, where approximation power is enabled entirely by depth, and featuring strong parameter sharing across layers, motivated by recent models such as the Looped Transformers. More specifically, we ask whether one can find a predefined finite set of parameters, each defining an attention block, such that the resulting finite set of transformations can map any collection of

N

sequences of

n

tokens to any other collection of

N

sequences of

n

tokens. Crucially, these transformations are \emph{fixed independently of the input and output} collections: only the order in which the blocks are applied, their signs, and their durations depend on the particular interpolation task. Our main result establishes it for residual softmax attention using only two frozen single-head blocks with Gaussian-initialized projection matrices. The result holds at both continuous and finite depth. We also characterize the restrictions imposed by causal masking and establish corresponding universal interpolation guarantees.

Sibylle Marcotte, Joan Bruna

Department of Computer Science New York University

Softmax Attention Transformer Self-Attention Neural Network Approximation Theory

Oct 1, 2026cs.AI

Contrastive Attention Mitigates Spectral Bias in Spiking Transformers

Spiking Transformers merge the energy-efficiency of spiking neural networks (SNNs) with the representational power of self-attention, creating a promising architecture for high-performance, energy-efficient computation. However, a performance gap persists versus its counterparts in artificial neural networks (ANNs). Unlike prior works attributing this to binary activations, we reveal that both spiking neurons and spiking self-attention (SSA) act as low-pass filters through multiscale spectral analysis. This characteristic leads to the dissipation of high-frequency components. To address this issue, we propose the Spiking Contrastive Attention (SCA) paradigm, which draw inspiration from the edge-detection and differential sensing properties of biological visual system. By extracting contrast prototypes via global contrastive aggregation and applying local differential refinement, SCA effectively enhances high-frequency information. Extensive experiments show that SCA is a general module that consistently boosts Spiking Transformers across image classification, semantic segmentation, and event-based tracking. Furthermore, it achieves lower complexity, offering superior efficiency over original SSA. These results establish its potential as a fundamental building block for energy-efficient Spiking Transformers.

Xiaoli Liu, Malu Zhang, Yang Yang

School of Computer Science and Engineering, University of Electronic Science and Technology of China

Spiking Transformers Self-Attention Spectral Bias Spiking Neural Networks

Sep 29, 2026cs.CL

Retrieval Capacity of Self-Attention Under Competition

How many tokens from its context does a language model actually use, and what determines that number? We study this question through self-attention. Without retraining, we retain only the tokens with the highest attention weights at each head, layer, and query, keeping their original weights unchanged. By varying the selected set size and measuring the increase in negative log-likelihood (NLL), we estimate the effective attention set size needed to stay within a chosen loss tolerance. Relatively small selected sets can keep NLL close to the full-attention baseline, although the required size varies across models. Attention-based selection substantially outperforms random selection. Selected sets exhibit geometric structure, although geometric separation alone does not establish that model loss is preserved. Extending context while evaluating the same prediction targets increases the required set size, while its fraction of context decreases over the tested range. Experiments with a fixed supporting fact show that additional background pushes its tokens down the attention ranking and reduces their attention mass. Renormalizing the retained weights can substantially reduce the required set size, showing that it also depends on how selected representations are combined. Conditional theoretical models explain how competition and attention-mass retention can produce growing set sizes without more distinct information to retrieve. These results provide a way to measure effective attention set size in language models and investigate its dependence on context, competition, and aggregation.

Timur Mudarisov, Mikhail Burtsev, Radu State

University of Luxembourg, Luxembourg

Self-Attention Long-Context Language Modeling Sparse Attention Attention Mechanisms

Sep 27, 2026cs.LG

PQ-HSA: Reusing Product-Quantized Scores for Hybrid Sparse-Approximate Attention

At each decoding step a language model attends over the key-value (KV) cache of every earlier token, so at long context the attention call is bounded by memory bandwidth. Sparse attention reads only a subset of keys chosen by a cheap score estimate, and most methods give the unread tokens zero weight. The output then draws on only a small fraction of the KV cache, and accuracy drops at small budgets, most on tasks that aggregate information across the context. An inverted-file product-quantization (IVF-PQ) index over the cached keys computes an approximate score for every indexed token in order to rank them; after ranking, those scores approximate the attention logits of the tokens left out. PQ-HSA (hybrid sparse-approximate attention) attends the selected tokens with their original keys and values, and the unselected tokens, the background, enter the same softmax through those scores, summed per inverted list and multiplied by the list's mean value. At 128K and a 1-2% retrieval budget, PQ-HSA is more accurate than Quest and SnapKV on Llama-3.1-8B and Qwen3-30B-A3B and stays close to full attention in macro accuracy; with the same selector, the background term raises macro accuracy on the 8B model from 0.71 to 0.83. In the same 128K setting, inside vLLM on one NVIDIA H20, the decode attention call runs 1.6x faster than the FlashAttention-3 kernel; the speedup grows with context length, and a cost model fitted on 8B to 30B models gives the context length at which it begins. A vLLM plugin runs PQ-HSA on two engine versions without changes to the engine source; code is available at https://github.com/KunmingSHAO/pqhsa_release.

