Transformer Attention

Architectural choices inside the Transformer feedforward network (FFN) block do not merely affect the block itself; they reshape the computations learned by the rest of the model. We study this effect in one-layer Transformers trained on digit addition with carry, modular arithmetic, and histogram counting. Comparing dense FFNs, gated linear units (GLUs), mixture-of-experts (MoE), and MoE-GLUs, we find that sparse MoE routing can shift computation from FFN to attention, with the strongest ablation-visible effect on carry-based addition. We decompose this redistribution into reduced per-token FFN capacity and sparse partitioning across experts. Critically, frozen random routing nearly matches learned routing, suggesting that redistribution is driven largely by architectural sparsity rather than router-learned specialization. As a secondary finding, GLU-style multiplicative gating rotates task-relevant Fourier structure out of the per-neuron basis and into distributed subspaces, making neuron-level interpretability less informative while preserving structured computation. We validate these conclusions with random-routing, narrow-FFN, and top-2 MoE controls, plus parameter-matching, activation-function, and width-scaling analyses. Together, these results show that local FFN design choices can have nonlocal consequences for Transformer computation.

Mechanistic accounts of in-context learning (ICL) have identified iterative algorithms for linear regression and related linear prediction tasks, often using linear or ReLU attention variants. For nonlinear ICL, prior work has related softmax and kernelized attention to functional-gradient-type dynamics, but it remains unclear whether a standard transformer with softmax attention can implement a convergent solver with an end-to-end prediction-error guarantee. In this paper, we study in-context kernel ridge regression (KRR) with Gaussian kernels and show that a standard softmax-attention transformer can approximate the KRR predictor during its forward pass by implementing preconditioned Richardson iteration on the associated kernel linear system. Under bounded-data assumptions, we construct a single-head transformer with $O(\log(1/ε))$ blocks and MLP width $O(\sqrt{N/ε})$ that achieves $ε$-accurate prediction for prompts of length $N$. Our construction reveals a functional decomposition within the transformer architecture: softmax attention produces a row-normalized Gaussian-kernel operator needed for cross-token interactions, while ReLU MLP layers act locally to approximate the intra-token scalar arithmetic required by the update. Empirically, we train GPT-2-style transformers on Gaussian-process regression tasks to further test the preconditioned Richardson interpretation. Through linear probing, we compare the transformer's layer-wise predictions with the step-wise outputs of classical KRR solvers and find that its error profiles align most consistently with preconditioned Richardson iteration. Ablation studies further support this interpretation. Together, our theory and experiments identify preconditioned Richardson iteration as a concrete mechanism that softmax-attention transformers can realize for nonlinear in-context Gaussian-kernel regression.

Certified verification of transformer attention requires bounding the softmax function over interval constraints on the pre-softmax scores. Existing verifiers relax softmax ndependently of the downstream objective, leaving avoidable slack. We prove that the exact optimum of this score-box problem is attained at a vertex of the constraint box, and establish a threshold structure theorem showing that, after sorting the objective coefficients, the optimum lies among only linearly many candidates, yielding the Vertex-Softmax primitive with log-linear complexity in the sequence length. We further prove a formal optimality result showing that Vertex-Softmax is the tightest sound bound obtainable from score intervals alone, characterizing precisely what additional structure (score correlations, score-value coupling) is needed for further improvement. Integrated into a CROWN Convex Relaxation based Optimization for Worst-case Neurons)-style verifier with a formal soundness guarantee, Vertex-Softmax significantly improves certified rates and substantially tightens lower bounds across MNIST, Fashion-MNIST, and CIFAR-10 attention models, while consistently matching or outperforming alpha-CROWN and branch-and-bound baselines at a fraction of their cost.

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.

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.

