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26 papers in the last four weeks, up 44% on the four weeks before. 0.3% of all new papers.

Jul 13Week of Sep 28

Latest papers 351

May 29, 2026cs.LG

Reachability and asymptotics of Gaussian Transformer dynamics

We formulate data propagation through the Transformer, the machine learning architecture powering large language models, as a nonlinear control system on the space of probability measures. For the mean-field Transformer model with self-attention and affine feed-forward layers, we prove that Gaussian distributions remain exactly Gaussian along the induced flow. This invariance reduces the infinite-dimensional measure dynamics to a finite-dimensional bilinear control system governing the evolution of the mean and covariance, reformulates the expressive capacity of Transformers as a reachability problem for prescribed Gaussian moments, and reveals a novel connection with Riccati-type equations from classical filtering and control. For time-varying controls, we prove exact finite-time reachability of any target Gaussian distribution whose covariance matrix has the same rank as the initial one, this rank constraint being an intrinsic invariant of the dynamics. For time-invariant parameters, we derive explicit spectral conditions leading either to asymptotic stability toward positive-definite equilibria or to finite-time blow-up of the covariance. Numerical experiments complement the theory by showing that practical Transformers with Gaussian inputs remain close to moment-matched Gaussian distributions through early and intermediate layers, while Transformers with prescribed attention matrices reproduce the predicted covariance regimes: bounded evolution in stabilizing configurations and blow-up in destabilizing ones.

Albert Alcalde, Zhengping Ji, Enrique Zuazua

†Chair for Dynamics, Control, Machine Learning & Numerics (Alexander von Humboldt Profes- sorship), Department of Mathematics, Friedrich–Alexander-Universität Erlangen–Nürnberg, 91058 Erlangen, Germany. · ‡Departamento de Matem´aticas, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain. · Chair of Computational Mathematics, Fundaci´on Deusto. Av. de las Universidades, 24, 48007 Bilbao, Basque Country, Spain.

May 28, 2026cs.LG

Revisiting Padded Transformer Expressivity: Which Architectural Choices Matter and Which Don't

Recent work describes what transformers can and cannot compute through connections to boolean circuits, but existing results lack exact characterizations and are sensitive to modeling choices. Padded transformers -- to whose input filler symbols such as ``...'' are appended -- emerge as a useful gadget for establishing equivalences to circuit classes by providing polynomial space for adaptive parallel computation. However, only a limited set of padded transformer idealizations has been studied, leaving open how robustly these equivalences hold under changes to attention type, model width, and uniformity. We find that, under practical assumptions, padded transformers are surprisingly robust to all of these, and identify numeric precision and model depth as the main factors affecting expressivity. Concretely, we prove that polynomially padded

\text{L-uniform}

constant-precision transformers are equivalent to

\text{L-uniform AC}^0

, while growing-precision ones achieve

\text{L-uniform TC}^0

regardless of width. Furthermore, looping enables sequential processing analogous to circuits:

\log^d N

-looped constant-precision transformers reach

\text{FO-uniform AC}^d

, and growing-precision ones reach

\text{FO-uniform TC}^d

. Interestingly, growing width or precision beyond logarithmic does not increase expressivity, and all our results hold for both softmax and average hard attention transformers.

Anej Svete, William Merrill, Ryan Cotterell +1

1. ETH Zürich · 2. Allen Institute for AI

Transformer Expressivity Transformer Neural Network Expressivity

May 28, 2026cs.LG

Anti Mode-Collapse in Mean-Field Transformer via Auxiliary Variables

We use a mean-field-based transformer model to theoretically investigate how auxiliary variables, such as positional encoding, prevent mode collapse of self-attention mechanisms. The use of mean-field transformers to analyze the properties of self-attention mechanisms has garnered significant attention in recent years due to their ability to comprehensively analyze token interactions. However, analysis of this simple model suggests that mode collapse, where token distributions degenerate to a single point, occurs during long inferences (i.e., many layers), indicating a discrepancy with reality. This study investigates this mean-field transformer model and demonstrates that the introduction of auxiliary variables, such as positional encoding, acts as a counterforce against theoretical mode collapse. Specifically, we show that in the theoretical scheme, the energy-maximizing distribution does not degenerate to a single point; instead, it is characterized by a pushforward of the auxiliary variable distribution, thereby avoiding concentration in the Dirac measure. Our main examples are the positional encoding and the fixed prompt insertion treated as a parallel auxiliary-variable mechanism. Furthermore, we demonstrate that positional encoding and prompt insertion possess universality of representation in the limit, meaning that the limit distribution of inference can exactly represent a wide class of distributions. We also analyze several key properties of positional encoding and metastability, and validate our theoretical results through mathematical experiments.

Masaaki Imaizumi, Masanori Koyama, Noboru Isobe +1

The University of Tokyo, Tokyo, Japan · RIKEN Advanced Intelligene Project, Tokyo, Japan · Kyoto University, Kyoto, Japan

Transformer Self-Attention Model Collapse Positional Encoding

May 27, 2026cs.LG

Transformers Provably Learn to Internalize Chain-of-Thought

Chain-of-Thought (CoT) prompting substantially improves the sample efficiency of transformers, reducing the complexity of tasks like parity learning from exponential to polynomial in the input length. However, generating explicit reasoning steps at inference is computationally expensive. Implicit Chain-of-Thought (ICoT) has emerged as a promising empirical remedy that trains models to internalize intermediate steps within their hidden states, but its theoretical foundations remain poorly understood. We give the first theoretical analysis of ICoT, proving that an

L

-layer transformer trained under our proposed Log-ICoT curriculum learns

k

-parity with

\mathsf{poly}(n)

samples and

L = \log_2 k

training stages. This matches the sample efficiency of explicit CoT while eliminating its inference overhead, and extends prior one-layer parity guarantees to multi-layer architectures. Compared to standard ICoT, which removes thinking tokens one at a time, Log-ICoT removes them in geometric chunks, reducing the number of stages from linear in

k

to logarithmic. Experiments on multi-layer transformers confirm the theory and visualize how reasoning is progressively absorbed into deeper layers.