Kunming Shao, Jierun Chen, Yanli Wang +4

The Hong Kong University of Science and Technology · Huawei Technologies Ltd. · Sun Yat-sen University +1

Self-Attention LLM Inference Acceleration Sparse Attention Hybrid Attention

Sep 23, 2026cond-mat.dis-nn

Nonequilibrium Phases of Repulsive Self-Attention: Chaos, Attention Condensation, and Emergent Locality

We study the nonequilibrium dynamics of a minimal recurrent transformer with

N

normalized tokens,

Q=K=I

, and a negative value map

V=-I

. Similarity-based attention selects nearby representations, while the negative value map drives tokens away from the selected field. This feedback can continually reorganize both the representation geometry and the attention network. For

d=2

, the tokens lie on a circle, where the regular polygon is an exact fixed point. As the attention feedback strength

γ

is increased, the polygon loses stability through a flip bifurcation, giving rise to period-two motion, chaos, and cluster-exchange or cluster-flip states. Despite this temporal complexity, attention remains diffuse as

N\to\infty

at finite fixed softmax sharpness

β

. Attention condensation instead emerges in the scaling regime

β\sim N^2

. In the hard-routing limit, repulsive updates amplify local perturbations and routing-partner switches transmit them ballistically, producing an emergent butterfly cone in representation space. High-dimensional geometry provides a distinct route to localization. For

d=N\to\infty

, simulations from Gaussian initial conditions provide evidence for a condensation transition at

β=O(1)

, driven by dynamically generated finite overlap gaps. Depending on

γ

, the resulting phases include diffuse simplex-like states, consensus flips, condensed active routing with signatures of chaos, and fragmented cluster flips. These results establish temporal activity, attention condensation, and geometric clustering as distinct collective phenomena, and show that sparse attention can sustain persistent dynamics rather than freeze it.

Qucheng Gao, Zuyi Yang, Xiao Chen

Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA

Self-Attention Chaotic Dynamical Systems Attention Mechanisms

Sep 21, 2026cs.LG

Prescriptive SVD-Inspired Attention via Spectral Energy Retention

Self-attention is central to modern Transformer architectures, but its dense dot-product formulation makes it difficult to identify which internal directions are structurally important and which can be modified without disrupting the model. SVD-Inspired Attention (SVDA) addresses part of this problem by introducing a learned diagonal spectrum into the query-key score interaction, making latent attention directions explicitly inspectable through indicators such as spectral entropy, effective rank, sparsity, alignment, selectivity, and perturbation response. This paper examines the transition from diagnostic interpretation to operational intervention. A diagnosis--intervention--verification framework is proposed, and one intervention is evaluated: spectral energy retention in the attention-score pathway. Across FashionMNIST, CIFAR-10, CIFAR-100, and Food-101, the

ρ=0.90

prescription removes 24.5--53.7% of score directions, reduces parameters by 2.6--4.3%, and reduces estimated MACs by 2.8--5.4%. The paired mean accuracy change of the dimension-reduced model ranges from

-0.03

to

+0.05

percentage points over three seeds. These results support SVDA as an intrinsically interpretable attention mechanism whose learned spectrum exposes an operational coordinate system for deterministic and verifiable modification of attention-score formation.

Vasileios Arampatzakis, Vasileios Sevetlidis, George Pavlidis

Athena Research Center, Greece

Transformer Interpretability Low-Rank Attention Self-Attention Singular Value Decomposition Neural Network Pruning

Sep 20, 2026cs.AI

Increasing Skill Level Recruits Deeper Attention Layers in a Frozen Chess Transformer

Chess involves complex reasoning in a deterministic environment, which makes it a useful setting for studying the mechanisms of computation inside transformers. The Maia-3 chess transformer takes Elo, a measure of competitive chess skill, as an input to the pre-trained network, so we can vary the skill the network is conditioned on with no change to its weights. Here we investigate how turning this skill dial affects self-attention. Ablating every attention head at every Elo from 700 to 2500, we find 1) increasing skill pushes the causal center of mass of the computation deeper, monotonically, for every chess piece and move type we measured; 2) the depth migration is much greater for specific tactics, especially knight forks, than for other move types; 3) the migration consists of deeper heads getting recruited for more specialized computations while one shared shallow head keeps a roughly constant contribution. These results may shed light on how conditioning inputs redistribute computation in larger transformers.

David Litman

Computational Neurobiology Laboratory, Salk Institute

Transformer Interpretability Attention Head Analysis Self-Attention

Sep 8, 2026cs.CL

It's Not RoPE that Creates Sinks: The Role of Self-Concentration and Value-Non-Mixing in Attention

Large Language Models (LLMs) often exhibit "Attention Sink" (AS) and the accompanying "Massive Activations" (MAs) at the initial position of a sequence. These phenomena frequently co-occur, and MAs can pose challenges for low-bit quantization. In this study, we analyze the factors underlying AS and MAs that emerge at the initial position regardless of the token occupying it. Our experiments suggest that self-concentration of attention, resulting from the causal mask, and the subsequent Value-non-mixing in attention outputs contribute to AS and MAs. These findings provide new empirical evidence on the internal dynamics of LLMs, offering insights that may inform future quantization strategies and advance our understanding of the internal mechanisms of attention layers.