We investigate the ability of transformers to perform in-context reinforcement learning (ICRL), where a model must infer and execute learning algorithms from trajectory data without parameter updates. We show that a linear self-attention transformer block can provably implement policy-improvement methods, including semi-gradient SARSA and actor-critic, via explicit parameter constructions. Beyond existence, we design a teacher-mimicking training procedure, analyze its gradient-flow dynamics, and establish the first convergence guarantee in the ICRL literature: under suitable richness conditions on the training MDP distribution, gradient flow converges locally and exponentially to an optimal parameter manifold corresponding to the desired RL update. Empirically, training transformers on randomly generated tabular MDPs confirms these predictions: the learned models recover the parameter structure of our explicit constructions and, when deployed on unseen MDPs, deliver strong in-context control performance. Together, these results illuminate how transformer architectures internalize and execute classical reinforcement learning algorithms in context, bridging mechanistic understanding and training dynamics in ICRL.

Transformers usually expose one inference cost per trained model, while deployed systems often need multiple cost-quality operating points. We study Budgeted Attention Allocation, a monotone head-gating mechanism conditioned on a requested attention budget. Dense warm-starting is important for stability: on a robust synthetic sequence task, one budgeted model reaches 99.7% accuracy at 0.303 estimated attention cost and 100.0% accuracy at 0.504 cost. On held-out AG News with a custom word-level transformer, hard-gate adaptation turns soft cost control into measured single-thread CPU speed, reaching 82.1% accuracy with 1.28x speedup at budget 0.50. In pretrained BERT-Mini AG News, budgeted structural pruning reaches 87.6% accuracy with 1.20x speedup at budget 0.50; a validation-ranked zero-shot dense post-hoc structural baseline reaches 86.1%, and one recovery epoch raises that per-budget specialist to 87.9%. On DBpedia14, BERT-Mini budgeted gates reach 97.4% at exact budget 0.50 versus 96.6% for dense full attention. Static fixed-budget gates and recovered dense specialists remain strong. The contribution is therefore not universal dominance, but a reproducible feasibility study of one controllable checkpoint across budgets that can trade attention cost for accuracy and be converted into measured structural speedups on small CPU benchmarks.

We analyse the computational power of transformer encoders as sequence-to-sequence functions on vectors. We show that average hard attention can be used to simulate arithmetic circuits if they are given as an input to an encoder. The circuit families that can be simulated this way have constant depth while using unbounded addition, binary multiplication and sign gates. The transformers we use have arithmetic circuits instead of feed-forward networks. With typical average attention the functions they compute are also computed by the same class of circuit families. Our results hold for transformers over the reals, rationals and any ring in between the two.

Transformer self-attention can be interpreted as a gradient flow on the unit sphere, in which tokens evolve under softmax interaction potentials and tend to form clusters. While prior work has established clustering behavior for single-head attention, the multi-head setting remains less understood due to geometric interference between heads, which invalidates standard monotonicity arguments. In this work, we develop a theoretical framework for multi-head self-attention dynamics and resolve several open questions. We show that, under suitable conditions on the score matrices, a natural multi-head energy functional is non-decreasing along both flat and spherical dynamics. We identify the key obstruction to per-head monotonicity as radial shadow terms, which are projections of each head's output onto token directions, persisting even under orthogonality assumptions. We introduce a sufficient condition ensuring monotonicity and establish robustness to approximate orthogonality. In a simplified scalar-head regime with equiangular token configurations, we derive a closed-form expression for the critical inverse temperature governing clustering behavior, and show that heterogeneous heads exhibit super-additive clustering rates. In this regime, we also prove a separation in clustering time between ReLU and softmax attention in the linearized dynamics. Finally, we establish an entropy production identity and show that attention entropy increases monotonically toward equilibrium as clustering progresses. Our results provide a unified perspective on the dynamics of multi-head attention and clarify the mechanisms underlying clustering and stability in transformer models.

Human attention is the gateway to conscious perception, memory and decision-making. However, its role in modern transformer models remains largely unexplored. As these systems increasingly influence what people see, prefer and buy, the question arises as to whether they encode principles of human interest or merely exploit large-scale correlations. Addressing this issue is crucial for understanding cognition and ensuring the responsible use of AI in communication and marketing. In order to address this issue, the concept of visual interest was examined within the multimodal vision-language-model Qwen3-VL-8B, using a pre-defined Common Interestingness (CI) score derived from large-scale human engagement data on the photo-sharing platform Flickr. Here, we analyzed internal representations across vision and language components using methods from the neurosciences. Our analyses revealed that CI information is linearly decodable from final-layer embeddings, indicating that it is aligned with human-derived measures of visual interestingness. Dimensionality reduction and Generalized Discrimination Value (GDV) analyses demonstrate that CI-related hidden representations emerge in intermediate vision transformer layers and becomes progressively more distinguishable across language model layers. Concept vectors derived using geometric, probe, and Sparse Auto-Encoder based methods converge in higher layers, as confirmed by representational similarity analysis. This indicates a robust and structured encoding of visual interestingness without explicit supervision. Future work will seek to identify shared computational principles linking human brain dynamics and transformer architectures, with the ultimate goal of uncovering the organizing mechanisms that give rise to attention and interest in both biological and artificial systems.