Yixiao Huang, Hanlin Zhu, Zixuan Wang +4

UC Berkeley · Princeton University

Transformer CoT Reasoning Latent CoT Reasoning

May 27, 2026cs.LG

Measure-to-measure Regression with Transformers

Many learning problems require predicting how populations evolve under an unknown transformation. A natural representation for such populations is a probability measure, with point clouds as a key example. In this work, we study the measure-to-measure (M2M) regression problem, in which one seeks to learn a map between probability measures from a finite collection of observed input-output pairs. In contrast to classical regression, where individual samples are transformed independently, M2M regression treats entire distributions as the data points. This perspective is vital in certain scientific applications, for example, cellular and molecular biology, where cells are known to evolve not as independent data points but as a collection. However, few existing approaches address the problem of M2M regression with sufficient expressivity and scalability. We present a formalization of nonlinear M2M regression and introduce two easy-to-use, expressive, and scalable approaches to learn such operators: transformers as static M2M maps and transformers as dynamic M2M velocity fields. Our approach leverages the natural measure-dependent and mean-field structure of transformers to learn nonlinear M2M maps on the space of probability distributions. We illustrate the effectiveness of our proposed method to generalize to unseen measures on synthetic experiments, interacting particle systems, and a large-scale patient-derived organoid dataset for predicting treatment response in colorectal cancer.

Matthew Vandergrift, Martha White, Yury Polyanskiy +2

University of Alberta · Canada CIFAR AI Chair · MIT +1

Transformer Operator Learning

May 27, 2026cs.GR

ClothTransformer: Unified Latent-Space Transformers for Scalable Cloth Simulation

Unified and scalable Transformers have recently achieved remarkable success in modeling diverse phenomena traditionally associated with computer graphics, such as 3D visual effects, rendering processes, and motion in videos. In this work, we take a step further by investigating whether modern Transformer techniques can tackle the challenging task of cloth simulation. To this end, we present ClothTransformer, a framework that reformulates cloth simulation as autoregressive sequence modeling in a learned latent space. Existing neural cloth simulators are largely specialized to single scenarios, intrinsically coupled to the mesh discretization, and lack robust collision handling. Our approach addresses these limitations through three contributions: (1) a unified Transformer architecture that handles diverse scenarios -- body-driven garments, robotic manipulation, and free-fall collisions -- under a single model and achieves approximately

4

--

9{\times}

lower error than prior state-of-the-art methods across all scenarios; (2) a scalable latent-space formulation that compresses arbitrary-resolution meshes into a fixed-size set of latent tokens, making temporal dynamics computation independent of mesh resolution; and (3) a diverse-scenario high-fidelity penetration-free dataset of

{\sim}

493.4k frames spanning all three settings, which enables a differentiable Continuous Collision Detection (CCD) module to suppress penetration artifacts. Project Page: https://yucrazing.github.io/clothtransformer/

Yu Zhang, Yidi Shao, Wenqi Ouyang +5

1S-Lab, Nanyang Technological University, Singapore · 2Feeling AI · University of Oxford +2

Transformer Physics-Based Simulation Deformable Object Simulation

May 26, 2026cs.LG

Customer Churn Prediction on Structured Data Using FT-Transformer and Stacking Ensembles

Customer churn prediction is essential across data-driven industries such as insurance, digital banking, eCommerce, and subscription platforms, where retaining existing customers is typically more cost-effective than acquiring new ones. Predicting churn on structured datasets remains challenging due to class imbalance, nonlinear feature interactions, and heterogeneous feature types. Tree-based ensemble methods consistently demonstrate strong performance in these contexts, often outperforming conventional neural networks. This study introduces a validated hybrid architecture that integrates feature-tokenized transformers (FT-Transformer) with gradient-boosted trees through calibration-aware stacking. The proposed framework addresses persistent gaps in statistical validation, probability calibration, and reproducibility found in prior research. The FT-Transformer captures higher-order feature interactions using self-attention, while XGBoost captures gradient-boosted decision boundaries with complementary inductive biases. Class imbalance is handled using class-weighted loss functions, thereby avoiding synthetic oversampling and preserving minority-class distributions. The models are ensembled using out-of-fold (OOF) stacking with a logistic regression meta-learner, which recalibrates overconfident base model outputs and learns optimal combination weights. On a public bank churn dataset, the hybrid model achieves 62.10% F1, 0.861 AUC-ROC, and 0.647 PR-AUC, outperforming the Multi-Layer Perceptron (MLP) baseline by 3.37 F1 points and 0.027 AUC under 5x5 cross-validation with 95% confidence intervals reported. Ablation studies demonstrate that both the transformer component and stacking strategy contribute materially to performance. The proposed methodology offers a reproducible and extensible reference architecture for contemporary churn prediction on structured tabular data.