Raito Kiya, Satoki Ohashi, Kosuke Sato +6

1Tohoku University · 2SOKENDAI · 3NINJAL +3

Self-Attention Attention Sinks Attention Mechanisms

Sep 3, 2026cs.CC

The Head Complexity of Boolean Functions in Single-Layer Attention

What can a single layer of self-attention compute? We study head complexity: the minimum number of attention heads required to compute a function in a one-layer attention-only model. We establish an exact hierarchy under this measure:

k

heads compute

k

-bit parity but cannot compute

(k+1)

-bit parity. The lower bound is unconditional in the two resources a transformer might otherwise exploit; it holds at unbounded embedding dimension and unbounded numerical precision. The proof rests on an alternating-sum obstruction: after clearing the softmax denominators, every monomial in the resulting decision polynomial omits at least one of the

k+1

input bits, forcing its correlation with parity to vanish. The same obstruction yields lower bounds for related tasks, including the well-studied multi-hop induction-head task. We also establish compactness bounds for embedding dimension and numerical precision. Specifically, a compactness theorem shows that any function computable at all can be computed with embedding dimension and precision bounded by the discrete data of the task, namely, head count, alphabet size, and length. Thus, potentially unbounded dimension or precision provably cannot substitute for heads. Finally, we derive nearly matching universal bounds for general binary functions:

2^n

heads suffice to compute every

n

-bit binary function, with one head per monomial in its multilinear expansion, while a counting argument shows almost all such functions require

Ω(2^n/n^2)

heads. This lower bound matches the upper bound to within a

\operatorname{poly}(n)

factor, even when dimension and precision are unbounded. Together, these results characterize head requirements for Boolean computation in this model.

Rajmohan Rajaraman, Ravi Sundaram, Amanuel Tesfaye

Northeastern University.

Self-Attention Attention Mechanisms Multi-Head Attention

Sep 1, 2026cs.LG

Multi-Head Self Attention is a Parameter Identification Mechanism

We prove that a multi-head scaled dot product attention can be viewed as a parameter identification strategy. The ratio of unidentified parameters to the total number of parameters scales like the reciprocal of the number of heads (

1/2 \to 1/(2H)

), meaning models with more heads are structurally more identified. A subtle side effect of the mathematics observation that attention can never be fully identified. Similarly we also show that some bias terms can have no effect on softmax-based attention layers in both the single- and multiple-head settings, though this is mostly a curiosity that should have a marginal effect on model size and model training/prediction efficiency. We also touch on modern improvements to transformers including RoPE and GQA from this perspective, illustrating how those as well can improve the ratio of

meaningful'' parameters to all parameters. Simple numerical examples demonstrate that training can indeed involve updates that overlap model-invariant subspaces that arise from a lack of identification. As part of our experiments we use a

rebalancing'' approach that can ``fix'' updates that overlap unindentified subspaces but do not try to present evidence this should actually be adopted. Instead we simply view our numerical results as exploring and confirming the theoretical results. As a whole we discuss a purely mathematical/statistical explanation, identification, for why specific architectural choices in transformers may have improved performance.

W. Ross Morrow

Softmax Attention Parameter Identifiability Self-Attention Multi-Head Attention

Aug 22, 2026cs.LG

TANGO: Treating Tokens as Operators

Transformers separate cross-token mixing in self-attention from token-wise transformation in feed-forward networks. We ask whether combining these operations can lower predictive loss under fixed data and parameter budgets. To do so, we introduce the Token-Aggregated Nonlinear Gating Operator (TANGO) model. TANGO computes a nonlinear feature-wise gate at each source token. Attention averages these gates for each destination. The average modulates a linear projection of the destination and forms the diagonal core of a source-conditioned linear operator. We test this proposal by comparing full-prefix and windowed TANGO with looped and untied Transformers, the Gated Attention Unit (GAU), and Fast Linear Attention with a Single Head (FLASH) on web text, Lean formal mathematics, DeepMind Mathematics, and code. The comparison uses two parameter scales, two depths, and three seeds. Checkpoints are selected on development data and evaluated on held-out test data. At matched parameters and training data, full-prefix TANGO has the lowest mean test negative log-likelihood in all 16 settings. In eight additional combinations of size and dataset, its development loss never increases as depth rises from 4 to 8 to 16, whereas the looped Transformer's loss increases in four. Full-prefix TANGO is computationally expensive because it averages wide gates over every visible source. To reduce this cost, we evaluate a variant with three narrower gated-projection sets assigned to the first, middle, and last applications. Across four FineWeb-Edu settings, this variant achieves 3.26 to 3.45 times the throughput of TANGO and 75% to 96% that of the looped Transformer. Its mean development negative log-likelihood is lower than TANGO's in three settings and 0.023 higher in the fourth, while remaining lower than both Transformer baselines in all four.