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 $l+1$.

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 \textbf{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 \texttt{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.

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.

The transformer is the most popular neural architecture for language modeling. The cornerstone of the transformer is its global attention mechanism, which lets the model aggregate information from all preceding tokens before generating the next token. One common variant of attention is called local attention, which restricts each token to aggregating information from a bounded window of predecessors, reducing the quadratic cost of global attention to linear. Although this restriction is usually motivated by efficiency, it has also been found to improve model quality, a phenomenon that has so far lacked a satisfactory explanation. We provide a formal account of this phenomenon in terms of recognizer expressivity. It has been shown that fixed-precision transformers with global attention correspond to a fragment of linear temporal logic containing a single past operator. We additionally prove that adding local attention introduces a second temporal operator, strictly enlarging the class of recognizable regular languages. Moreover, global and local attention are expressively complementary: neither subsumes the other, and combining them yields the richest fragment. Experiments on formal language recognition and natural language modeling corroborate the theory, showing that hybrid global--local transformers outperform their global-only counterparts.

Every Transformer architecture dedicates enormous capacity to learning rich representations in semantic embedding space -- yet the rotation manifold acted upon by Rotary Positional Embeddings (RoPE) has been treated as a fixed, hand-crafted structure, populated only by discrete ordinal indices. We argue that this rotation space is a largely overlooked second dimension of expressivity in the attention mechanism, one whose systematic exploration may open a new door for attention-based architectures. The analogy to complex numbers is instructive: just as introducing the imaginary axis -- orthogonal to and independent of the real line -- unlocked new algebraic structure once believed impossible, treating the rotation manifold as a learnable, signal-conditioned space opens an orthogonal degree of freedom in attention. In this framing, the token embedding encodes the semantic (real) component of a representation -- what a token means -- while the rotation encodes its dynamic (imaginary) component -- how it relates to every other token across time, position, and context. We introduce SIREN-RoPE, a concrete instantiation of this idea, which populates the rotation dimension with heterogeneous signals -- continuous timestamps, cyclical temporal patterns, and categorical metadata -- via a dual-branch Sinusoidal Representation Network (SIREN). As a proof of concept, we evaluate on a production-scale news feed dataset from a major social network using a generative recommender as the ranking model, demonstrating that activating this hidden dimension yields consistent improvements across calibration and ranking objectives with negligible computational overhead. We invite the community to view the rotation space not as a solved positional-encoding detail, but as an untapped axis whose rich structure may prove as consequential for attention as the imaginary unit proved for algebra.

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.

Device-free fall detection utilizing WiFi Channel State Information (CSI) has emerged as a promising, privacy-preserving solution for elderly health monitoring in the Internet of Things (IoT) era. However, existing deep learning approaches suffer from severe performance degradation when deployed in unseen environments due to static background overfitting and Non-Line-of-Sight (NLoS) signal attenuation. To address these critical bottlenecks, we propose a robust, domain-generalizable framework featuring a novel Attention-Enhanced CNN-Transformer hybrid architecture. First, we design a physics-driven \textbf{Dynamic Variance Gate (DVG)} to dynamically calculate local temporal variance, acting as a soft-attention mask that eliminates static environmental DC components while amplifying dynamic human motion. Second, we introduce a Physics-Aware Data Augmentation strategy to force the network to learn invariant morphological signatures rather than environment-specific noise. Furthermore, a Convolutional Block Attention Module (CBAM) is integrated to refine spatiotemporal features prior to Transformer-based sequence modeling. Extensive cross-domain evaluations across four distinct indoor environments demonstrate that our method achieves 97.6% accuracy in NLoS scenarios and 98.8% in completely unseen environments without target-domain fine-tuning. Finally, we deploy the proposed framework on an edge computing system equipped with commercial WiFi NICs. Real-world live inference field tests confirm the system's robustness against unseen environmental layouts and its capability for continuous, low-latency whole-home safety monitoring.

