Toward Complex-Valued Neural Networks for Waveform Generation
Authors: Hyung-Seok Oh
Deok-Hyeon Cho
Seung-Bin Kim
Seong-Whan Lee
Venue: ICLR 2026 Poster (2026)
Keywords: waveform generation
complex-valued neural networks
iSTFT-based vocoder
generative adversarial network
Primary area: applications to computer vision, audio, language, and other modalities
TL;DR: ComVo is a complex-valued iSTFT vocoder that jointly models the real and imaginary spectrogram components, stabilizes training with a phase-quantization nonlinearity, and uses block-matrix operations for efficient complex computation.

Abstract: Neural vocoders have recently advanced waveform generation, yielding natural and expressive audio. Among these approaches, iSTFT-based vocoders have recently gained attention. They predict a complex-valued spectrogram and then synthesize the waveform via iSTFT, thereby avoiding learned upsampling stages that can increase computational cost. However, current approaches use real-valued networks that process the real and imaginary parts independently. This separation limits their ability to capture the inherent structure of complex spectrograms. We present ComVo, a Complex-valued neural Vocoder whose generator and discriminator use native complex arithmetic. This enables an adversarial training framework that provides structured feedback in complex-valued representations. To guide phase transformations in a structured manner, we introduce phase quantization, which discretizes phase values and regularizes the training process. Finally, we propose a block-matrix computation scheme to improve training efficiency by reducing redundant operations. Experiments demonstrate that ComVo achieves higher synthesis quality than comparable real-valued baselines, and that its block-matrix scheme reduces training time by 25%. Audio samples and code are available at https://hs-oh-prml.github.io/ComVo/.

Published as a conference paper at ICLR 2026
TOWARD COMPLEX- VALUED NEURAL NETWORKS FOR WAVEFORM GENERATION
Hyung-Seok Oh, Deok-Hyeon Cho, Seung-Bin Kim & Seong-Whan Lee"
Department of Artificial Intelligence
Korea University
Seoul, Republic of Korea {hs_oh, dh,cho, sb-kim, sw www.leejkkree.acckk
ABSTRACT
Neural vocoders have recently advanced waveform generation, yielding natural and expressive audio. Among these approaches, iSTFT-based vocoders have re- cently gained attention. They predict complex-valued spectrogram and then syn- thesize the waveform via iSTFT, thereby avoiding learned upsampling stages that can increase computational cost. However, current approaches use real-valued networks that process the real and imaginary parts independently. This separa- tion limits their ability to capture the inherent structure of complex spectrograms. We present Com Vo, Complex-valued neural Vocoder whose generator and dis- criminator use native complex arithmetic. This enables an adversarial training framework that provides structured feedback in complex-vali valuud representations. To guide phase transformations in structured manner, we introduce phase quan- tization, which discretizes phase values and regularizes the training process. Fi- nally, propose block-matrix computation scheme to improve training effi- ciency by reducing redundant operations. Experiments demonstrate that Com achieves higher synthesis quality than comparable real-valued baselines, and that its block-matrix scheme reduces training time by 25%. Audio samples and code are available at Ittps: //tp:://hs-aae http://ssshhppppee oh-prml. gi github...... wtt.ioo/oooo ComVo/
INTRODUCTION
Deep tearning-basee vocoders have significantly advanced speech synthesis, producing more natural and expressive synthetic speech. Recent developments include models based on generative adver-
sarial networks (GANs) (Kumar et al. 2019 Yamamoto et al.) (0220 Kong et al. 20200 Lee et al.)
2023), normalizing flow-based models (van den Oord al.) [2018] Ping al.][2200
Lee al. 2020).
and diffusion-basee models (Kong et al. |021 Lee et al., 2022 Chen et al. 2021 Lee et al. 2025
Although these approaches achieve high-fidelity speech generation, some neural vocoders still rely on sequential sample prediction learned upsampling, thereby increasing model complexity and
inference latency.
An alternative is to synthesize speech in the spectral domain using the inverse short-time Fourier transform (iSTFT). Operating directly on complex spectrograms (Oyamada et al. [2018] | Neekhara ct al.) 2019 Gritsenko al. (2020) Kancko et al.] 2022 2023 Siuzdak 2024 Yoneyama et al. 2024
Liu et al. 2025) avoids the need for sample-by-san baampl generation and learned upsampling. To our knowledge, current iSTFT-based vocoders rely real-valued neural networks (RVNNs) that process real and imaginary parts as separate channels. This separation limits their ability to model the coupling between these components.
