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Selection Mechanisms for Sequence Modeling using Linear State Space Models

Umberto Casti, Sandro Zampieri, Fabio Pasqualetti

TL;DR

The paper addresses efficient, selective sequence modeling and introduces a control-theory–inspired residual selection mechanism for Linear State Space Models (SSMs) that preserves an LTI structure, in contrast to Mamba’s LTV dynamics. It formalizes the sequence-to-sequence problem, designs two synthetic tasks (Induction Head and Extended Induction Head) to probe selectivity and memory, and proposes a Residual SSM architecture with three LTI blocks and a gating mechanism. Through simulations, the authors show that the Residual SSM achieves robust performance on both tasks, while a time-varying Selective SSM struggles, highlighting memory-enabled selectivity as a key advantage. The work demonstrates that integrating control-theoretic ideas with SSMs can yield efficient, scalable sequence models suitable as modular building blocks for larger architectures.

Abstract

Recent advancements in language modeling tasks have been driven by architectures such as Transformers and, more recently, by Selective State Space Models (SSMs). In this paper, we introduce an alternative selection mechanism inspired by control theory methodologies. Specifically, we propose a novel residual generator for selection, drawing an analogy to fault detection strategies in Linear Time-Invariant (LTI) systems. Unlike Mamba, which utilizes Linear Time-Varying (LTV) systems, our approach combines multiple LTI systems, preserving their beneficial properties during training while achieving comparable selectivity. To evaluate the effectiveness of the proposed architecture, we test its performance on synthetic tasks. While these tasks are not inherently critical, they serve as benchmarks to test the selectivity properties of different cores architecture. This work highlights the potential of integrating theoretical insights with experimental advancements, offering a complementary perspective to deep learning innovations at the intersection of control theory and machine learning.

Selection Mechanisms for Sequence Modeling using Linear State Space Models

TL;DR

The paper addresses efficient, selective sequence modeling and introduces a control-theory–inspired residual selection mechanism for Linear State Space Models (SSMs) that preserves an LTI structure, in contrast to Mamba’s LTV dynamics. It formalizes the sequence-to-sequence problem, designs two synthetic tasks (Induction Head and Extended Induction Head) to probe selectivity and memory, and proposes a Residual SSM architecture with three LTI blocks and a gating mechanism. Through simulations, the authors show that the Residual SSM achieves robust performance on both tasks, while a time-varying Selective SSM struggles, highlighting memory-enabled selectivity as a key advantage. The work demonstrates that integrating control-theoretic ideas with SSMs can yield efficient, scalable sequence models suitable as modular building blocks for larger architectures.

Abstract

Recent advancements in language modeling tasks have been driven by architectures such as Transformers and, more recently, by Selective State Space Models (SSMs). In this paper, we introduce an alternative selection mechanism inspired by control theory methodologies. Specifically, we propose a novel residual generator for selection, drawing an analogy to fault detection strategies in Linear Time-Invariant (LTI) systems. Unlike Mamba, which utilizes Linear Time-Varying (LTV) systems, our approach combines multiple LTI systems, preserving their beneficial properties during training while achieving comparable selectivity. To evaluate the effectiveness of the proposed architecture, we test its performance on synthetic tasks. While these tasks are not inherently critical, they serve as benchmarks to test the selectivity properties of different cores architecture. This work highlights the potential of integrating theoretical insights with experimental advancements, offering a complementary perspective to deep learning innovations at the intersection of control theory and machine learning.

Paper Structure

This paper contains 15 sections, 10 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Synthetic Tasks
  4. Induction Head
  5. Extended Induction Head
  6. State Space Models
  7. Vector Embedding
  8. Selective & Residual SSMs
  9. Selective SSMs
  10. Residual SSMs
  11. Simulations & experimental setup
  12. Implementation Details
  13. Induction Head
  14. Extended Induction Head
  15. Conclusions

Figures (3)

  • Figure 1: In this figure, we present the two tasks considered in this work. On the left, we illustrate the Induction Head Task, where a trigger token appears in the sequence, and the goal is to recall the token following it at the end of the input sequence. On the right, we depict the Extended Induction Head Task, which represents a natural extension of the Induction Head problem. In this task, the objective is to recall the token following a trigger sequence comprising more than just a single trigger token. This new task differs from the previous one as it requires a dynamic and selective mechanism with memory to handle the extended context.
  • Figure 2: Aggregated Selective SSM AG-TD:23
  • Figure 3: This figure shows the control scheme relative to the proposed architecture: the Residual SSM. Here, $\Sigma_f$, $\Sigma_M$, and $\Sigma_r$ represent LTI systems, i.e., Linear SSMs, while $\Sigma_g$ represents a nonlinear gating mechanism.

Theorems & Definitions (2)

  • Remark 3.1
  • Remark 3.2