We present AttentionBender, a tool that manipulates cross-attention in Video Diffusion Transformers to help artists probe the internal mechanics of black-box video generation. While generative outputs are increasingly realistic, prompt-only control limits artists' ability to build intuition for the model's material process or to work beyond its default tendencies. Using an autobiographical research-through-design approach, we built on Network Bending to design AttentionBender, which applies 2D transforms (rotation, scaling, translation, etc.) to cross-attention maps to modulate generation. We assess AttentionBender by visualizing 4,500+ video generations across prompts, operations, and layer targets. Our results suggest that cross-attention is highly entangled: targeted manipulations often resist clean, localized control, producing distributed distortions and glitch aesthetics over linear edits. AttentionBender contributes a tool that functions both as an Explainable AI style probe of transformer attention mechanisms, and as a creative technique for producing novel aesthetics beyond the model's learned representational space.

Dropout is a widely used regularization technique in deep learning, but its effects are typically realized through stochastic masking rather than explicit optimization objectives. We propose a deterministic formulation that expresses dropout as an additive regularizer directly incorporated into the training loss. The framework derives explicit regularization terms for Transformer architectures, covering attention query, key, value, and feed-forward components with independently controllable strengths. This formulation removes reliance on stochastic perturbations while providing clearer and fine-grained control over regularization strength. Experiments across image classification, temporal action detection, and audio classification show that explicit dropout matches or outperforms conventional implicit methods, with consistent gains when applied to attention and feed-forward network layers. Ablation studies demonstrate stable performance and controllable regularization through regularization coefficients and dropout rates. Overall, explicit dropout offers a practical and interpretable alternative to stochastic regularization while maintaining architectural flexibility across diverse tasks.

For state-of-the-art image understanding, Vision Transformers (ViTs) have become the standard architecture but their processing diverges substantially from human attentional characteristics. We investigate whether this cognitive gap can be shrunk by fine-tuning the self-attention weights of Google's ViT-B/16 on human saliency fixation maps. To isolate the effects of semantically relevant signals from generic human supervision, the tuned model is compared against a shuffled control. Fine-tuning significantly improved alignment across five saliency metrics and induced three hallmark human-like biases: tuning reversed the baseline's anti-human large-object bias toward small-objects, amplified the animacy preference and diminished extreme attention entropy. Bayesian parity analysis provides decisive to very-strong evidence that this cognitive alignment comes at no cost to the model's original classification performance on in- (ImageNet), corrupted (ImageNet-C) and out-of-distribution (ObjectNet) benchmarks. An equivalent procedure applied to a ResNet-50 Convolutional Neural Network (CNN) instead degraded both alignment and accuracy, suggesting that the ViT's modular self-attention mechanism is uniquely suited for dissociating spatial priority from representational logic. These findings demonstrate that biologically grounded priors can be instilled as a free emergent property of human-aligned attention, to improve transformer interpretability.

Vision Transformers (ViTs) have become a dominant architecture in computer vision, yet their prediction process remains difficult to interpret because information is propagated through complex interactions across layers and attention heads. Existing attention based explanation methods provide an intuitive way to trace information flow. However, they rely mainly on raw attention weights, which do not explicitly reflect the final decision and often lead to explanations with limited class discriminability. In contrast, gradient based localization methods are more effective at highlighting class specific evidence, but they do not fully exploit the hierarchical attention propagation mechanism of transformers. To address this limitation, we propose Decision-Aware Attention Propagation (DAP), an attribution method that injects decision-relevant priors into transformer attention propagation. By estimating token importance through gradient based localization and integrating it into layer wise attention rollout, the method captures both the structural flow of attention and the evidence most relevant to the final prediction. Consequently, DAP produces attribution maps that are more class sensitive, compact, and faithful than those generated by conventional attention based methods. Extensive experiments across Vision Transformer variants of different model scales show that DAP consistently outperforms existing baselines in both quantitative metrics and qualitative visualizations, indicating that decision aware propagation is an effective direction for improving ViT interpretability.

