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Prox-Friendly Log-Magnitude Prior on Complex-Valued Signal

Kazuki Matsumoto, Keidai Arai, Kohei Yatabe

arXiv:2609.34445v1cs.SD

TL;DR

Directly imposing priors on log-magnitude is challenging for proximal splitting because the log transform composed with a linear operator complicates operator derivation. EPILOG regularizes an auxiliary variable linked to log-magnitude and supplies variable-wise proximity operators for an ADMM-based method. In speech dereverberation, its methods outperform the alternatives, with DCT-based EPILOG achieving the highest scores across all reported metrics.

  • Problem

    Directly imposing log-magnitude priors such as cepstral sparsity is challenging for proximal splitting because the log-magnitude operation composed with a linear operator complicates operator derivation.

    Source: §1 · p. 1

  • Method

    EPILOG regularizes an auxiliary variable linked to log-magnitude, derives variable-wise proximity operators, and uses them in an ADMM-based algorithm.

    Source: §1 · p. 1

  • Results

    EPILOG methods (c-i) and (c-ii) outperformed the other methods, and DCT-based (c-ii) achieved the highest performance across all metrics.

    Source: §4.2 · p. 4

  • Takeaways & Limitations

    The speech-dereverberation experiment demonstrates the effectiveness of cepstral-domain sparsity realized by EPILOG.

    Source: §5 · p. 4

    The algorithm's convergence analysis is left for future work.

    Source: §3.3 · p. 3

Abstract

from arXiv · show

The logarithmic transform is essential in audio signal processing since human auditory perception is approximately logarithmic with respect to magnitude. However, directly incorporating prior knowledge about signals (e.g., harmonic structure) in the log-magnitude domain into optimization problems solved by standard proximal splitting algorithms remains challenging. To address this issue, this paper proposes a novel regularizer termed EPILOG (Exponential Penalty for Imposing priors on LOG-magnitude). EPILOG indirectly imposes prior knowledge on the log-magnitude of a complex-valued signal through regularization of an auxiliary variable that is shown to be linked with the log-magnitude. Furthermore, we derive its variable-wise proximity operators and develop a proximal splitting algorithm using these operators. Experiments on speech dereverberation demonstrate the effectiveness of the proposed regularizer, particularly in promoting cepstral-domain sparsity.

1. INTRODUCTION

Log-magnitude representations are important in audio, but directly imposing priors such as cepstral sparsity there complicates proximal splitting. The paper introduces EPILOG to transfer structure to log-magnitude through an auxiliary variable and develops its proximity operators and an ADMM-based algorithm.

1. INTRODUCTION

  • Optimization-based audio methods use regularization to incorporate prior knowledge, including low-rankness, smoothness, and harmonic structure.Proximal splitting algorithms are widely used to solve associated problems.Source: §1 · p. 1
  • Directly imposing priors on log-magnitude, including cepstral sparsity, complicates proximal splitting because log-magnitude is composed with a linear operator.Log-magnitude representations also underlie classical features such as the cepstrum and MFCCs and are motivated by auditory sensitivity to magnitude.Source: §1 · p. 1
  • EPILOG imposes log-magnitude priors indirectly by regularizing an auxiliary variable linked to the log-magnitude of a complex-valued signal.The paper demonstrates cepstral-domain sparsity by imposing sparsity on DCT coefficients of the log-magnitude representation.Source: §1 · p. 1
  • The paper derives variable-wise proximity operators for EPILOG and develops an ADMM-based algorithm using them.Its stated contributions also include a speech-dereverberation application that experimentally confirms the effectiveness of EPILOG and cepstral-domain sparsity.Source: §1 · p. 1

2. REGULARIZATION OF MAGNITUDE SPECTROGRAM

Magnitude-spectrogram priors can encode structures such as low-rankness, smoothness, and harmonicity, but composing magnitude with a linear operator obstructs direct proximal splitting. A perspective-function framework instead couples an auxiliary variable to magnitude so structure imposed on that variable is reflected in the spectrogram.

