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

cs.SD cs.LG Kazuki Matsumoto, Ren Uchida, Kohei Yatabe · Mar 23, 2026

What it does

Why it matters

This paper proves that AMs are generally not Lipschitz continuous, derives sufficient conditions for Lipschitz continuity (Assumption 3), and proposes LipsAM architectures that enforce these bounds via element-wise minimum and ReLU...

Main concern

Community signal

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Plain-language introduction

Existing Lipschitz-constrained DNNs don't directly apply to audio amplitude modifiers (AMs) because the complex-valued reconstruction breaks continuity. This paper proves that AMs are generally not Lipschitz continuous, derives sufficient conditions for Lipschitz continuity (Assumption 3), and proposes LipsAM architectures that enforce these bounds via element-wise minimum and ReLU operations. The work matters because it enables certified robust amplitude modification and stabilizes Plug-and-Play algorithms where conventional AMs diverge.

Critical review

Verdict

Bottom line

The paper presents a rigorous theoretical solution to a genuine gap in audio DNN robustness. The core result (Theorem 4) that AMs require both Lipschitz-continuous amplitude mapping and element-wise boundedness ($0 \leq (\mathcal{A}(\mathbf{x}))_{n} \leq L_{2}x_{n}$) to achieve overall Lipschitz continuity is novel and well-supported. While the application to speech dereverberation demonstrates practical utility, the work's primary value lies in establishing design principles for certifiably robust audio processing architectures.

“$\mathrm{Lip}(\mathcal{D}_{\mathcal{A}}) \leq \max(L_{1}, L_{2})$”

Matsumoto et al., Sec. 3.2 · Theorem 4

“The AMs $\mathcal{D}_{\mathcal{A}}$, $\mathcal{D}_{\mathcal{S}}$ and $\mathcal{D}_{\mathcal{R}}$ in Eqs. (3)-(5) are not Lipschitz continuous in general, even when the amplitude-modifying parts $\mathcal{A}$, $\mathcal{S}$ and $\mathcal{R}$ are Lipschitz continuous.”

Matsumoto et al., Sec. 3.1 · Section 3.1

What holds up

The theoretical framework is sound. Theorem 4 provides a clean sufficient condition with a tight bound, and the proposed LipsAM-SE/LipsAM-RE architectures are elegant plug-in modifications requiring only element-wise minimum or ReLU layers. The numerical validation in Section 3.4 empirically confirms that conventional AMs exceed the termination threshold while LipsAMs respect the theoretical bounds. In the PnP application (Section 4.3), the stability improvement is striking: "The conventional AMs were not robust to the choice of $\lambda$. Specifically, AM-SE tended to diverge and caused NaN for most $\lambda$." In contrast, "LipsAMs (solid lines) successfully avoided such divergence and ran stably."

Main concerns

The experimental evaluation is limited in scope and scale. Only 10 test signals were used for the final evaluation metrics in Table 1, and the dereverberation task is the sole application demonstrated. The paper does not provide a convergence proof for the specific ADMM-based PnP algorithm used (Eq. 15); while stability is empirically shown via $\|\Delta\mathbf{x}\|_2 \to 0$, the theoretical guarantee mentioned in Section 4.1 ("convergence of such algorithms typically requires a non-expansive DNN") remains unproven for this particular setup. Additionally, the trade-off between the Lipschitz constraint and reconstruction quality is not quantified.

“10 additional signals”

Matsumoto et al., Sec. 4.3 · Table 1 caption

“convergence of such algorithms typically requires a non-expansive DNN, i.e., its Lipschitz constants should be 1 or less”

Matsumoto et al., Sec. 4.1 · Section 4.1

Evidence and comparison

The evidence supports the primary claims about Lipschitz continuity and algorithmic stability. Figure 2 validates that the Jacobians of LipsAMs remain bounded by the theoretical limits whereas conventional AMs diverge. The comparison to $\ell_1$-norm baselines and unconstrained AMs in Table 1 and Figure 3 is fair but narrow; the paper does not compare against other recent robust audio denoisers or Lipschitz-regularized training methods, leaving open the question of whether structural constraints outperform regularization approaches.

“the value of $B$ for the conventional AMs ($\mathcal{D}_{\mathcal{S}}$ and $\mathcal{D}_{\mathcal{R}}$) exceeded the termination threshold... In contrast, the values $B$ for LipsAMs... were tightly bounded by the theoretical bound”

Matsumoto et al., Fig. 2 · Figure 2 caption

Reproducibility

Experimental details are reasonably complete—LibriTTS-R and BUT reverb database are public, architecture specifications (3-layer Conv1D, kernel size 5, 512 channels) are provided, and training hyperparameters are listed. However, the authors do not indicate whether code will be released, and the exact stopping criteria and initialization for the PnP algorithm (beyond 2000 iterations) are not specified. The small test set size (10 signals) limits statistical confidence in the reported metrics (SI-SNR, PESQ, STOI, ViSQOL), and no standard deviations are reported.

“The architecture for their learnable part $\mathcal{S}$ and $\mathcal{R}$ includes one-dimensional convolution (Conv1D) layers... three Conv1D layers with a kernel size of 5... intermediate feature dimension was set to 512 channels”

Matsumoto et al., Sec. 4.2 · Section 4.2

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.

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