Joshua Nunley

Department of Informatics Luddy School of Informatics, Computing, and Engineering Cognitive Science Program Indiana University Bloomington

Transformer FFNs Transformer Self-Attention Autoregressive Language Modeling

Aug 20, 2026cs.LG

Ask Self, Ask Others: Relation Is All You Need

Attention dominates token mixing, but it collapses relation formation and flow allocation into a single score-to-flow step. We introduce Relation, which separates them by first organizing pairwise evidence into explicit Self and Exchange relations and deriving information flow afterward. Relation first decides whether a token should rely on itself or draw from its history, and if it draws from history, where to look. This relational organization gives rise to Full Relation, FlashRelation, Linear Relation, and Hybrid Relation. Across matched decoder-only models, Full Relation achieves lower mean final-validation NLL than MHA and reaches the paired MHA final training loss with 4.5-7.3% fewer tokens. Structural diagnostics further show that Relation learns a distinct depth organization: the first layer acts as a current-token anchor and a high-rank router, while later layers shift strongly toward history. In a fixed-context reference benchmark, FlashRelation is 4.17-5.28x faster than the materialized Full Relation implementation. Across scale-matched production workloads, it reaches 89.7-92.9% of PyTorch FlashAttention throughput while executing the exact Full Relation operator. Hybrid Relation demonstrates that Full and Linear Relation layers can be composed within a single decoder. These results support a relation-first view of token mixing: ask Self, ask Others, then let Flow follow Relation.

Yuting Ge, Pengju Yang, Mingkai Nie

Department of Electrical Engineering City University of Hong Kong · Jilin University · National University of Singapore

Self-Attention Decoder-Only Language Models Attention Mechanisms Efficient Language Model Inference Hybrid Attention

Aug 10, 2026cs.LG

Power law graph attention: exact generalization of scaled dot-product attention, empirical collapse at inference

The Large Language Model from Power Law Decoder Representations (PLDR-LLM) and its attention, Power Law Graph Attention (PLGA), replace the fixed bilinear form of scaled dot-product attention (SDPA) with a learned, input-generated bilinear operator

G_{LM}

, built from a positive tensor

A_{LM}

by elementwise power laws. The architecture is fully specified, verified against pinned reference releases; claims are labeled theorem, conditional theorem, measurement, or conjecture. Unconditionally: PLGA contains SDPA exactly at

G_{LM}=I

;

A_{LM}

and

A_P

are strictly entrywise positive, with Perron-Frobenius structure on

A_{LM}

; the DAG regularizer has the NOTEARS walk-counting form and positivity obstructs exact acyclicity; and, under nonresonance (satisfied by standard rotary frequencies), a commutant criterion identifies which operators preserve relative-position dependence. An inference-collapse theorem: exact input invariance of deductive outputs collapses inference to generalized SDPA with a constant operator. Measured invariance: relative fluctuations of

10^{-6}

and below; perturbation bounds quantify but do not certify cached inference; the assembled proxy misses the decoding margin. A conditional three-stage mechanism (rotary twirl, concentration, row-map contraction) is measured on a released checkpoint. Blockwise training and scoring under the global Gram are stated with explicit target exposure; on tested samples, block and sequential scoring select identical answers and agree on the published TruthfulQA probability-mass metric within

5\times 10^{-5}

per item. Self-organized criticality enters as a phenomenological framework with an intrinsic order parameter; open claims become falsifiable conjectures. Selected proof cores are machine-checked in Lean 4.

Burc Gokden

Fromthesky Research Labs LLC, Oregon, USA

Graph Attention Networks LLM Inference Self-Attention

Aug 10, 2026cs.LG

Why Post-Norm Transformers Collapse: Attention Amplification and Gradient Repair Failure

Deep decoder-only Transformers often replace the original Post-Norm architecture with Pre-Norm variants because Post-Norm training is highly sensitive to warmup and learning rate under conventional initialization schemes. Although prior work has identified rank collapse and gradient vanishing as related symptoms, it remains poorly understood how causal attention creates high-similarity representations and why training dynamics fail to repair them. We give a two-stage analysis of Post-Norm rank collapse using token similarity as a scalar state variable. First, at initialization, causal attention acts approximately as a prefix-averaging operator that increases token similarity across depth, while the SwiGLU branch contributes only a smaller damping effect. Second, once training enters a high-similarity regime, growth of pre-normalization residual norms makes the RMSNorm backward factor contractive; under mild conditions, gradients to earlier layers decay geometrically. As a complementary result, we characterize the properties of a collapsed network: its best predictor is frequency distribution with relatively high loss floor, and gradients in collapsed layers vanish at frequency distribution. Experiments on 48-layer decoder-only Transformers trained on C4 dataset match the predicted initialization-time similarity growth and collapse-time gradient contraction, and show that collapsed runs stay near the predicted frequency loss. Together, these results distinguish the forward similarity amplification and backward repair incapacity in Post-Norm collapse, while also characterizing the behavior of collapsed networks.

Xingjian Wang, Qingyu Han, Xiaodong Luo +1

The Chinese University of Hong Kong, Shenzhen · Shenzhen Research Institute of Big Data

Transformer Self-Attention Decoder-Only Language Models Layer Normalization

Aug 9, 2026cond-mat.dis-nn

Clustered Attractor Manifolds and Dynamical Condensation in Self-Attention

Transformer layers generate state-dependent interaction networks: token representations determine the attention matrix, which in turn updates the representations. We study this feedback in a minimal normalized self-attention dynamics and identify the overlap gap as the central quantity governing its attractor structure in the thermodynamic limit. When tokens form internally aligned clusters and their similarity to members of the same cluster exceeds that to every other cluster by a nonvanishing amount, inter-cluster attention is exponentially suppressed as the dimension increases. This mechanism produces a high-dimensional manifold of clustered fixed points, ranging from a few macroscopic clusters to extensive microscopic fragmentation, and also controls their stability against perturbations. Starting from an unstructured Gaussian state, we find that clustered states nucleate from the diffuse background only above a finite threshold in attention sharpness, giving rise to a dynamical attention-condensation transition.

