How Many Heads Make an SSM? A Unified Framework for Attention and State Space Models

• URL: http://arxiv.org/abs/2512.15115v1
• Date: Wed, 17 Dec 2025 06:15:24 GMT
• Title: How Many Heads Make an SSM? A Unified Framework for Attention and State Space Models
• Abstract summary: We introduce a unified framework that represents a broad class of sequence maps via an input-dependent effective interaction operator $W_ij(X)$.<n>We prove an Equivalence (Head-Count) Theorem showing that, within our multi-head factorized class, representing a linear SSM whose lag operators span a $k$-dimensional subspace on length-$n$ requires and is achievable with $H=k$ heads.
• Score: 2.1693096503777003
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Sequence modeling has produced diverse architectures -- from classical recurrent neural networks to modern Transformers and state space models (SSMs) -- yet a unified theoretical understanding of expressivity and trainability trade-offs remains limited. We introduce a unified framework that represents a broad class of sequence maps via an input-dependent effective interaction operator $W_{ij}(X)$, making explicit two recurring construction patterns: (i) the Unified Factorized Framework (Explicit) (attention-style mixing), in which $W_{ij}(X)$ varies through scalar coefficients applied to shared value maps, and (ii) Structured Dynamics (Implicit) (state-space recurrences), in which $W_{ij}$ is induced by a latent dynamical system. Using this framework, we derive three theoretical results. First, we establish the Interaction Rank Gap: models in the Unified Factorized Framework, such as single-head attention, are constrained to a low-dimensional operator span and cannot represent certain structured dynamical maps. Second, we prove an Equivalence (Head-Count) Theorem showing that, within our multi-head factorized class, representing a linear SSM whose lag operators span a $k$-dimensional subspace on length-$n$ sequences requires and is achievable with $H=k$ heads. Third, we prove a Gradient Highway Result, showing that attention layers admit inputs with distance-independent gradient paths, whereas stable linear dynamics exhibit distance-dependent gradient attenuation. Together, these results formalize a fundamental trade-off between algebraic expressivity (interaction/operator span) and long-range gradient propagation, providing theoretical grounding for modern sequence architecture design.