Global self-attention drives modern graph transformers, yet the softmax at its core imposes a structural constraint rarely examined directly: every attention row is non-negative and sums to one, so each per-head output is a mass-conserving convex combination of value vectors. A node can never "attend to nothing." We argue this conservation constraint is a single root cause behind three pathologies usually studied in isolation: the collapse of node representations with depth (over-smoothing), a low-rank bottleneck on per-head outputs, and brittle optimization in deep stacks. Drawing on how sigmoid gating removes analogous attention sinks in language models, we introduce SigGate-GT, a graph transformer that applies a learned, per-head, input-conditioned sigmoid gate to the attention output inside the GraphGPS framework. The gate is a smooth, per-dimension "volume control" that can drive head outputs toward zero, relaxing the constraint without abandoning attention's probabilistic interpretation. Analytically and through synthetic experiments, we show the gate strictly increases the stable rank of per-head outputs, and connect this rank gain to all three manifestations. On five molecular and long-range benchmarks, SigGate-GT matches the prior best on ZINC (0.059 MAE), records the strongest result among the graph-transformer baselines we evaluate on ogbg-molhiv (82.47% ROC-AUC), and is competitive on ogbg-molpcba and the Long-Range Graph Benchmark, with statistically significant gains over GraphGPS on all five datasets (p < 0.05). Mechanism analyses confirm the diagnosis: gating slows over-smoothing (a 30% mean relative gain in representation diversity across 4-16 layers), keeps attention entropy from collapsing, and stabilizes training across a 10x learning-rate range, at about 1% parameter overhead on OGB and under 3% wall-clock cost.

To quantify the geometric expressivity of transformers, we introduce a tropical geometry framework to characterize their exact spatial partitioning capabilities. By modeling self-attention as a vector-valued tropical rational map, we prove it evaluates exactly to a Power Voronoi Diagram in the zero-temperature limit. Building on this equivalence, we establish a combinatorial rationale for Multi-Head Self-Attention (MHSA): via the Minkowski sum of Newton polytopes, multi-head aggregation expands the polyhedral complexity to $\mathcal{O}(N^H)$, overcoming the $\mathcal{O}(N)$ bottleneck of single heads. Extending this to deep architectures, we derive the first tight asymptotic bounds on the number of linear regions in transformers ($Θ(N^{d_{\text{model}}L})$), demonstrating a combinatorial explosion driven intrinsically by sequence length $N$, ambient embedding dimension $d_{\text{model}}$, and network depth $L$. Importantly, we guarantee that this idealized polyhedral skeleton is geometrically stable: finite-temperature soft attention preserves these topological partitions via exponentially tight differential approximation bounds.

Understanding the internal mechanisms of transformer-based large language models (LLMs) is crucial for their reliable deployment and effective operation. While recent efforts have yielded a plethora of attribution methods attempting to balance faithfulness and computational efficiency, dense component attribution remains prohibitively expensive. In this work, we introduce Dual Path Attribution (DPA), a novel framework that faithfully traces information flow on the frozen transformer in one forward and one backward pass without requiring counterfactual examples. DPA analytically decomposes and linearizes the computational structure of the SwiGLU Transformers into distinct pathways along which it propagates a targeted unembedding vector to receive the effective representation at each residual position. This target-centric propagation achieves O(1) time complexity with respect to the number of model components, scaling to long input sequences and dense component attribution. Extensive experiments on standard interpretability benchmarks demonstrate that DPA achieves state-of-the-art faithfulness and unprecedented efficiency compared to existing baselines.

