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<p>Wavelet theory pioneered by Ingrid Daubechies has transformed signal processing, with applications ranging from digital imaging to remote sensing. Recently, the tools of signal processing have inspired the development of sequence modeling in machine learning, in particular State Space Models (SSMs). In this expository note, we introduce SSMs from the lens of signal processing and their connections to wavelets. We present the results by Huang et. al. (Forty-second International Conference on Machine Learning, 2025. <a href="https://openreview.net/forum?id=rnMH9njZxb">https://openreview.net/forum?id=rnMH9njZxb</a>) revealing how Mamba — a modern SSM — can mimic projections onto Haar wavelets, which helps explaining its performance. We then explore consequences of these results by showing that Mamba can also approximate Daubechies wavelet, albeit at a higher cost in terms of state size.</p>

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Wavelets and State Space Models

  • 期刊论文

  • 发表日期：

    2026年1月23日

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  • Ningyuan Huang,
  • Federico Danieli

摘要

Wavelet theory pioneered by Ingrid Daubechies has transformed signal processing, with applications ranging from digital imaging to remote sensing. Recently, the tools of signal processing have inspired the development of sequence modeling in machine learning, in particular State Space Models (SSMs). In this expository note, we introduce SSMs from the lens of signal processing and their connections to wavelets. We present the results by Huang et. al. (Forty-second International Conference on Machine Learning, 2025. https://openreview.net/forum?id=rnMH9njZxb) revealing how Mamba — a modern SSM — can mimic projections onto Haar wavelets, which helps explaining its performance. We then explore consequences of these results by showing that Mamba can also approximate Daubechies wavelet, albeit at a higher cost in terms of state size.