Joyjit Roy, Samaresh Kumar Singh, Laxmi Shaw

Independent Researcher, Austin, TX, USA · Independent Researcher, Leander, TX · Texas A & M University-Victoria, Victoria, TX

Transformer Ensemble Learning Tabular Prediction

May 26, 2026cs.LG

Kan Extension Transformers: A Categorical Unification of Attention, Diffusion, and Predict-Detach Self-Conditioning

We propose Kan Extension Transformers (KETs) as a unifying categorical framework for a diverse group of Transformer implementations. The core claim is that a Transformer layer can be viewed as a weighted structured extension operator: standard attention is the singleton-neighborhood case, Geometric Transformer style incidence mixing is a sparse edge-restricted case, and KET is the higher-order simplicial case. This lens also clarifies a bridge to diffusion-style completion. When the extension operator acts on detached predictive carriers instead of teacher-forced hidden states, it becomes a valid self-conditioning mechanism that exposes noncausal structure without leaking gold future tokens. We include a comprehensive experimental validation of 12 different Transformer implementations varying across strict-causal and predict-detach regimes on Penn Treebank, WikiText-2, and WikiText-103. In the strict-causal setting, quadratic KET is the strongest model among the compared causal architectures on WikiText-2 and WikiText-103. Across all datasets, however, the largest gains come from the predict-detach regime rather than from changing the neighborhood family alone.

Sridhar Mahadevan

Adobe Research San Jose, CA · Academic affiliation: Research Professor, University of Massachusetts, Amherst

Transformer Autoregressive Language Modeling Transformer Attention

May 26, 2026cs.LG

More Expressive Feedforward Layers: Part I. Token-Adaptive Mixing of Activations

Feedforward network (FFN) layers account for a large fraction of parameters and nonlinear expressivity in Transformer-based large language models (LLMs). Despite the evolution from ReLU and GELU to gated variants such as SwiGLU, most FFN designs still use a single fixed activation function, applying the same nonlinear transformation to all tokens. In this work, we propose Mixture of Activations (MoA), a token-adaptive FFN design that mixes a dictionary of activation functions using lightweight input-dependent gates while sharing the same linear projections. As an input-independent counterpart, we also introduce learnable activations (LA), which form linear combinations of activation functions for both ReLU-type and SwiGLU-type FFNs. Theoretically, we establish strict finite-width expressive separations among fixed-activation FFNs, LA, and MoA: LA strictly contains fixed-activation FFNs, while MoA strictly contains LA, with the additional expressivity arising from input-dependent nonlinear hybridization. Empirically, we evaluate MoA through extensive pre-training experiments on dense and MoE language models ranging from 0.12B to 2B parameters under different token budgets, optimizers, and learning rate schedules. MoA consistently achieves lower terminal loss and exhibits more favorable scaling behavior than well-tuned baselines, with minimal parameter and computational overhead. These results suggest that token-adaptive activation mixing is a simple and effective mechanism for improving FFN expressivity in LLMs.

Mingze Wang, Jinbo Wang, Yikuan Xia +2

ByteDance Seed · Peking University

Transformer FFNs Transformer Expressivity Transformer Neural Network Activation Functions Feedforward Neural Networks Neural Network Expressivity

May 26, 2026cs.LG

PRISM: Position-encoded Regressive Inverse Spectral Model for Multilayer Thin-Film Design

The inverse problem of multilayer thin-film optical coatings design represents a complex combinatorial-continuous optimization challenge. We present PRISM (Position-encoded Regressive Inverse Spectral Model), a unified decoder-only autoregressive transformer that streamlines this process by jointly predicting discrete material selection and continuous thickness regression within a single backbone. PRISM introduces two primary architectural innovations: (1) spectrum prefix conditioning, which utilizes standard prefix tokens for in-context target injection, and (2) cumulative-depth Rotary Position Embeddings, which encode continuous thickness directly into the positional representation to preserve the physical spatial relationships of the stack. Our benchmarks demonstrate that a PRISM-13M model reduces MAE by over 50% compared to other transformer baselines while utilizing only one-fifth of the parameters. Furthermore, a 44M-parameter variant achieves state-of-the-art performance (MAE = 0.010) on our in-distribution validation benchmark and operates significantly faster than simulated annealing, offering a highly efficient alternative to classical optimization methods.

Runtian Wang, Renhao Xue, Baige Chen +1

Independent Researcher · Amazon

Transformer Materials Science Inverse Design

May 25, 2026cs.LG

The Quantization Benefits of Residual-Free Transformers

Large-scale transformer training and deployment are increasingly constrained by the transfer of activations, gradients, and optimizer states across accelerators. Low-bit quantization offers a natural remedy, but transformer activations are often heavy-tailed and outlier-dominated, making simple quantization highly lossy. We show that this difficulty is not only a property of the quantizer, but also of the architecture. Specifically, residual connections can drive transformer activations away from Gaussianity during training. Using controlled comparisons between residual and residual-free transformers, we demonstrate that this effect leads to substantially higher quantization error and accuracy degradation at low precision in residual models. We explain the phenomenon through an excess kurtosis analysis, showing that residual mixing can amplify non-Gaussianity, whereas dense mixing in residual-free contracts non-Gaussianity. We then show that residual-free transformers can be made trainable using orthogonal initialization, spectral or second-order optimization, and depth-aware scaling of attention temperature. In language tasks, while there is a small drop in full precision performance, these models retain near-Gaussian activations and exhibit significantly improved robustness to low-bit quantization. Our results identify an accuracy--compressibility trade-off in transformer design and motivate architecture-level approaches to quantization-friendly foundation models.

