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

Kazuki Matsumoto; Keidai Arai; Kohei Yatabe

cs.SDSep 28, 2026

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

Authors: Kazuki Matsumoto, Keidai Arai, Kohei Yatabe

Organizations: Tokyo University of Agriculture and Technology (TUAT), Tokyo, Japan

Abstract

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.

Figures & tables

Figure 1: Visualization of

\psi^{(\alpha,\epsilon)}(z,\sigma)

in Eq. ( 6 ) for

(\alpha,\epsilon)=(1,0.1)

, where

z\in\mathbb{R}

here. The left panel shows the values of

\psi^{(\alpha,\epsilon)}(z,\sigma)

, where brighter shades represent larger values. The middle and right panels show its slices at

z=z_{0}=1

(blue dotted line) and

\sigma=\sigma_{z_{0}}^{\star}=\log(\sqrt{1+\epsilon})

(green dotted line). The red star in the middle panel shows

\sigma_{z_{0}}^{\star}=\arg\min_{\sigma\in\mathbb{R}}\psi^{(\alpha,\epsilon)}(z_{0},\sigma)

. The red curve in the left panel is the trajectory of

(z_{0},\sigma_{{z}_{0}}^{\star})

obtained by varying

z_{0}\in\mathbb{R}

, illustrating the log-magnitude-link property in Prop. 2 .

Figure 2: Examples of the variables obtained by each method. The left six columns show the results of (a-i) – (c-ii) , where (c-ii) is the proposed method that promotes cepstral-domain sparsity . The above numbers indicate the SI-SNR scores in dB, where bold and underlined numbers represent the best and second-best scores, respectively. The top and middle rows show the log-magnitude spectrograms $\log(|\mathbf{z}|)$ of the estimated signals $\mathbf{z}$ and its corresponding auxiliary variables $\boldsymbol{\sigma}$ , respectively; methods (a) have no auxiliary variables and hence “N/A” is shown. The bottom row shows the magnitudes of the DCT coefficients (i.e., the magnitudes of the cepstral coefficients) $|\mathbf{C}\log(|\mathbf{z}|)|$ , where $\mathbf{C}$ denotes the DCT operator along the frequency direction. The rightmost column shows the ground-truth signal in the log-magnitude domain $\log(|\mathbf{G}\mathbf{s}|)$ (top), in the magnitude domain $|\mathbf{G}\mathbf{s}|$ (middle), and in the cepstral domain $|\mathbf{C}\log(|\mathbf{G}\mathbf{s}|)|$ (bottom), where $\mathbf{G}$ is the DGT operator.

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Sparsity Regularization

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Sparse training reduces the memory and computational costs of deep neural networks. However, sparse optimization methods, e.g., those adding an

\ell_1

penalty, often control sparsity only indirectly through a regularization parameter

λ

, whose mapping to the final sparsity rate is non-trivial. In our experiments, we found this parameter sensitivity to be particularly pronounced for Bregman-based optimizers. Specifically, the two variants LinBreg and AdaBreg reach the same sparsity at

λ

values that differ by up to two orders of magnitude, requiring expensive trial-and-error sweeps to achieve a user-specified sparsity. To address this, we propose an adaptive regularization scheme that updates

λ

based on the difference between the model's current sparsity and the target sparsity. We analyze the resulting algorithm and evaluate it on automatic speaker verification with ECAPA-TDNN and ResNet34 on VoxCeleb and CNCeleb. The proposed method reliably achieves sparsity targets ranging between 75% and 99%. It also converges faster than the oracle-tuned non-adaptive baseline during early training and matches or surpasses its final performance in equal error rate. We further show that the adaptive scheme inherits key properties from its non-adaptive counterpart, including improved out-of-distribution robustness over the dense baselines.

Abstract

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