2. REGULARIZATION OF MAGNITUDE SPECTROGRAM

  • The recovery formulation minimizes data fidelity plus regularization subject to the spectrogram variable equaling the DGT of the signal estimate.The regularizer encodes prior knowledge about the signal to be recovered.Source: §2 · p. 1
  • Regularizers of the form R(L|z|) can promote magnitude structure, but the composition of magnitude and a linear operator makes direct proximal splitting difficult.The choice of L and R determines the promoted structure; for example, a difference operator with a squared ℓ2-norm promotes smoothness.Source: §2 · p. 2
  • A perspective-function framework enables flexible magnitude regularization by introducing an auxiliary variable and coupling it to magnitude.The coupling is provided by Φ^(α)(z, σ), while γR(Lσ) imposes structure on σ.Source: §2 · p. 2
  • Minimizing Φ^(α)(z, σ) encourages σ to match |z| up to scaling by 1/α, transferring structure on σ to |z| through a weighted quadratic term.The weighted term is |z_n|²/(2σ_n).Source: §2 · p. 2

3. PROPOSED METHOD

EPILOG regularizes an auxiliary variable linked to the log-magnitude, enabling structural priors while supporting variable-wise proximity operators and an ADMM-based algorithm. The paper illustrates this approach with cepstral sparsity and describes how leaving the DC coefficient unregularized reduces dependence on global signal scale.

3.1. EPILOG regularizer

  • EPILOG defines a regularizer that imposes structure on an auxiliary variable linked to the log-magnitude of a complex-valued signal.The link transfers the auxiliary variable's structure to log(|z|).Source: §3.1 · p. 2
  • For sufficiently small ϵ, minimizing Ψ(α,ϵ)(z, σ) encourages σ to match log(|z|) up to the offset −log(α).The function is quadratic in |z| for fixed σ and contains exponential and linear terms in σ for fixed z; ϵ > 0 is required for it to remain bounded below.Source: §3.1 · p. 2

3.2. Example: Cepstral-domain sparsity via EPILOG

  • For cepstral sparsity, a frequency-direction DCT and a sparsity-inducing function R make γR(Lσ) indirectly regularize the cepstrum of the log-magnitude.The example uses the typical sparsity of the cepstrum of clean speech as prior knowledge.Source: §3.2 · p. 2
  • Leaving the DCT's DC coefficient unregularized reduces EPILOG's dependence on global signal scale.The scale and α contribute an additive constant to σ⋆, which the DCT maps to its DC coefficient; the corresponding weight is set to 0 while other weights are 1.Source: §3.2 · pp. 2–3

3.3. ADMM-based algorithm for EPILOG

  • The ADMM-based algorithm uses variable-wise proximity operators of ψ and alternately minimizes the augmented Lagrangian over primal variables, assuming a Parseval-tight DGT.The authors leave convergence analysis of the algorithm for future work.Source: §3.3 · p. 3

4. APPLICATION TO SPEECH DEREVERBERATION

The speech-dereverberation experiment compares regularization domains and DCT settings, finding that EPILOG with DCT-based cepstral sparsity achieves the highest performance across the reported metrics.

4. APPLICATION TO SPEECH DEREVERBERATION

  • The experiment used LibriTTS-R test-clean speech, BUT Reverb Database room responses, 8 kHz signals, and noise set to −20 dB relative to As.Ten random speech/RIR pairs were evaluated; five distinct pairs were used for parameter tuning.Source: §4 · p. 3, §4.1 · pp. 3–4

4.1. Algorithms and parameters

  • Six methods compared regularization domains (a)–(c), with settings (i) excluding DCT and (ii) applying DCT-based regularization; all used variable splitting and 1,000 iterations.Setting (c-ii) is the proposed EPILOG method promoting cepstral-domain sparsity.Source: §4.1 · p. 3

4.2. Results

  • The (c-ii) auxiliary variable σ closely resembled log(|z|), and its estimated log-magnitude was sparser and smoother than (c-i), with fewer artifacts.Its cepstral representation was also sparser than (c-i).Source: §4.2 · p. 4, Figure 2 · p. 4
  • EPILOG with DCT-based cepstral sparsity (c-ii) consistently achieved the highest performance across all metrics.EPILOG methods (c-i) and (c-ii) outperformed the other methods.Source: §4.2 · p. 4, Table 1 · p. 4
  • EPILOG without the additional γR(Lσ) term (c-i) still performed strongly, indicating that Ψ(α,ϵ)(z, σ) itself acts as an effective sparsity-inducing regularizer.The log-magnitude spectrogram from (c-i) was sparser than those from methods (a) and (b).Source: §4.2 · p. 4

5. CONCLUSION

The conclusion recaps EPILOG’s log-magnitude prior mechanism, its proximity operators and algorithm, and the dereverberation evidence for cepstral sparsity; it also identifies convergence guarantees and broader applications as future work.

  • EPILOG links an auxiliary variable to log-magnitude, and the paper derives variable-wise proximity operators and a proximal splitting algorithm tailored to the regularizer.Source: §5 · p. 4
  • The speech-dereverberation experiment demonstrated the effectiveness of cepstral-domain sparsity realized by EPILOG.Source: §5 · p. 4
  • Future work includes developing an algorithm with a convergence guarantee and extending EPILOG to more signal-processing tasks.Source: §5 · p. 4
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