Yiping Ji, Mahalakshmi Sabanayagam, Peyman Moghadam +2

Australian Institute for Machine Learning, Adelaide University · DATA61, CSIRO

Transformer Activation Outliers Activation Quantization Low-Bit Quantization

May 25, 2026cs.LG

Analogies between Transformer Layers and Power Method

In the paper we show that there is an analogy between the operations occurring in a layer of a transformer (projections and layer normalizations, disregarding the feedforward neural network) and a step in the power method. Coherently with this analogy, we show that passing through a layer the tokens tend to be tilted towards the principal eigenvector of a matrix which is the product of the output and value weight matrices of that layer. In the special case of a transformer with shared weights (i.e., in which all layers have identical weights) then the alignment with this principal eigenvector is particularly evident empirically, and can also be shown analytically. The analogy also suggests a method to steer the output of the transformer towards an arbitrary desired direction in token space.

Chenglong Li, Claudio Altafini

Transformer Interpretability Transformer Activation Steering

May 23, 2026cs.LG

Polymorphism Is Rotation: Operational Mechanistic Interpretability from a Two-Layer Transformer to Pythia-70m

Independently trained transformers compute the same function in residual-stream bases that differ by a uniform random rotation on

\mathrm{SO}(d_{\mathrm{model}})

. We call this phenomenon polymorphism: same function, mutually unintelligible interior coordinates. One matrix multiplication per model pair removes it: an orthogonal Procrustes fit on a single batch of activations transfers sparse-autoencoder feature dictionaries and steering vectors between independently trained models, with no retraining. The phenomenon is invisible to the standard SAE universality metric. Decoder-column cosine similarity matches across seeds at 98%, the SAE-universality headline number, while an SAE trained on one seed reconstructs another seed's activations at negative explained variance, worse than predicting the constant mean. The decoder columns align; the encoder reads from a rotated frame. A single Procrustes rotation

R

restores reconstruction to within 0.025 EV of the within-seed ceiling at every internal site.

R

is Haar-distributed:

\|R - I\|_F

matches the random-orthogonal prediction

\sqrt{2 d_{\mathrm{model}}}

to 0.1% at

d_{\mathrm{model}} = 512

, and a Kolmogorov-Smirnov test of

R

's eigenvalue spectrum against Haar

\mathrm{SO}(d_{\mathrm{model}})

returns

p \approx 1.000

pooled and per-pair. Diff-of-means steering vectors transfer in three regimes by alignment with

R

's invariant subspace: clean when pinned by shared output weights, partial when overlapping the rotated subspace, inverted otherwise. With no shared I/O (Pythia), all three collapse to universally inverted. The same rotation account holds across training checkpoints within a single run. Validated on a 104k-parameter Dyck-3 transformer and nine independently-trained Pythia-70m seeds on The Pile, via a pre-registered four-bar operational framework. Frontier-scale (10B+) replication remains open.

Jordan F. McCann

Independent Researcher

Transformer Interpretability Transformer Sparse Autoencoders Mechanistic Interpretability Representation Alignment Latent Space Alignment

May 23, 2026cs.LG

Momentum Streams for Optimizer-Inspired Transformers

The residual update of a pre-norm Transformer layer admits an interpretation as one step of a first-order optimizer acting on a surrogate token energy, wherein the attention and MLP sublayers function as gradient oracles. Based on this observation, we build a family of optimizer-inspired Transformers (triple-momentum, Adam/AdamW, Muon, SOAP) and compare them under matched compute. In our main pretraining experiment, the triple-momentum TMMFormer achieves the lowest validation loss, outperforming the vanilla Transformer and prior architectural variants. A controlled ablation and supporting theory show that momentum, not preconditioning, is the main source of the gain. We further show that TMMFormer and other momentum-based designs reach flatter minima than the vanilla Transformer, which leads to less forgetting and better generalization.

Jingchu Gai, Nai-Chieh Huang, Jiayun Wu

Carnegie Mellon University

Deep Learning Optimization Transformer Momentum Methods Neural Network Optimization Flat Minima

May 22, 2026cs.LG

Complete-muE: Optimal Hyperparameter Transfer and Scaling for MoE Models

We propose Complete-muE, a framework which targets hyperparameter transfer across dense FFN and any Mixture-of-Experts (MoE) setups in transformer blocks. Existing tools such as

μ

P (requires fixed architectue) or SDE (requires fixed per-step token count) cannot directly solve the hyperparameter transfer problem in MoE setups because Dense to MoE transfer or MoE total experts scaling changes both architecture and tokens per expert. Complete-muE solves this challenge with a two-bridge system: Bridge~~I maps between dense FFN and Dense MoE by active-width $μ$ P with a normalized router scale. Bridge~~II maps between Dense MoE and sparse MoE by activated-expert scaling, where the first-order SDE LR/WD correction cancels while a bounded residual

σ_0

shift remains. The resulting transfer rule, which we term as Complete muE, covers changes in activated experts, total capacity, granularity, and shared/group-balanced hybrids for MoE models as well as network width/depth, batch size, and duration changes for general Transformer models. Extensive language model and diffusion model pretraining experiments confirm that complete-muE yields relatively stable hyperparameter optima across model architectures and parameter counts -- with only minor drift consistent with the non-strict SDE behavior of Bridge~II. In practice this drift is small enough that hyperparameters tuned on a single dense reference transfer near-optimally to all MoE configurations -- \emph{tune dense once, transfer to all} is the practical recipe at the core of Complete-muE. This enables MoE models to achieve accelerated convergence speedup over dense models when scaling model capacity without costly hyperparameter search.