Qucheng Gao, Zuyi Yang, Xiao Chen

Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA · Department of Physics, Tsinghua University, Beijing 100084, China

Dynamical Systems Self-Attention Transformer Attention

Aug 9, 2026cs.CV

CDGC-Net: 3D Medical Image Segmentation with Cooperative Dual-Scale Self-Attention and Grouped Channel Modeling

Accurate 3D medical image segmentation requires the integration of long-range anatomical context with fine boundary detail. Existing methods often model global and local features in separate modules or feature levels and perform channel recalibration independently. This may cause semantic mismatch between global context and local boundaries, insufficient channel relationship modeling, weak spatial-channel interaction, and redundant representations. We propose CDGC-Net, a 3D medical image segmentation network that combines cooperative dual-scale spatial attention with grouped hierarchical channel modeling. With-in each CDGC block, Cooperative Dual-Scale Self-Attention (CDSA) assigns attention heads to parallel local-window and global-sparse branches. The two branches capture fine spatial details and long-range anatomical context at the same feature level. Their outputs are concatenated into an

N\times C

spatial representation and directly passed to Grouped Hierarchical Channel Attention (GHCA). GHCA organizes the channels into

r

groups and models both within-group and cross-group dependencies. CDSA and GHCA reuse a shared key projection to maintain a consistent feature reference. Residual feature alignment subsequently integrates the refined features with the original representation. On the Synapse, ACDC, BraTS, and LA datasets, CDGC-Net achieved mean DSC values of 86.96%, 92.91%, 82.56%, and 93.52%, respectively, exceeding the next-highest reported values by 0.39, 0.47, 0.17, and 0.32 percentage points. CDGC-Net contains 25.83M parameters and 28.62G FLOPs for an input size of

64\times128\times128

, reducing these quantities by 39.87% and 40.30%, respectively, relative to UNETR++. These results indicate a favorable trade-off between segmentation accuracy and computational complexity.

Zheyang Jing, Qin Lu, Jianwang Li +3

Nanchang Hangkong University, Nanchang, China

Self-Attention Multi-Scale Feature Fusion 3D Medical Image Segmentation Attention Mechanisms

Aug 8, 2026cs.CV

Gated Spatial Redundancy Projection for Pathology Transformer Attentions

Transformer models are increasingly used for whole-slide image analysis in computational pathology. Yet, WSIs differ fundamentally from natural images: neighbouring patches often contain highly similar tissue type, stain, texture, and cellular composition. We identify this local spatial redundancy as a pathology-specific failure mode of self-attention, where dominant neighbourhood features can be repeatedly mixed into patch-tokens and weaken subtle diagnostic or prognostic deviations. We propose Gated Spatial Redundancy Projection (Gated SRP), a lightweight drop-in correction module for self-attention layers. For each patch token and attention head, Gated SRP estimates a local redundancy axis from neighbouring value vectors, projects the attention output onto this axis, and applies a learned signed gate to correct the redundancy-aligned component geometrically. Across five TCGA survival cohorts, Gated SRP obtains the highest mean C-index among the compared attention variants in all cohorts, with an average improvement over the base attention, while adding only +0.02% parameters. Across five slide-level classification datasets, it improves the base attention on 12 of 16 reported metrics and achieves the best AUC on three datasets. Code is publicly available at https://github.com/AtlasAnalyticsLab/GatedSRP.

Zhiyuan Yang, Jiahao Cheng, Vincent Quoc-Huy Trinh +1

Department of Computer Science and Software Engineering (CSSE), Concordia University, Montreal, Canada · Axe Cancer, Centre de recherche du CHUM, Université de Montréal, Montréal, Canada · Mila - Quebec AI Institute, Montreal, Canada

Cancer Survival Prediction Self-Attention Attention Mechanisms Histopathology Image Classification Whole-Slide Image Analysis

Aug 7, 2026cs.LG

Faster Query-Key Learning Sharpens Attention in Self-Attention Models

A standard self-attention layer consists of two interacting circuits: the query-key circuit that governs attention allocation, and the output-value circuit that maps attended representations to predictions. Collapsed and factorized parameterizations of the query-key and output-value circuits lead to qualitatively different attention patterns. In particular, some parameterizations give sharper attention to task-relevant tokens, at a similar training loss. We analyze how the parameterizations of these circuits shape the parameter trajectories in single-layer self-attention models trained for next-token prediction. Through gradient-flow analysis, we show that factorization induces implicit rescaling of the two circuits' learning rates. We derive closed-form dynamics showing that output-value and query-key parameters move along a line, with relative speeds determined by their learning rates. Faster query-key learning relative to output-value learning thus produces sharper attention, as the model compensates for slower output-value learning by increasing attention mass on relevant tokens. Experiments show that differences in the relative learning rates of the two circuits govern attention concentration. This improves attention interpretability proxies while maintaining comparable predictive performance.

