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LipsAM: Lipschitz-Continuous Amplitude Modifier for Audio Signal Processing and its Application to Plug-and-Play Dereverberation

Kazuki Matsumoto, Ren Uchida, Kohei Yatabe

Abstract

The robustness of deep neural networks (DNNs) can be certified through their Lipschitz continuity, which has made the construction of Lipschitz-continuous DNNs an active research field. However, DNNs for audio processing have not been a major focus due to their poor compatibility with existing results. In this paper, we consider the amplitude modifier (AM), a popular architecture for handling audio signals, and propose its Lipschitz-continuous variants, which we refer to as LipsAM. We prove a sufficient condition for an AM to be Lipschitz continuous and propose two architectures as examples of LipsAM. The proposed architectures were applied to a Plug-and-Play algorithm for speech dereverberation, and their improved stability is demonstrated through numerical experiments.

LipsAM: Lipschitz-Continuous Amplitude Modifier for Audio Signal Processing and its Application to Plug-and-Play Dereverberation

Abstract

The robustness of deep neural networks (DNNs) can be certified through their Lipschitz continuity, which has made the construction of Lipschitz-continuous DNNs an active research field. However, DNNs for audio processing have not been a major focus due to their poor compatibility with existing results. In this paper, we consider the amplitude modifier (AM), a popular architecture for handling audio signals, and propose its Lipschitz-continuous variants, which we refer to as LipsAM. We prove a sufficient condition for an AM to be Lipschitz continuous and propose two architectures as examples of LipsAM. The proposed architectures were applied to a Plug-and-Play algorithm for speech dereverberation, and their improved stability is demonstrated through numerical experiments.
Paper Structure (14 sections, 2 theorems, 11 equations, 4 figures, 1 table)

This paper contains 14 sections, 2 theorems, 11 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Lipschitz Continuity of DNNs
  4. Amplitude Modifier (AM)
  5. Proposed Method
  6. AMs are not Lipschitz Continuous
  7. LipsAM: Lipschitz-continuous Amplitude Modifier
  8. Proposed Architectures: LipsAM-SE and LipsAM-RE
  9. Numerical Validation of Lipschitz Bound
  10. Application to Plug-and-Play Speech Dereverberation
  11. Problem Setting and Plug-and-Play Algorithm
  12. Experimental Settings
  13. Results
  14. Conclusion

Key Result

Theorem 4

Let $\mathcal{D}_{\mathcal{A}}:\mathbb{C}^N\to\mathbb{C}^N:\mathbf{z}\mapsto \mathcal{A}(|\mathbf{z}|)\odot\mathrm{sign}(\mathbf{z})$, and $\mathcal{A}:\mathbb{R}_+^N\to\mathbb{R}_+^N$ satisfy Eqs. eq:cond1 and eq:cond2 in Assumption asm:assumption. Then, $\mathcal{D}_{\mathcal{A}}$ is Lipschitz con

Figures (4)

  • Figure 1: Amplitude-Modifying DNNs. Red layers are trainable, blue layers are introduced to enforce Lipschitz continuity. Layers "mul" represent the element-wise multiplication. While the popular architectures, AM-SE and AM-RE, are generally not Lipschitz continuous, Lipschitz constants of our LipsAM-SE and LipsAM-RE can be bounded by $\sqrt{(\mathrm{Lip}(\mathcal{S}))^2+1}$ and $\mathrm{Lip}(\mathcal{R})+1$, respectively.
  • Figure 2: Numerical estimate of the value $B$ in Eq. \ref{['eq:B']} for each architecture. Upper rows are that of AM-SE in Eq. \ref{['eq:signal']} and LipsAM-SE in Eq. \ref{['eq:signallim']}. Lower rows are that of AM-RE in Eq. \ref{['eq:residual']} and LipsAM-RE in Eq. \ref{['eq:residuallim']}. Dots indicate results of 100 trials, and maximum result is marked with large circle. Solid lines show the theoretical bound in Theorem \ref{['thm:signallim']}. Areas exceeding termination threshold are darkened.
  • Figure 3: Parameter $\lambda$ vs SI-SNR after 500 iterations. Solid colored lines represents the proposed LipsAMs, and dotted colored lines represent the conventional AMs. Use of orthogonal layers is indicated by (Ortho). Gray lines show $\ell_1$-norm-based method. Best parameters for each model are shown by the markers. A close-up view of left is on the right. Missing points indicate that algorithms diverged.
  • Figure 4: Transition of amount of update $\|\Delta \mathbf{x}\|_2$. In each box, results of LipsAMs with best $\lambda$ in Fig. \ref{['fig:parameters']} were compared with that of AMs with the same $\lambda$ as LipsAMs.

Theorems & Definitions (6)

  • Example 1: Bias
  • Example 2: Permutation
  • Theorem 4
  • proof
  • Theorem 5
  • proof