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.

翻译：状态空间模型（SSMs）已成为建模序列数据中长期依赖关系的强大框架。与传统循环神经网络（RNNs）和卷积神经网络（CNNs）不同，SSMs 利用控制理论和动态系统的原理，为序列建模提供了一种结构化且稳定的方法。然而，序列建模中的一个关键挑战在于将长期依赖关系压缩为紧凑的隐藏状态表示，同时不丢失关键信息。本文为理解选择性状态空间模型中的记忆压缩建立了一个严格的数学框架。我们引入了一种选择性门控机制，该机制根据输入相关性动态过滤和更新隐藏状态，从而实现高效的记忆压缩。我们使用互信息和率失真理论等信息论工具，形式化了记忆效率与信息保留之间的权衡关系。我们的分析为在不牺牲模型性能的前提下可压缩的信息量提供了理论界。我们还推导了定理，证明了选择性 SSMs 中隐藏状态的稳定性和收敛性，确保了可靠的长期记忆保留。计算复杂度分析表明，与传统的基于 RNN 的模型相比，选择性 SSMs 在记忆效率和处理速度方面提供了显著改进。通过在时间序列预测和自然语言处理等序列建模任务上的实证验证，我们证明选择性 SSMs 在消耗更少内存和计算资源的同时，实现了最先进的性能。