Rahul Vashisht, Harish G. Ramaswamy

Department of CSE, IIT Madras, India · Department of DSAI, IIT Madras, India.

Aug 2, 2026cs.LG

Riemannian Attention Mechanisms for Transformers: A Theoretical Framework and Architecture Design

All Transformer-based large language models compute attention via the Euclidean inner product, an architectural choice that Dong et al. (2021) proved causes representational rank to decay doubly exponentially with depth in pure self-attention stacks. We develop a theoretical framework that targets this structural limitation at the mathematical level by replacing the flat Euclidean metric with learned per-token Riemannian metrics. Our contributions are threefold. (1) We prove that Riemannian attention scores with heterogeneous per-token metrics are non-Gram---they cannot be factorized as QK^T with factorization dimension O(d). We are explicit that this is a structural observation, not a proof of rank preservation. (2) We establish that low-rank metric factors render all geometric operations tractable: geodesic distance in O(dr) per token and metric inversion in O(dr^2) via the Woodbury identity---both far below the O(d^3) cost of a general matrix---making Riemannian attention feasible at billion-parameter scale with negligible overhead. (3) We present the Fiber Bundle Transformer, a complete architecture specification in which each token position carries its own Riemannian metric, attention is geodesic distance computation, feed-forward updates use metric-preconditioned steps, and the connection carries explicit curvature and torsion proxies. We derive formal predictions about correctly implemented geometric architectures and identify the central open problem: proving or disproving that heterogeneous Riemannian metrics prevent the rank collapse that row-stochastic attention matrices otherwise cause. This paper presents theoretical analysis and architectural design; empirical validation is the subject of future work.

Sen Song

Independent Researcher, Guangzhou, China

Transformer Self-Attention Attention Mechanisms Representation Geometry in Language Models

Jul 25, 2026cs.LG

Training with (Swap) Regret Loss in a Single-Layer Self-Attention Model: A Case Study on the Probability Simplex

We revisit the regret loss framework introduced in Park et al. (2025), which uses decision-theoretic regret as a direct loss function for training models to make better decisions, through the lens of probability-simplex policies. Our first result shows that a single-layer self-attention model trained with regret loss admits a stationary point whose forward-pass exactly matches smoothed fictitious play with the appropriate stepsize that ensures no-regret behavior-i.e., for any given policy input, the model outputs the same update that smoothed fictitious play would produce. In parallel, we also newly introduce a swap-regret loss function, which extends the regret-loss framework beyond external regret and enables models to directly optimize for swap-deviation robustness. We further show that this swap-regret loss admits a stationary point whose forward pass implements the corresponding swap-regret update induced by classical Blum-Mansour no-pass implementation algorithm, with each head implementing an external-regret update via smoothed fictitious play. Together, these results show that regret-trained attention can realize differentiable mechanisms whose deployment induces equilibrium behavior in games: external-regret dynamics lead to coarse correlated equilibrium, while swap-regret dynamics lead to correlated equilibrium. Thus, regret-based objectives steer minimal attention architectures toward online-learning dynamics with game-theoretic guarantees, without supervised traces of those algorithms.

Chanwoo Park, Asuman Ozdaglar

1MIT EECS

Self-Attention Regret Minimization

Jul 14, 2026cs.CV

Inhibited Self-Attention: Sharpening Focus in Vision Transformers

Vision Transformers (ViTs) have demonstrated remarkable performance in computer vision tasks. However, their self-attention mechanism often diffuses focus across background regions, relying on spurious correlations rather than object-relevant cues. Inspired by inhibitory mechanisms observed in biological vision systems, we propose the Inhibited Self-Attention (ISA), a novel self-attention that integrates inhibitory signals to enhance feature selectivity and suppress spurious responses. In contrast to conventional self-attention, which relies solely on positive attention values due to softmax normalization, our approach retains and utilizes negative attention scores to suppress irrelevant features and sharpen focus on objects of interest. Experiments across multiple datasets, including ImageNet-1k and COCO, and several robustness benchmarks demonstrate that ISA enhances object-centric selectivity, reduces shortcut reliance, and improves out-of-distribution generalization. Our analysis of relevance maps confirms that ViTs with ISA exhibit sharper, more localized focus on object-relevant regions while reducing distractions from non-relevant (background) features, enabling more reliable models. We release our code at https://github.com/prdvanderwal/inhibited-self-attention

Peter R. D. van der Wal, Nicola Strisciuglio, George Azzopardi

University of Groningen · University of Twente · 3Stellenbosch University +1

Self-Attention Vision Transformer Shortcut Learning

Jul 13, 2026cs.CV

LoSA-Net: A Localized and Scale-Adaptive Network for Boundary-Sensitive Prediction of Perineural Invasion in 3D MRI

Perineural invasion (PNI) is a clinically relevant indicator of tumor aggressiveness and can influence surgical decision-making, motivating interest in reliable preoperative assessment. The subtle MRI features of PNI, however, often resemble nearby anatomy, complicating noninvasive prediction. These fine perineural cues are easily attenuated by routine downsampling or overly global feature aggregation, reducing the effectiveness of conventional volumetric models. We present LoSA-Net, a localized and scale-adaptive architecture for boundary-sensitive PNI prediction in 3D MRI. Talking Neighborhood Attention (TNA) preserves nerve-aligned detail through localized self-attention with head-wise mixing, and Scale-Adaptive Feature Mixing (SAFM) modulates the receptive field using multi-scale depthwise processing. Cross-Scale Refinement and Alignment (CSRA) maintains consistency between semantic context and high-resolution boundaries across stages. In contrast-enhanced MRI scans from 168 patients with cholangiocarcinoma, LoSA-Net achieves an AUC of 0.7567 and outperforms representative convolutional and transformer baselines under matched preprocessing and optimization settings.