Complex-valued neural networks (CVNNs) extend standard neural networks to the complex domain by allowing both inputs and parameters be complex-valued. Operating entirely in the complex domain enables these models to capture the intrinsic dependencies between the real and imaginary components. CVNNs have been applied in domains such as radar signal classification (Yang al. 2022 MRI reconstruction (Vasudeva et al. 2022), and wireless communication (Xu et al. 2022
*Corresponding author

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Published as a conference paper at ICLR 2026
where measurements carry both magnitude and phase information and naturally form complex- valued data. In speech processing, CVNNs have been explored for tasks including speech enhance- ment (Nustede & Anemüller 2024 Mamun & Hansen (023), speech recognition (Hayakawa al. 2018). and even statistical parametric speech synthesis (Hu et al. 2016). These studies demonstrate the potential of CVNNs to better capture spectral structure.
Although some recent vocoders produce complex spectrograms, they still use real-valued networks that handle each spectrogram channel independently, CVNNs, by jointly processing complex coeffi- cients, could overcome this limitation. By treating each spectrogram coefficient as unified complex entity, CVNN- based models can capture cross cossocommpnent interactions that real-valued models miss. Motivated by this, we adopt CVNNs to better capture structure in the complex domain, yielding higher-quality synthesis.
In this work, we propose Com Vo, Complex-valued neural Vocoder that performs iSTFT-based waveform generation entirely in the complex domain with a GAN-based architecture. The generator uses CVNN layers to jointly model the real and imaginary components of spectrograms, thereby better capturing their algebraic structure. We then design a complex multi-resolution discrimina- tor (cMRD) that operates directly complex spectrograms. Together, these components form complex- domain adversarial training framework in which both the generator and discriminator op- erate complex-valued representations. This design allows feedback that respects the structure of the complex domain. Inspired by recent studies on complex activation functions (Vasudeva et al.) 2022), introduce phase quantization, nonlinear transformation that discretizes phase angles to serve as an inductive bias for stable learning. Finally, to reduce redundant computations in complex- valued operations, develop a block-matrix computation scheme that improves overall training efficiency.
CVNN-based architecture with complex adversarial training: We introduce Com Vo, which, to our knowledge, the first iSTFT-based vocoder to employ complex-vale -valued neu- ral networks in both its generator and discriminator. We design the discriminator losses in the complex domain, thus establishing an adversarial framework that operates on complex- valued representations.
Structured nonlinear transformation: We propose phase quantization, a tailored nonlin- ear operation that discretizes phase angles and serves as an inductive bias.
Block mateixx computation scheme: We present an efficient implementation that fuses the four real-valued multiplications required for each complex operation into single block- matrix multiplication, reducing training time by 25%.
Improved synthesis performance: Com outperforms real-valued vocoders, as demon- strated in our experiments.
RELATED WORKS
2.1
COMPLEX-VALUUE NEURAL NETWORKS
CVNNs represent inputs, activations, and weights directly complex numbers. They have been applied in a range of domains where signals are naturally expressed in the complex field, including radar classification Yang et al. 2022). MRI reconstructio (Vasudeva et al. (2022), wireless commu- nication (Xu et al. 2022). and audio analysis (Sarroff 2018). Several studies report that CVNNs can exhibit favorlle learning behavior
approximation properties compared to real-valued networks in various settings (Barrachina et al.] 2021 Voigtlaender 2023) Geuchen & Voigtlaender (023). This prior work suggests that complex-valuee modeling be viable choice when dealing with data or transformations formulated in the complex domain.
2.2 ISTFT-BASED VOCODER
The short-time Fourier transform (STFT) decomposes a waveform into overlapping frames of com- plex spectral coefficients. The iSTFT reconstructs the time-domain signal using the overlap-ade method. This fully differentiable analysis-synthesis pipeline enables end-to-end training on frame- level spectra while generating sample-level waveforms in single pass. This approach eliminates
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Ground truth
CVNN
with ERITpNL
-2
-3
-2
Figure 1: Ground touund-truth distribution compared with samples generated by RVNN and CVNN.
any explicit upsampling or autoregressive generation, thereby reducing latency. Early methods, such the Griffin-Lim algorithm (Griflin & Lim 1984), used iterative phase reconstruction but of- ten yielded suboptimal coherence between magnitude and phase. GLA-Grad (Liu et al. 2024) later
combined Griffin-Lim with neural diffusion models to improve phase accuracy.