Hongwu Peng, Ohiremen Dibua, Yuanjun Xiong +3

Adobe Research

Transformer Neural Network Optimization Hyperparameter Transfer Mixture-of-Experts Models

May 22, 2026cs.LG

Preisach Attention: A Hysteretic Model of Sequential Memory

We introduce the Preisach Attention Layer (PAL), a novel sequence modelling architecture grounded in the classical Preisach hysteresis operator from mathematical physics. PAL replaces the softmax attention mechanism with a binary relay operator parameterised by learned activation and deactivation thresholds, maintaining a stack of local extrema as its internal state. A single-layer PAL-Transformer with O(1) depth is Turing-complete under arbitrary precision arithmetic, achievable through simulation of a two-stack pushdown automaton -- in contrast to the O(log n) depth required by standard hard-attention transformers. Second, we prove that the function classes computable by PAL and by the transformer are incomparable: PAL computes historical range statistics in O(1) layers that require O(log n) layers for transformers, while transformers support random-access retrieval that PAL cannot perform without auxiliary state. The separating property is rate-independence -- PAL responds only to the sequence of local extrema, not to absolute token positions or temporal spacing. Third, we show that the extremum stack constitutes a minimal sufficient statistic of the input history for all rate-independent functionals, providing a formal analogue of the wiping property in classical hysteresis theory. PAL is thus an efficient architecture for tasks with long episodic memory and weak positional dependence, with O(n log n) total inference cost versus O(n^2) for standard attention.

Piotr Frydrych

The Metrology and Biomedical Engineering Institute · Faculty of Mechatronics, Warsaw University of Technology

Transformer Efficient Attention Neural Network Expressivity

May 21, 2026cs.LG

How Many Different Outputs Can a Transformer Generate?

We study how we can leverage only a handful of characteristics of a transformer's architecture to closely predict the number of different sequences it can output, both qualitatively and quantitatively. We provide an upper bound depending on the length of the prompt, which we show empirically to be tight up to a factor less than 10, across architectures and model sizes. Our analysis also provides a theoretical explanation for previously observed empirical failures of transformers on simple sequence tasks, such as copying and cramming. Formally, we prove that (i) the maximal length of accessible sequences (those that the transformer can output for some prompt) grows linearly with the prompt length, (ii) beyond a critical threshold, the proportion of accessible sequences decays exponentially with sequence length, and (iii) the linear coefficient relating prompt length to accessible sequence length admits a theoretical upper bound. Notably, these results hold even with unbounded context and computation time.

Maxime Meyer, Mario Michelessa, Caroline Chaux +1

Department of Mathematics, National University of Singapore, Singapore, 117543 · IPAL, IRL2955, Singapore · School of Computing, National University of Singapore, Singapore, 117543 +2

Transformer Expressivity Transformer Neural Network Expressivity

May 21, 2026cs.LG

Towards Verifiable Transformers: Solver-Checkable Circuit Explanations

Mechanistic interpretability often identifies circuits inside Transformer models, but explanations of those circuits are usually validated through examples, ablations, and manual reasoning. This leaves a gap between finding a plausible circuit and proving what the circuit does. We introduce Verifiable Transformers, a framework for converting task-localized Transformer circuits into bounded, solver-checkable claims. Given a behavior, a finite task domain, and a candidate-token projection, we extract a task circuit and verify properties such as projected functional equivalence, edge necessity, task-relevant invariance, and final-residual robustness. Direct verification encodes the extracted circuit itself into an SMT solver. When a circuit contains operators that are not exactly or tractably encodable, surrogate-mediated verification fits an SMT-encodable surrogate, validates it against the extracted circuit over the bounded domain, and verifies symbolic explanations against the surrogate. We instantiate direct verification with a GPT-style architecture using Signed L1 BandNorm, sparsemax attention, and LeakyReLU. On small symbolic sequence tasks, we train an SMT-representable Transformer, extract sparse circuits for quote closing and bracket type tracking, and exhaustively verify projected functional equivalence, content invariance, edge necessity, and final-residual robustness. At GPT-2 scale, the same operator stack trains stably on OpenWebText, although naive direct SMT verification remains intractable. We also demonstrate surrogate-mediated verification on task-localized circuits with hard-to-encode attention, showing both verified symbolic explanations and solver-generated counterexamples. The goal is not full-model verification, but a concrete path for turning mechanistic circuit explanations into formal propositions that can be proven or refuted.

Neel Somani

Independent Researcher

Neural Network Verification Transformer Mechanistic Interpretability

May 20, 2026stat.ML

Large-Step Training Dynamics of a Two-Factor Linear Transformer Model

Gradient-flow analyses show that simplified linear transformers can learn the in-context linear-regression algorithm, but they do not explain the finite-step behavior of gradient descent at large learning rates. Motivated by empirical work on high-learning-rate transformer instabilities and by the cubic-map phase diagram for quadratic regression, we study an exactly reducible one-prompt linear-transformer training problem. After normalization, the dynamics reduce to a two-factor product map with an effective step-size parameter

μ

. On the balanced slice, this map recovers the known scalar cubic transition from monotone convergence to catapult convergence, periodic and chaotic bounded nonconvergence, and divergence. We then analyze the full two-dimensional system and show that, for

0<μ<2

, it has an explicit invariant Chebyshev ellipse separating forward-invariant regions; this ellipse carries off-balanced chaotic dynamics but is transversely repelling, while balanced scalar attractors can be transversely attracting. These results show that large constant learning rates can change the training attractor of the learned transformer rather than merely accelerating convergence: beyond sharp stability thresholds, finite-step training may settle into cycles, bounded chaos, or divergence instead of a single in-context linear-regression solution. We also discuss the consequences for mini-batch gradient descent based training methods.