Transformer models have redefined sequence learning, yet dot-product self-attention introduces a quadratic token-mixing bottleneck for long-context time-series. We introduce the Phasor Transformer block, a phase-native alternative representing sequence states on the unit-circle manifold $S^1$. Each block combines lightweight trainable phase-shifts with parameter-free Discrete Fourier Transform (DFT) token coupling, achieving global $\mathcal{O}(N\log N)$ mixing without explicit attention maps. Stacking these blocks defines the Large Phasor Model (LPM). We validate LPM on autoregressive time-series prediction over synthetic multi-frequency benchmarks against honest baselines: it beats a zero-parameter persistence baseline and, with the corrected gradient path, improves monotonically with depth before saturating, while remaining competitive-but-not-superior to self-attention at a fraction of the parameter count. Our results establish an explicit efficiency--accuracy frontier, showing that scalable temporal modeling in oscillatory domains can emerge from geometry-constrained phase computation with deterministic global coupling.

This paper studies simple transformers trained on a high-order Markov chain, where the model must incorporate information from multiple past positions, each with different statistical importance. We show that transformers learn the task incrementally, with each stage corresponding to learning how to copy information from a subset of positions via a sparse attention pattern. Notably, the learning dynamics transition from a competitive phase, where all heads focus on the statistically most important positions, to a cooperative phase, where different heads specialize in different patterns. We model these dynamics with simplified differential equations and prove stage-wise convergence of the resulting system. Functionally, these stages correspond to a sequence of increasingly expressive misspecified models, with the full model class reached only at the end. Overall, we give a theoretical account of how structured attention patterns and head specialization emerge in stages without an explicit curriculum, with implications for generalization in sequential tasks.

Mechanistic interpretability aims to reverse-engineer large language models (LLMs) into human-understandable computational circuits. However, the complexity of pretrained models often obscures the minimal mechanisms required for specific reasoning tasks. In this work, we train small, attention-only transformers from scratch on a symbolic version of the Indirect Object Identification (IOI) task, a benchmark for studying coreference-like reasoning in transformers. Surprisingly, a single-layer model with only two attention heads achieves perfect IOI accuracy, despite lacking MLPs and normalization layers. Through residual stream decomposition, spectral analysis, and embedding interventions, we find that the two heads specialize into additive and contrastive subcircuits that jointly implement IOI resolution. Furthermore, we show that a two-layer, one-head model composes information from the previous layer primarily through query-key interactions. These results demonstrate that task-specific training induces highly interpretable, minimal circuits, offering a controlled testbed for probing the computational foundations of transformer reasoning.

Whether artificial neural networks organize information comparably to the human brain remains unclear. Prior brain--AI alignment studies are constrained by specific inputs and tasks, limiting cross-modal comparison. Here we introduce a brain--model topological alignment space, mapping Transformer attention topology onto human intrinsic connectivity networks (ICNs) to enable task-free, modality-agnostic comparison. Analyzing 151 Transformer-based models with 62,480 attention head graphs, we observe a continuous arc-shaped distribution reflecting varying alignment. Models optimized for global semantics aligned with higher-order ICNs, while local-detail models aligned with sensory ICNs. Non-intuitive findings include reduced alignment in DINOv2 compared to its predecessors and a counterintuitive scaling inversion in distilled DeiT models, while fine-tuning and instruction tuning had limited effect. Alignment scores showed no significant correlation with ImageNet accuracy (r = 0.266, p = 0.156). This work offers a quantitative framework for comparing the organizational principles of artificial and biological systems.

The Transformer architecture has achieved remarkable success in natural language processing and high-level vision tasks over the past few years. However, the inherent complexity of self-attention is quadratic to the size of the image, leading to unaffordable computational costs for high-resolution vision tasks. In this paper, we introduce Concertormer, featuring a novel Concerto Self-Attention (CSA) mechanism designed for image deblurring. The proposed CSA divides self-attention into two distinct components: one emphasizes generally global and another concentrates on specifically local correspondence. By retaining partial information in additional dimensions independent from the self-attention calculations, our method effectively captures global contextual representations with complexity linear to the image size. To effectively leverage the additional dimensions, we present a Cross-Dimensional Communication module, which linearly combines attention maps and thus enhances expressiveness. Moreover, we amalgamate the two-staged Transformer design into a single stage using the proposed gated-dconv MLP architecture. While our primary objective is single-image motion deblurring, extensive quantitative and qualitative evaluations demonstrate that our approach performs favorably against the state-of-the-art methods in other tasks, such as deraining and deblurring with JPEG artifacts. The source codes and trained models will be made available to the public.