More recent neural iSTFT-based vocoders, such as iSTFTNet (Kaneko et al. 2022). iSTFTNet2 (Kaneko al., 2023), APNet (Ai & Ling. 2023), APNet2 (Du et al. 2024), Free (Lv al. 2024),
Vacos (Suuzakk 2024), and RFWave (Liu et al. 2025), employ diverse architectural designs for iSTFT-based waveform generation.
In these systems, the STFT-domain coefficients are generated directly for frame-level synthesis, enabling efficient inference without waveform upsampling. Our work retains this benefit but ad- ditionally focuses on how this representation is modeled within the network. For this reason, we use complex-valuee layers that operate directly in the complex domain rather than separating each coefficient into real and imaginary channels.
3 PRELIMINARY ANALYSIS OF REAL- AND COMPLEX- VALUED NETWORKS
Recent work on complex-valuee neural networks suggests that operating directly in the complex field can better capture interactions between a variable's 's magnitude and phase than relying on real- valued parameterizations that treat two components independently (Barrachina et al.) 2021 Dou al. (2025). Motivated by this perspective, conduct controlled generative experiment designed to isolate the effect of complex-domann modeling from architectural factors specific to waveform generation.
We train a lightweight MLP-based GAN on a synthetic complex distribution and compare two models: RVNN, which represents complex num- bers real channels, and CVNN, which pro- cesses each coefficient as single complex entity. Because the CVNN stores real and imaginary pa- rameters separately, it requires roughly twice the memory for given layer width; to match mem- ory usage fairly, the RVNN is assigned twice the hidden dimension.
Table 1: JSD between the generated and ground-truth magnitude (mag.) and phase dis- tributions for RVNN and CVNN.
Model
ISD (mag.)
JSD (phase)
RVNN CVNN
0.018350 + 0.014
0.021110 0.036
0.006548 1 0.003
0.003911 1 0.002
Figure I presents sample visualizations across multiple training seeds, and Table I reports the Jensen-Shannon divergence (JSD) between the generated and target magnitude and phase distri- butions, computed using kernel density-based estimator. Both models recover the broad structure of the target distribution, but the CVNN yields samples that adhere more closely to the underlying trajectory and exhibit lower JSD in both magnitude and phase.
These observations provide a simple, controlled example in which modeling directly in the complex domain offers representational advantages when the data possess inherent ceal-imaginary dependen- cies. This motivates our use of CVNNs in the proposed method that follows. Extended analysis and additional visualizations included in the Appendix B
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Input Foature
Complex Convld
Complex Depthwise Conv
STPT;
Reshape
Phase Quantization
Complex LayerNorm
Layer
Complex Conv2d
Conv2d
Complex
Complex Pointwise Cozv
ConvNed Block
Complex LeakyReLU
LeakyReL.
Complex
Complex GELU
Head
Complex Conv2d
Complex
Com22d
ISTFT
Poknlww iso Conv
+
Real/Fake
Real/Fake
udio
(a) Generator
(b) Complex ConvNext Block
(c) cMRD
(d) MPD
Figure Overview of the Com Vo architecture.
METHOD
We present ComVo, an based GAN vocoder whose generator and discriminator operate en- tirely in the complex domain, preserving real-imaginary interactions end to end. The model uses an iSTFT synthesis pipeline with adversarial training objectives. We also include a phase quanti- zation layer inductive bias and adopt block-matrix formulation for efficient complex-valued computation. Figure provides an overview of the architecture.
4.1 GENERATOR
Figure 2 2aa) depicts our generator, which is adapted from the Vocos architecture (Siuzdak] 2024 starting point because it synthesizes via frame-level iSTFT without requiring
We chose Vocos learned upsampling, features a compact feed-forward structure, and serves as a widely used base- line for comparison. All convolutions and normalizations in our generator complex domain. We use split GELU activation (Hendrycks & Gimpel (016)) to maintain the
implemented in the
ConyNeXt-style block layout the complex setting. After the initial complex convolution, phase quantization layer discretizes phase values to stabilize training. Figure 2b) details the complex ConvNeXt block used at each generator stage.
