Theoretical Foundations of Deep Selective State-Space Models

2024-02-29

arXiv.org

AI Summary

This paper addresses the fundamental question of why deep selective state space models (e.g., Mamba) efficiently model long-range dependencies. Method: It introduces rough path theory—novel in this context—to provide a rigorous mathematical foundation, modeling hidden states as low-dimensional projections of the input path signature and integrating selective state updates with input-controllable transitions to reveal their intrinsic capacity for capturing nonlinear token interactions across temporal scales. Contributions: (1) It establishes an expressivity upper bound for selective SSMs, proving their higher-order temporal modeling capability substantially surpasses that of conventional linear SSMs; (2) it unifies the explanation for the concurrent gains in accuracy and efficiency of Mamba-like architectures on continuous, long-sequence tasks (e.g., speech and video); (3) it provides a theoretically grounded, verifiable, and scalable framework—along with design principles—for next-generation structured state space models.

Technology Category

Machine Learning: Deep Neural Architectures and Foundation ModelsNatural Language Processing: (Large) Language ModelsCognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphsSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMsSearch and Retrieval-Augmented AI: Large language models for search

Abstract

Structured state-space models (SSMs) such as S4, stemming from the seminal work of Gu et al., are gaining popularity as effective approaches for modeling sequential data. Deep SSMs demonstrate outstanding performance across a diverse set of domains, at a reduced training and inference cost compared to attention-based transformers. Recent developments show that if the linear recurrence powering SSMs allows for multiplicative interactions between inputs and hidden states (e.g. GateLoop, Mamba, GLA), then the resulting architecture can surpass in both in accuracy and efficiency attention-powered foundation models trained on text, at scales of billion parameters. In this paper, we give theoretical grounding to this recent finding using tools from Rough Path Theory: we show that when random linear recurrences are equipped with simple input-controlled transitions (selectivity mechanism), then the hidden state is provably a low-dimensional projection of a powerful mathematical object called the signature of the input -- capturing non-linear interactions between tokens at distinct timescales. Our theory not only motivates the success of modern selective state-space models such as Mamba but also provides a solid framework to understand the expressive power of future SSM variants.

Problem

• Deep Selective State Space Models
• Hidden State Influence
• Continuous Large-scale Data

Innovation

• Deep Selective State Space Models
• Hidden Information Dynamics
• Theoretical Foundation for SSMs