Youngung Han, Hyunsu Go, Kyeonghun Kim +9

Seoul National University, Seoul, Republic of Korea · OUTTA, Seoul, Republic of Korea · Chung-Ang University, Seoul, Republic of Korea +3

Self-Attention Multi-Scale Feature Fusion Medical Image Classification Clinical Prediction Oncology

Jul 12, 2026cs.LG

From Self-Attention to Connection Laplacian: A Unified Operator View of Transformers

Self-attention is a ubiquitous primitive in modern sequence models, yet its operator-level geometry is only partially understood. We view a token sequence as a vector field over the token-position graph and identify attention as a connection walk: messages are aggregated by a nonnegative walk matrix while being transported along each edge by a learned linear map. Within this framework, we prove that single-head attention (SHA) is exactly a connection propagation step with constant transport, and that multi-head attention (MHA) is exactly a single edge-dependent connection walk whose effective transport is an attention-gated mixture of headwise transports. We further clarify the conditions under which the corresponding generator reduces to a random-walk connection Laplacian, highlighting the roles of stochasticity, reversibility, and metric-compatible transports. Empirically, we find that trained Transformers across scales (from 124M to 8B) and structures (encoder/decoder) exhibit geometric structure consistent with our theory: effective attention graphs converge to stable geometric operators in deeper layers, learned transports self-organize into approximate scaled isometries, and both phenomena strengthen consistently with scale. Overall, the paper provides a precise connection-walk formalism that links self-attention to classical geometric operators, along with a set of operator-level tools for analyzing transformer models from a geometric perspective.

Binbin Lin, Wei Chen, Yalun Li +3

School of Software Technology, Zhejiang University, China · College of Computer Science and Technology, Zhejiang University, China · Alibaba Cloud, China

Self-Attention Transformer Attention Representation Geometry in Language Models Multi-Head Attention

Jun 24, 2026cs.LG

Towards Robust EEG Decoding Based on Riemannian Self-Attention

Brain-Computer Interface (BCI) based on electroencephalography (EEG) enables direct interaction between the brain and external environments and has significant applications in assistive technologies, medical rehabilitation, and entertainment. Recently, EEG decoding methods based on Symmetric Positive Definite (SPD) learning have demonstrated superior performance. However, these methods typically employ basic network architectures and do not explicitly capture local relationships between EEG signals. This limitation is problematic for EEG signals due to their inherently low Signal-to-Noise Ratio (SNR). Moreover, most existing Riemannian manifold-based methods are restricted to specific metrics. The most widely used is the Affine-Invariant Metric (AIM). However, it has a quadratic dependency on the SPD matrices and cannot handle ill-conditioned SPD matrices, which hinders the effectiveness of networks. In contrast, the Bures-Wasserstein Metric (BWM) exhibits linear dependence on SPD matrices and demonstrates superior performance for ill conditioning. To overcome these challenges, we propose a Riemannian self-attention network based on the BWM. Additionally, the recently introduced power-deformed generalized Bures-Wasserstein metric reveals a nonlinear relationship between SPD matrices and matrix power deformation. This metric provides a more nuanced representation of the geometric structure of the SPD manifold. Consequently, we extend our model to a learnable version. For simplicity, we refer to it as GBWAtt. Experimental results on three EEG benchmarking datasets validate the robustness and effectiveness of our proposed method. The code is available at https://github.com/jissc/GBWAtt.

Shaocheng Jin, Tao Zhou, Rui Wang +4

School of Artificial Intelligence and Computer Science Jiangnan University Wuxi, China · Department of Information Engineering and Computer Science University of Trento Trento, Italy · Centre for Vision, Speech and Signal Processing (CVSSP) University of Surrey Guildford, UK

Brain-Computer Interfaces Self-Attention EEG Decoding

Jun 24, 2026cs.LG

Communicability-Inspired Positional Encoding (CIPE)

Positional encodings (PEs) are essential for Transformers. Yet designing effective PEs for non-Euclidean graphs remains challenging. Such encodings should ideally induce an Attention-Compatible Geometry for self-attention: not merely describing graph structure, but defining a geometry whose inner products reflect meaningful structural relatedness. To realize this geometry, we propose Communicability-Inspired Positional Encoding (CIPE), built from communicability, a measure between pairs of nodes that aggregates contributions from paths of all lengths. By construction, CIPE inner products recover communicability, converting global multi-path connectivity into an attention-ready similarity geometry. For practical Transformer training, we introduce dimensionality alignment, mapping graph-size-dependent CIPE representations to prescribed dimensions while faithfully preserving the induced geometry. Empirically, CIPE improves structure-agnostic Transformers by 35.5% on average across seven benchmarks, outperforming representative PEs; it also consistently improves structure-biased graph Transformers, where competing PEs often yield only marginal benefits. These results position CIPE as a principled framework for attention-compatible graph positional encodings.