Krishnakumar Balasubramanian

Department of Statistics, University of California, Davis

Transformer Phase Transitions Neural Network Training Dynamics Gradient Descent Dynamics

May 20, 2026cs.CV

UniT: Unified Geometry Learning with Group Autoregressive Transformer

Recent feed-forward models have significantly advanced geometry perception for inferring dense 3D structure from sensor observations. However, its essential capabilities remain fragmented across multiple incompatible paradigms, including online perception, offline reconstruction, multi-modal integration, long-horizon scalability, and metric-scale estimation. We present UniT, a unified model built upon a novel Group Autoregressive Transformer, which reformulates these seemingly disparate capabilities within a single framework. The key idea is to treat groups of sensor observations as the basic autoregressive units and predict the corresponding point maps in an anchor-free and scale-adaptive manner. More specifically, diverse view configurations in both online and offline settings are naturally unified within a single group autoregression process. By varying the group size, online mode operates over multiple autoregressive steps with single-frame groups, whereas offline mode aggregates a multi-frame group in a single forward pass. Meanwhile, a queue-style KV caching mechanism ensures bounded autoregressive memory over long horizons. This is enabled by reducing long-range dependencies on early frames through anchor-free relational modeling, thereby allowing outdated memory to be discarded on the fly. To improve metric-scale generalization across scenes, a scale-adaptive geometry loss is further introduced within this framework. It couples relative geometric constraints with a partial absolute scale term, implicitly regularizing global scale and inducing a progressive transition from scale-invariant geometry to metric-scale solutions. Together with a dedicated modal attention module for integrating auxiliary modalities, UniT achieves state-of-the-art performance in unified geometry perception, as validated on ten benchmarks spanning seven representative tasks.

Haotian Wang, Yusong Huang, Zhaonian Kuang +4

Intelligent Transportation Thrust of the Systems Hub, The Hong Kong University of Science and Technology (GZ), Guangzhou, P.R.China. · The National Key Laboratory of Human-Machine Hybrid Augmented Intelligence, Xi’an Jiaotong University, Xi’an, P.R.China. · Applied Science, Amazon.com, Inc., USA.

Transformer Depth Estimation Multi-View 3D Reconstruction Geometric Representation Learning Feed-Forward 3D Reconstruction

May 20, 2026cs.LG

HORST: Composing Optimizer Geometries for Sparse Transformer Training

Sparsifying transformers remains a fundamental challenge, as standard optimizers fail to simultaneously encourage sparsity and maintain training stability. Effective adaptive optimizers exhibit an implicit

L_{\infty}

bias favoring stability, yet, sparsity requires an

L_1

bias. To integrate sparsity, we propose a composition of optimizer steps, which we cast as non-commutative operators to analyze and combine their optimization geometry in a principled way. This yields HORST (Hyperbolic Operator for Robust Sparse Training), a modular optimizer that inherits stability from adaptive methods while inducing

L_1

sparsity bias through a hyperbolic mirror map. Our experiments demonstrate its utility for sparse training of transformers on both vision and language tasks. HORST consistently and significantly outperforms AdamW baselines across all sparsity levels, with large gains at higher sparsity.

Tom Jacobs, Rohan Jain, Rebekka Burkholz

CISPA Helmholtz Center for Information Security

Deep Learning Optimization Transformer Neural Network Optimization Mirror Descent Sparse Neural Networks

May 20, 2026cs.LG

Markovian Circuit Tracing for Transformer State Dynamic

Many sequence computations are easier to study as movement through internal states than as isolated local circuits. We introduce Markovian Circuit Tracing (MCT), a diagnostic pipeline for testing whether transformer activations contain coarse state-transition structure. The benchmark uses synthetic Hidden Markov Model (HMM) tasks where latent states, transition matrices, Bayesian belief vectors, Bayes-optimal predictions, and forced-state counterfactual targets are known exactly. Across six HMM families and three seeds per family, tiny causal transformers learn near-Bayes next-token predictors, with mean excess loss over Bayes of 0.0138. Residual activations contain partial Bayesian belief information in this controlled synthetic benchmark. State abstractions extracted from these activations recover coarse transition signal, strongest in persistent and lower-state regimes, and weaker in ambiguous-emission and six-state regimes. The clearest result comes from state forcing. Patching a recovered-state centroid reduces KL to the exact HMM counterfactual target from 0.1957 in the unpatched model to 0.0532 on average, beating wrong-state, mean-activation, random-activation, and shuffled-label controls. The contribution is a controlled benchmark and evaluation framework for transformer state-dynamics interpretability, with MCT as a simple reference pipeline

Abdullah X

Project AWARE and Zephara AI

Transformer Interpretability Transformer State Abstraction State Tracking Sequence Modeling Hidden-State Probing

May 20, 2026cs.LG

Most Transformer Modifications Still Do Not Transfer at 1-3B: A 2020-2026 Update to Narang et al. (2021) with Downstream Evaluation and a Noise Floor

Narang et al. (2021) evaluated 40+ Transformer modifications at T5-base scale and concluded that most did not transfer. Five years later, the typical working regime has moved to 1-3B parameters, downstream evaluation has replaced pretraining perplexity, and a substantially different catalogue of modifications has emerged. We revisit their question by testing 20 post-2021 Transformer modifications at 1.2B and 3B under strict iso-data, iso-compute, iso-recipe control, with a multi-seed baseline noise floor and CLIMB-12 downstream evaluation as the primary metric. The central finding reproduces theirs at this curated set: most modifications do not transfer. Of the 20 modifications, only two clear Bonferroni correction at 1.2B; one of those two further fails to train stably at 3B under the shared recipe. We also find that the loss-downstream gap reported by Tay et al. (2023) enlarges several-fold for attention-output modifications: two significant failures converge to within 2-3% of baseline validation loss yet drop 6-16 CLIMB-points. We conclude that noise-floor reporting, downstream evaluation, and cross-scale stability testing are now prerequisites for architecture comparisons at 1-3B.

