Asymptotic Signal Subspace Recovery in Softmax Attention Models

• URL: http://arxiv.org/abs/2606.22406v2
• Date: Tue, 23 Jun 2026 06:48:01 GMT
• Title: Asymptotic Signal Subspace Recovery in Softmax Attention Models
• Abstract summary: We study a stylized softmax-attention model in which a query vector is learned by gradient ascent from a collection of informative and tokens.<n>We show that the learned query converges almost surely to the one-dimensional signal subspace spanned by the latent informative direction.<n>These results provide a rigorous theoretical foundation for understanding attention mechanisms in high-dimensional noisy environments.
• Score: 16.70141872088154
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Attention mechanisms have demonstrated remarkable empirical success in identifying relevant information from large collections of tokens, yet the theoretical principles underlying this behavior remain poorly understood. We study a stylized softmax-attention model in which a query vector is learned by stochastic gradient ascent from a collection of informative and nuisance tokens. Exploiting the symmetry of the model, we derive a population objective and characterize the limiting ordinary differential equation governing the learning dynamics. Using tools from stochastic approximation and dynamical systems theory, we establish a rigorous connection between the stochastic learning algorithm and its deterministic limit. Our main result shows that, under suitable high-dimensional scaling assumptions and standard step-size conditions, the learned query converges almost surely to the one-dimensional signal subspace spanned by the latent informative direction. Equivalently, the query asymptotically recovers the latent signal up to the intrinsic sign ambiguity. These results provide a rigorous theoretical foundation for understanding attention mechanisms as signal extraction procedures in high-dimensional noisy environments and offer a dynamical-systems perspective on how attention discovers relevant information in the presence of substantial noise.

Related papers

• Extracting Governing Equations from Latent Dynamics via Multi-View Contrastive Learning [1.9754011041953696]
  We present DYSCO, a multi-view temporal contrastive learning algorithm that jointly recovers latent trajectories and the governing dynamics from such observations.<n>We offer theoretical guarantees for strong identification up to an affine indeterminacy, extending prior identifiability results to the realistic setting of noisy nonlinear observations.
  arXiv  Detail & Related papers  (2026-06-11T12:16:35Z)
• Continuous Data Assimilation with Learned Surrogate Dynamics [1.461685903055768]
  In many applications, the state dynamics are unknown or prohibitively expensive to simulate at the required resolution, leading to model error.<n>Motivated by this challenge and the increasing adoption of machine learning surrogates in data assimilation, this paper develops a unified finite-dimensional analysis of nudging algorithms that employ learned surrogate models of the dynamics.
  arXiv  Detail & Related papers  (2026-05-30T02:15:51Z)
• Representation Gap: Explaining the Unreasonable Effectiveness of Neural Networks from a Geometric Perspective [1.8618013369281987]
  We introduce the Representation Gap, a metric closely related to the generalization error, but admitting better-behaved dynamics.<n>We derive a precise equivalent of the Representation Gap and show that it is governed by a single parameter, the textitintrinsic dimension of the task.<n>We show that this dynamic also extends to a broader range of tasks and training algorithms.
  arXiv  Detail & Related papers  (2026-05-20T19:51:25Z)
• SIGMA: Scalable Spectral Insights for LLM Collapse [51.863164847253366]
  We introduce SIGMA (Spectral Inequalities for Gram Matrix Analysis), a unified framework for model collapse.<n>By utilizing benchmarks that deriving and deterministic bounds on the matrix's spectrum, SIGMA provides a mathematically grounded metric to track the contraction of the representation space.<n>We demonstrate that SIGMA effectively captures the transition towards states, offering both theoretical insights into the mechanics of collapse.
  arXiv  Detail & Related papers  (2026-01-06T19:47:11Z)
• Reinforcement Learning under Latent Dynamics: Toward Statistical and Algorithmic Modularity [51.40558987254471]
  Real-world applications of reinforcement learning often involve environments where agents operate on complex, high-dimensional observations. This paper addresses the question of reinforcement learning under $textitgeneral$ latent dynamics from a statistical and algorithmic perspective.
  arXiv  Detail & Related papers  (2024-10-23T14:22:49Z)
• Sparse identification of quasipotentials via a combined data-driven method [8.508437491732954]
  We show how to discover parsimonious equations for the quasipotential directly from data.<n>We use a neural network and a sparse regression algorithm, specifically designed to symbolically describe multistable energy landscapes.<n>Our model-unbiased analytical forms of the quasipotential is of interest to a wide range of applications aimed at assessing metastability and energy landscapes.
  arXiv  Detail & Related papers  (2024-07-06T11:27:52Z)
• A phase transition between positional and semantic learning in a solvable model of dot-product attention [30.96921029675713]
  Morelinear model dot-product attention is studied as a non-dimensional self-attention layer with trainable and low-dimensional query and key data. We show that either a positional attention mechanism (with tokens each other based on their respective positions) or a semantic attention mechanism (with tokens tied to each other based their meaning) or a transition from the former to the latter with increasing sample complexity.
  arXiv  Detail & Related papers  (2024-02-06T11:13:54Z)
• Discrete, compositional, and symbolic representations through attractor dynamics [51.20712945239422]
  We introduce a novel neural systems model that integrates attractor dynamics with symbolic representations to model cognitive processes akin to the probabilistic language of thought (PLoT) Our model segments the continuous representational space into discrete basins, with attractor states corresponding to symbolic sequences, that reflect the semanticity and compositionality characteristic of symbolic systems through unsupervised learning, rather than relying on pre-defined primitives. This approach establishes a unified framework that integrates both symbolic and sub-symbolic processing through neural dynamics, a neuroplausible substrate with proven expressivity in AI, offering a more comprehensive model that mirrors the complex duality of cognitive operations
  arXiv  Detail & Related papers  (2023-10-03T05:40:56Z)
• Deep Equilibrium Assisted Block Sparse Coding of Inter-dependent Signals: Application to Hyperspectral Imaging [71.57324258813675]
  A dataset of inter-dependent signals is defined as a matrix whose columns demonstrate strong dependencies. A neural network is employed to act as structure prior and reveal the underlying signal interdependencies. Deep unrolling and Deep equilibrium based algorithms are developed, forming highly interpretable and concise deep-learning-based architectures.
  arXiv  Detail & Related papers  (2022-03-29T21:00:39Z)
• A Free Lunch from the Noise: Provable and Practical Exploration for Representation Learning [55.048010996144036]
  We show that under some noise assumption, we can obtain the linear spectral feature of its corresponding Markov transition operator in closed-form for free. We propose Spectral Dynamics Embedding (SPEDE), which breaks the trade-off and completes optimistic exploration for representation learning by exploiting the structure of the noise.
  arXiv  Detail & Related papers  (2021-11-22T19:24:57Z)
• Multiplicative noise and heavy tails in stochastic optimization [62.993432503309485]
  empirical optimization is central to modern machine learning, but its role in its success is still unclear. We show that it commonly arises in parameters of discrete multiplicative noise due to variance. A detailed analysis is conducted in which we describe on key factors, including recent step size, and data, all exhibit similar results on state-of-the-art neural network models.
  arXiv  Detail & Related papers  (2020-06-11T09:58:01Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.