Mathematical Formalism for Memory Compression in Selective State Space Models

TL;DR

This paper introduces a mathematical framework for memory compression in selective state space models, combining control theory and information theory to improve long-term sequence modelling efficiency.

Contribution

The paper develops a rigorous mathematical formalism for memory compression in selective SSMs, including a gating mechanism and theoretical bounds on information retention.

Findings

Selective SSMs outperform RNNs in memory efficiency and speed.

Theoretical bounds on information compression without performance loss.

Empirical results show state-of-the-art performance with less memory.

Abstract

State space models (SSMs) have emerged as a powerful framework for modelling long-range dependencies in sequence data. Unlike traditional recurrent neural networks (RNNs) and convolutional neural networks (CNNs), SSMs offer a structured and stable approach to sequence modelling, leveraging principles from control theory and dynamical systems. However, a key challenge in sequence modelling is compressing long-term dependencies into a compact hidden state representation without losing critical information. In this paper, we develop a rigorous mathematical framework for understanding memory compression in selective state space models. We introduce a selective gating mechanism that dynamically filters and updates the hidden state based on input relevance, allowing for efficient memory compression. We formalize the trade-off between memory efficiency and information retention using information-theoretic tools, such as mutual information and rate-distortion theory. Our analysis provides theoretical bounds on the amount of information that can be compressed without sacrificing model performance. We also derive theorems that prove the stability and convergence of the hidden state in selective SSMs, ensuring reliable long-term memory retention. Computational complexity analysis reveals that selective SSMs offer significant improvements in memory efficiency and processing speed compared to traditional RNN-based models. Through empirical validation on sequence modelling tasks such as time-series forecasting and natural language processing, we demonstrate that selective SSMs achieve state-of-the-art performance while using less memory and computational resources.