Related papers

• Toward a mechanistic understanding of inference in visual cortex and diffusion models [46.86106767256413]
  We describe a model of perceptual inference in primary visual cortex (V1)<n>We efficiently train these recurrent dynamics using a denoising score-matching objective and implicit differentiation.<n>This model exhibits exceptionally good denoising performance, restoring image features such as extended contours amid extreme visual ambiguity.
  arXiv  Detail & Related papers  (2026-07-17T07:11:57Z)
• Decoupled and Divergence-Conditioned Prompt for Multi-domain Dynamic Graph Foundation Models [59.8742208800447]
  We propose DyGFM, a Dynamic Graph Foundation Model over multiple domains based on decoupled and divergence-conditioned prompting.<n>We show that DyGFM consistently outperforms 12 state-of-the-art baselines on both node classification and link prediction tasks.
  arXiv  Detail & Related papers  (2026-05-13T13:50:03Z)
• Non-monotonic causal discovery with Kolmogorov-Arnold Fuzzy Cognitive Maps [0.40611352512781856]
  Fuzzy Cognitive Maps constitute a neuro-symbolic paradigm for modeling complex dynamic systems.<n>This research proposes the Kolmogorov-Arnold Fuzzy Cognitive Map (KA-FCM), a novel architecture that redefines the causal transmission mechanism.
  arXiv  Detail & Related papers  (2026-04-06T19:56:12Z)
• TopoOR: A Unified Topological Scene Representation for the Operating Room [54.50897207203292]
  Surgical Scene Graphs abstract the complexity of surgical operating rooms (OR) into a structure of entities and their relations.<n>We introduce TopoOR, a new paradigm that models multimodal operating rooms as a higher-order structure.<n>We also propose a higher-order attention mechanism that explicitly preserves manifold structure and modality-specific features.
  arXiv  Detail & Related papers  (2026-03-10T10:19:42Z)
• UniG2U-Bench: Do Unified Models Advance Multimodal Understanding? [50.92401586025528]
  Unified multimodal models have recently demonstrated strong generative capabilities, yet whether and when generation improves understanding remains unclear.<n>We introduce UniG2U-Bench, a comprehensive benchmark categorizing generation-to-understanding (G2U) evaluation into 7 regimes and 30 subtasks.
  arXiv  Detail & Related papers  (2026-03-03T18:36:16Z)
• PRISM: Parallel Residual Iterative Sequence Model [52.26239951489612]
  We propose PRISM (Parallel Residual Iterative Sequence Model) to resolve this tension.<n>PRISM introduces a solver-inspired inductive bias that captures key structural properties of multi-step refinement in a parallelizable form.<n>We prove that this formulation achieves Rank-$L$ accumulation, structurally expanding the update manifold beyond the single-step Rank-$1$ bottleneck.
  arXiv  Detail & Related papers  (2026-02-11T12:39:41Z)
• Lag Operator SSMs: A Geometric Framework for Structured State Space Modeling [3.3864018929063477]
  We introduce a framework for constructing discrete-time Structured State Space Models (SSMs) that is both flexible and modular.<n>Our approach is based on a novel lag operator, which geometrically derives the discrete-time recurrence by measuring how the system's basis functions "slide"
  arXiv  Detail & Related papers  (2025-12-22T02:25:26Z)
• Decoupling and Damping: Structurally-Regularized Gradient Matching for Multimodal Graph Condensation [3.2987327415317895]
  We propose Structurally-Regularized Gradient Matching (SR-GM), a novel condensation framework tailored for multimodal graphs.<n> SR-GM significantly improves accuracy and accelerates convergence compared to baseline methods.<n>This research provides a scalable methodology for multimodal graph-based learning in resource-constrained environments.
  arXiv  Detail & Related papers  (2025-11-25T11:50:34Z)
• The Neural Differential Manifold: An Architecture with Explicit Geometric Structure [8.201374511929538]
  This paper introduces the Neural Differential Manifold (NDM), a novel neural network architecture that explicitly incorporates geometric structure into its fundamental design.<n>We analyze the theoretical advantages of this approach, including its potential for more efficient optimization, enhanced continual learning, and applications in scientific discovery and controllable generative modeling.
  arXiv  Detail & Related papers  (2025-10-29T02:24:27Z)
• Non-Hermitian $\mathrm{sl}(3, \mathbb{C})$ three-mode couplers [0.0]
  We introduce a general $mathrmsl(N,mathbbC)$ framework for arbitrary $N$-mode couplers in classical and quantum regimes.<n>An exact Wei--Norman propagator captures the full dynamics and makes crossing exceptional points explicit.<n>We study the family spanning $mathcalPT$-symmetric and non-Hermitian cyclic couplers, where two exceptional points of order three lie within a continuum of exceptional points of order two.
  arXiv  Detail & Related papers  (2025-10-28T04:04:11Z)
• Graded Transformers [0.0]
  We introduce the Graded Transformer framework, a new class of sequence models that embed inductive biases through grading on vector spaces.<n>The framework enables adaptive feature prioritization, overcoming limitations of fixed grades in earlier models.<n>The Graded Transformer provides a mathematically principled approach to hierarchical learning and neuro-symbolic reasoning.
  arXiv  Detail & Related papers  (2025-07-27T02:34:08Z)
• Sequential-Parallel Duality in Prefix Scannable Models [68.39855814099997]
  Recent developments have given rise to various models, such as Gated Linear Attention (GLA) and Mamba.<n>This raises a natural question: can we characterize the full class of neural sequence models that support near-constant-time parallel evaluation and linear-time, constant-space sequential inference?
  arXiv  Detail & Related papers  (2025-06-12T17:32:02Z)
• State-space models can learn in-context by gradient descent [1.3087858009942543]
  We show that state-space models can perform gradient-based learning and use it for in-context learning in much the same way as transformers.<n>Specifically, we prove that a single structured state-space model layer, augmented with multiplicative input and output gating, can reproduce the outputs of an implicit linear model.<n>We also provide novel insights into the relationship between state-space models and linear self-attention, and their ability to learn in-context.
  arXiv  Detail & Related papers  (2024-10-15T15:22:38Z)
• Theoretical Foundations of Deep Selective State-Space Models [13.971499161967083]
  Deep SSMs demonstrate outstanding performance across a diverse set of domains.<n>Recent developments show that if the linear recurrence powering SSMs allows for multiplicative interactions between inputs and hidden states.<n>We show that when random linear recurrences are equipped with simple input-controlled transitions, then the hidden state is provably a low-dimensional projection of a powerful mathematical object.
  arXiv  Detail & Related papers  (2024-02-29T11:20:16Z)
• SEGNO: Generalizing Equivariant Graph Neural Networks with Physical Inductive Biases [66.61789780666727]
  We show how the second-order continuity can be incorporated into GNNs while maintaining the equivariant property. We also offer theoretical insights into SEGNO, highlighting that it can learn a unique trajectory between adjacent states. Our model yields a significant improvement over the state-of-the-art baselines.
  arXiv  Detail & Related papers  (2023-08-25T07:15:58Z)
• Intensity Profile Projection: A Framework for Continuous-Time Representation Learning for Dynamic Networks [50.2033914945157]
  We present a representation learning framework, Intensity Profile Projection, for continuous-time dynamic network data. The framework consists of three stages: estimating pairwise intensity functions, learning a projection which minimises a notion of intensity reconstruction error. Moreoever, we develop estimation theory providing tight control on the error of any estimated trajectory, indicating that the representations could even be used in quite noise-sensitive follow-on analyses.
  arXiv  Detail & Related papers  (2023-06-09T15:38:25Z)
• Understanding and Constructing Latent Modality Structures in Multi-modal Representation Learning [53.68371566336254]
  We argue that the key to better performance lies in meaningful latent modality structures instead of perfect modality alignment. Specifically, we design 1) a deep feature separation loss for intra-modality regularization; 2) a Brownian-bridge loss for inter-modality regularization; and 3) a geometric consistency loss for both intra- and inter-modality regularization.
  arXiv  Detail & Related papers  (2023-03-10T14:38:49Z)

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.