Yang Zhao, Jiahao Lu, Bin Huang +2

Tencent

LLM Evaluation Transformer

May 19, 2026cs.AI

Position: The Turing-Completeness of Autoregressive Transformers Relies Heavily on Context Management

Many works make the eye-catching claim that Transformers are Turing-complete. However, the literature often conflates two distinct settings: (i) a fixed Transformer system setting, in which a fixed autoregressive Transformer is coupled with a fixed context-management method to process inputs of different lengths step by step, and (ii) a scaling-family setting, in which a family of different models (with increasing context-window length or numerical precision) is used to handle different input lengths. Existing proofs of Transformer Turing-completeness are frequently established in setting (ii), whereas real-world LLM deployment and the standard notion of Turing-completeness correspond more naturally to setting (i). In this paper, we first formalize the fixed-system setting, thereby providing a concrete characterization of how real-world LLMs operate. We then argue that results proved in the scaling-family setting provide theoretically meaningful resource bounds but do not establish Turing-completeness, thereby clarifying a common misinterpretation of existing results. Finally, we show that different context-management methods can yield sharply different computational power, and we advocate the position that context management is a central component that critically determines the computational power of real-world autoregressive Transformers.

Guanyu Cui, Zhewei Wei, Kun He

Gaoling School of Artificial Intelligence, Renmin University of China, Beijing, China · DEKE Lab, Renmin University of China, Beijing, China

Transformer Expressivity Transformer LLM Context Management

May 18, 2026cs.LG

Chessformer: A Unified Architecture for Chess Modeling

Chess has long served as a canonical testbed for artificial intelligence, but modeling approaches for its central tasks have diverged. Maximizing playing strength, predicting human play, and enabling interpretability are typically solved with disparate architectures, and these designs are often misaligned with the geometry of the domain. This raises the natural question of whether these objectives require separate modeling paradigms, or if there exists a single architecture that supports them simultaneously. We introduce Chessformer, a unified architecture that advances the state of the art on all three central goals in chess modeling. Chessformer is an encoder-only transformer that represents board squares as tokens, augments self-attention with a novel dynamic positional encoding called Geometric Attention Bias (GAB) that adapts to domain-specific geometry, and predicts actions with an attention-based source-destination policy head. We evaluate Chessformer on each front. First, we develop \maiathree, a family of models for human move prediction that reaches 57.1% move-matching accuracy, significantly surpassing the previous state of the art with fewer than a quarter of the parameters. Second, we integrate Chessformer into Leela Chess Zero, a leading open-source engine, adding over 100 Elo of playing strength and resulting in tournament victories over Stockfish in major computer chess competitions. Third, we show that Chessformer's square-token design makes attention patterns and activations directly attributable to board squares, enabling granular interpretability analyses that prior architectures do not naturally support. More broadly, our results demonstrate that aligning a model's tokenization, positional encoding, and output design with the underlying structure of a domain can yield simultaneous gains in performance, human compatibility, and interpretability.

Daniel Monroe, George Eilender, Philip Chalmers +2

University of Toronto · Williams College

Transformer Interpretability Game-Playing Agents Transformer Positional Encoding Chess

May 18, 2026cs.LG

The Expressive Power of Low Precision Softmax Transformers with (Summarized) Chain-of-Thought

Existing expressivity results for transformers typically rely on hardmax attention, high precision, and other architectural modifications that disconnect them from the models used in practice. We bridge this gap by analyzing standard transformer decoders with softmax attention and rounding of activations and attention weights, while allowing depth and width to grow logarithmically with the context length. As an intermediate step, we construct hardmax transformers with ternary activations and well-separated attention scores that simulate Turing machines using Chain-of-Thought (CoT). This lets us convert the constructions to equivalent softmax transformers without the unrealistic parameter magnitudes or activation precision that prior approaches would require. Using the same technique, we analyze a recently proposed summarized CoT paradigm and show that it simulates Turing machines more efficiently, with model size scaling logarithmically in a space bound rather than a time bound. We empirically test predictions made by our results on a Sudoku reasoning task and find better alignment with learnability than for prior high-precision results. Our code is available at https://github.com/moritzbroe/transformer-expressivity.

Moritz Brösamle, Stephan Eckstein

Department of Mathematics, University of Tübingen, Germany.

Transformer Expressivity Transformer CoT Reasoning Neural Network Expressivity

May 18, 2026cs.LG

InfoFlow: A Framework for Multi-Layer Transformer Analysis

While the approximation properties of single-layer Transformer architectures have been studied in recent works, a rigorous theoretical understanding of the multi-layer setting remains limited. In this work, we establish that multi-layer Transformers possess fundamentally different approximation capabilities from single-layer ones: for certain retrieval tasks, any single-layer Transformer requires least

Ω(\varepsilon^{-k})

parameters to achieve precision

\varepsilon

, where

k

grows linearly with sequence length

T

, whereas a two-layer Transformer with a single head per layer achieves the same approximation precision with at most

O (\varepsilon^{-1})

parameters. To understand this separation, we identify two structural mechanisms underlying multi-layer approximation. Specifically, softmax attention can only efficiently retrieve the token attaining the maximum attention score, incurring exponential-in-length parameter cost for

k

-th largest retrieval with

k \geq 2

. Moreover, the parameter cost of decoding coupled information scales with the size of the retrieved token set. Motivated by these findings, we propose InfoFlow, a framework for multi-layer Transformers. The framework tracks an information set of accessible input positions at each token and layer, assigning an explicit approximation rate to each mode of information propagation. This abstraction recovers known approximation bounds, remains consistent with experimental observations on trained networks, and yields concrete predictions in settings where direct theoretical analysis is currently intractable. Our results provide a principled framework for reasoning about the approximation efficiency of multi-layer Transformers.