Yipeng Zhang, Zhongtian Sun, Pietro Liò +1

Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore · School of Computing, University of Kent, Canterbury, Kent CT2 7NZ, UK · Department of Computer Science and Technology, University of Cambridge, 15 JJ Thomson Avenue, Cambridge CB3 0FD, UK

Graph Representation Learning Self-Attention Positional Encoding Graph Transformer

Jun 22, 2026cs.CE

Attention mechanism for scalable mesh-based neural surrogates of free-surface fluids

High-fidelity simulations of free-surface flows using Lagrangian methods such as the Particle Finite Element Method (PFEM) are computationally demanding due to continuous domain updates and repeated solution of the governing equations. This challenge is further amplified by non-Newtonian rheologies, where material nonlinearities increase computational cost. These limitations motivate the development of efficient surrogate models to approximate PFEM dynamics at reduced cost. While data-driven deep learning approaches are promising, a key challenge is designing models that operate on arbitrary and evolving geometries. We propose a self-attention-based neural surrogate for PFEM simulations of free-surface flows. The architecture leverages attention mechanisms to model node interactions and capture complex spatial dependencies, while preserving the PFEM mesh discretization. This provides a geometric and topological framework for remeshing and node redistribution, maintaining high-quality spatial discretization during rollouts, improving long-term stability, and enabling reconstruction of derived mechanical quantities via standard finite element operators. Two attention formulations are considered: a standard self-attention mechanism and a linear variant that reduces computational cost and improves scalability. The models are evaluated on two- and three-dimensional free-surface flow benchmarks with evolving geometries, varying material parameters, and non-Newtonian fluids. Results show accurate prediction of transient dynamics and final configurations, with significantly improved scalability. The mesh-based formulation also enables direct reconstruction of quantities such as stress fields. Overall, the framework provides an accurate and scalable surrogate strategy for PFEM simulations in engineering-scale applications.

Federico Lanteri, Massimiliano Cremonesi

Department of Civil and Environmental Engineering, Politecnico di Milano Piazza Leonardo da Vinci, 32, 20133, Milano, Italy

Neural Surrogate Modeling Self-Attention PDE Surrogate Modeling Finite Element Method Computational Fluid Dynamics

Jun 10, 2026cs.LG

Kuramoto Attention: Synchronizing Self-Attention on the Torus

Transformer models are increasingly used as computational models of cognition and neural representation, so the mechanism implemented by self-attention is of interest beyond engineering performance. A complementary tradition in cognitive science models coordination, binding, and memory through dynamical interactions such as oscillator synchrony; we bring this mechanism into self-attention by introducing the Kuramoto Attention layer, whose value update is a synchronization step. Each token carries a bank of phase oscillators, so its hidden state lives on a high-dimensional torus. The attention weights form an adaptive coupling graph, and using the raw phase states as values makes the value update exactly the Kuramoto coupling direction for fixed attention weights. The softmax selects which oscillators couple, while the value path moves each token toward the attention-weighted circular mean of the tokens it selects. We train Kuramoto Attention on enwiki8 and CodeParrot against parameter-matched RoPE and SwiGLU transformers. At 5M parameters on CodeParrot, it improves on the transformer by both median and mean, with mean gaps of 0.012 validation and 0.010 test bits per byte. At 5M on enwiki8, all six runs have lower validation/test medians than the transformer and all-seed means within 0.01 BPC; five of six also form a tight lower-mean cluster. At 1M, it trails by about 0.02 BPC on enwiki8 and by 0.013-0.015 bits per byte on CodeParrot. Ablations and phase diagnostics show how the layer's synchronization and geometry-motivated components shape model performance. The result is a self-attention mechanism whose learned computation can be read directly as adaptive synchronization on phase states.

Joshua Nunley

Department of Informatics, Luddy School of Informatics, Computing, and Engineering Cognitive Science Program Indiana University Bloomington

Dynamical Systems Self-Attention

Jun 9, 2026cs.LG

RoVE: Rotary Value Embeddings Attention for Relative Position-dependent Value Pathways

Rotary Position Embeddings (RoPE) make attention scores position-relative but leave the value pathway position-blind: the message sent by a value token is the same regardless of its distance from the query. We propose RoVE, a parameter-free modification that makes values position-sensitive by rotating them simultaneously with keys, and show that it turns RoPE attention into attentive convolution. This new perspective unifies several independent formulations of the same operation across computer vision, robotics, and modern LLM architectures. Trained 124M and 354M GPT-2 models show consistent empirical gains over RoPE on few-shot in-context learning, out-of-distribution perplexity, and long-context retrieval, with the clearest improvements on tasks that require long-range aggregation.

Alejandro García-Castellanos, Maurice Weiler, Erik J Bekkers

AMLab, University of Amsterdam · MIT CSAIL

Rotary Positional Embeddings Self-Attention