Penghao Yu, Haotian Jiang, Zeyu Bao +1

Department of Mathematics National University of Singapore · Institute for Functional Intelligent Materials National University of Singapore · National University of Singapore +1

Transformer Expressivity Transformer Self-Attention Transformer Attention Neural Network Approximation Theory

May 17, 2026math.OC

Training Infinitely Deep and Wide Transformers

Transformers have become the dominant architecture in modern machine learning, yet the theoretical understanding of their training dynamics remains limited. This paper develops a rigorous mathematical framework for analyzing gradient-based training of transformers in the mean-field regime, where both the depth (number of layers) and width (number of attention heads) tend to infinity. While ResNet training can be understood as controlling a neural ODE, transformer training corresponds to controlling a neural PDE, due to the coupling of multiple token distributions through the attention mechanism. Our mean-field model features two types of measure representations: token distributions evolving through layers and attention parameters at each layer. We establish well-posedness of the forward pass through infinitely deep transformers, characterizing token evolution via flow maps that satisfy ODEs in function spaces. Using adjoint sensitivity analysis, we derive an explicit formula for the conditional Wasserstein gradient of the training risk, involving adjoint variables governed by backward ODEs. We prove the existence and uniqueness of gradient flow curves in the conditional Wasserstein metric space, establishing a rigorous foundation for gradient-based transformer training. A key technical contribution is providing necessary and sufficient conditions for injectivity of the Neural Tangent Kernel (NTK) for attention mechanisms: we show that NTK injectivity is equivalent to linear independence of log-sum-exp functions modulo affine functions, a condition satisfied by diverse token distributions, including discrete distributions, uniform distributions, and Gaussian mixtures. Under this NTK injectivity assumption, we prove that gradient flow converges to global minima when the initial loss is sufficiently small, eliminating spurious local minima from the optimization landscape.

Raphaël Barboni, Maarten V. de Hoop, Takashi Furuya +1

Bocconi University · Rice University · Doshisha University, RIKEN AIP +1

Transformer Wasserstein Gradient Flows Neural Network Training Dynamics Neural Tangent Kernel

May 17, 2026cs.LG

A Two-Parameter Weibull Framework for Diagnosing Transformer Weight Distributions

We apply the Weibull distribution -- a two-parameter family from extreme-value theory -- as a diagnostic framework for element-wise weight magnitude distributions in transformers. At initialization, i.i.d. Gaussian weights give |w| ~ HalfNormal, yielding k ~ 1.20 via middle-80% probability-plot fit (the protocol used throughout this work). This anchor makes k a principled, architecture-independent measuring stick for training dynamics; fitting each weight matrix independently at every layer at every checkpoint enables per-component, per-layer, and per-step diagnostics that aggregate statistics cannot resolve. Applying this framework to 12 model entries spanning 7 architectural families (Pythia, OLMo-1/2, LLaMA-3, Mistral, Qwen2.5/3) reveals three findings. First, FFN modules and the attention output projection W_o -- the Transmission Class -- fall in a narrow k band: median terminal k in [1.186, 1.204] across 12 entries (cross-family CV = 0.51%), shared across SwiGLU/GeLU activations, Pre-LN/QK-Norm placements, and 70M-14B sizes. Second, the attention input projections W_q, W_k -- the Selection Class -- depart from the Weibull family, with severity shaped by storage: separately-stored Q/K (OLMo-1, OLMo-2) yields k in [0.76, 0.99] (deep); GQA models yield k in [1.10, 1.16] (mild); Pythia's merged W_qkv occupies a transitional zone tracking training budget T/tau monotonically. Third, lambda grows substantially during training and scales with sqrt(eta/lambda_wd) within the Pythia family (Pearson r = 0.94, three Transmission kinds), directionally consistent with Fan et al. (2025). The two parameters carry independent information: k labels the functional class, lambda labels training progress. We release npm-weibull-py v0.4 (Python library) and DATABASE_v9_1 at https://github.com/tiexinding/NPM-Weibull-public .

Tiexin Ding

Independent Researcher

Transformer Weight-Space Analysis Neural Network Training Dynamics

May 16, 2026cs.LG

OPTNet: Ordering Point Transformer Network for Post-disaster 3D Semantic Segmentation

Post-disaster damage assessment requires rapid and accurate semantic segmentation of 3D point clouds to identify critical infrastructure such as damaged buildings and roads. Early Point Transformers (e.g., PTv1, PTv2) relied on computationally expensive neighbor searching (k-NN) and Farthest Point Sampling (FPS). To improve efficiency, recent architectures like Point Transformer V3 (PTv3) adopted static serialization methods, such as Hilbert curves or Z-order, to organize unstructured points for window-based attention. However, these fixed orderings are not optimal for capturing the complex geometry of disaster scenes. In this paper, we propose OPTNet (Ordering Point Transformer Network), which introduces a learnable Point Sorter module. OPTNet utilizes a self-supervised ordering loss to dynamically predict an optimal permutation that maximizes the locality of the attention mechanism. We evaluate our method on the 3DAeroRelief dataset, significantly outperforming state-of-the-art baselines.

Nhut Le, Ehsan Karimi, Maryam Rahnemoonfar

Computer Science and Engineering, Lehigh University, Bethlehem PA 18015, US · Civil and Environmental Engineering, Lehigh University, Bethlehem PA 18015, US

Disaster Damage Assessment Transformer Point Cloud Learning 3D Semantic Segmentation