4.2 DISCRIMINATOR
We propose a complex multi-resolution discriminator (cMRD), as shown in Figure 2 Prior work on spectrogram-based discriminators typically used either only magnitude spectra concatenated the real and imaginary spectrogram channels as independent inputs to a real-valued network (Jang et al. 222211 [Suzzaak (024). In contrast, cMRD uses valuue layers and operates directly complex spectrogram inputs. It comprises multiple sub-discrininators each operating at a different STFT resolution. During training, we apply the adversarial loss separately the real and imaginary parts. We also include a multi-period discriminator (MPD), shown in Figure dd) which consists of multiple sub-discriminators operating over different periods and processing reshaped waveform segments (Kong et al. 2020). Because the MPD operates at the waveform level, it remains a real- valued network. The overall training objective combines the adversarial losses from cMRD and MPD, along with feature matching and reconstruction losses. Full loss definitions and weights are provided in Appendix
4.3 PHASE QUANTIZATION LAYER
Complex-values networks remain largely unexplored in terms of nonlinear transformations since any nonlinearity must jointly handle the real and imaginary components. We represent each Mel- spectrogram as a complex value by initializing the imaginary part to zero. We then introduce a phase quantization layer that discretizes phase angles into fixed set of levels. This provides structured
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nonlinearity that preserves relative phase relationships and mitigates phase drift during training. For a complex feature 2 = re" where r 2 0 denotes the magnitude and E (-T, т] denotes the principal phase, the quantized phase is defined as:
N 2n
0g
round
0
(1)
N₂
where Ng is the number of quantization levels. The quantized complex value is reconstructed as
2q re
Quantizing the phase by mapping continuous angles to a fixed set of levels introduces inherent discontinuities that would normally block gradient propagation. To preserve end-to-end differen- tiabiliiy, we adopt the straight-through estimator (STE) (Bengio 2013), in which the quantization operation is applied in the forward pass, while its gradient is approximated by identity function during backpropagation. This preserves gradient propagation through the phase quantization layer and improves optimization stability in practice. Furthermore, by restricting phase values dis- crete set, phase quantization acts as form of regularization: it limits unwarranted phase variability intermediate representations and guides the network toward learning more coherent and structured phase patterns.
4.4 OPTIMIZING COMPLEX COMPUTATION WITH BLOCK MATRICES
To improve efficiency in both the forward and backward passes, we reformulate CVNN operations real-valued block-matrix multiplications. In many autodifferentiation systems, complex-valued layers are implemented by explicitly tracking real and imaginary components as separate real-valued tensors. This leads to redundant operations and inefficient memory access during both the forward and backward passes. We address this by adopting a block-wise formulation that represents com- plex values as structured pairs of real values and processes them jointly through unified matrix operations. This approach reduces component-wist operations and enhances parallelism on modern GPU architectures by enabling matrix-base execution throughout the computational graph. The forward complex operation can be expressed as:
Re(2") Im
W. W.
WW Wr
(3)
y
where = = + y (with & and denoting the real and imaginary input vectors), W = W, + wi, the complex weight matrix (with Wr. W; its real and imaginary parts), and is the resulting complex output. The backward gradient computation uses the same block matrix structure:
aL
W W
W
9r g
=
W
(4)
84
where and 9J are the real and imaginary components of the gradient from the next layer. This unified formulation implemented for all parameterized CVNN layers via custom autograd func- tions. It reduces the number of separate operations and improves parallelism on GPUs by replacing four independent real-valued multiplies with single block-matre multiply, thereby eliminating redundant computation and allowing more efficient gradient evaluation.
5 RESULTS
5.1
EXPERIMENTAL SETUP
We train our model on the LibriTTS corpus (Zen et al., 2019), using the traan--clean crain-clean- 360, and oeein-other-Spe
subsets for training, and evaluating on test-clean and test-other sets. All audio sampled at 24 kHz. The STET uses an FFT size of 1024, hop size of 256, and Hann window of length 1024. Mel-spectrograms
computed with 100 Mel-bins and a maximum frequency of 12 kHz. We compare Com Vo against several repre- sentative vocoders: HiFi-GAN (yl) (Kong al. 2020). iSTFTNet (Kaneko et al. 2022), Big BGAN

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Table 2: Objective and subjective evaluation on the LibriTTS dataset.
Model
UTMOS
MR-STFT + PESQ 1 Periodicity
VUVV FI
MOS
CMOS 1
GT
3.8712
4.08 0.04
0.14
HiFi-GAN
3,3453
1.0455
2.9360
0.1554
0.9174
4.00
0.09
iSTFTNet
3,3591
1.1046
2.8136
0.1476
0.9243
3.98
0.04
BigVGAN
3,5197
0.8994
3.6122
0.1181
0,9418
4.05
0..5
Vocos
3.6025
0.8856
3.6266
0.1061
0,9522
4.05
00.2
Com Vo
3.6901
0.8439
3.8239
0.0903
0.9609
4.07
0
Table 3: Objective evaluation on the MUSDB18-HQ.
Model
MR-STFT
PESQ
Periodicity
VUV FI
HiFi-GAN
1.1909
2.3592
0.1804
0,9004
iSTFTNet
1,2388
2.2357
0.1815
0,9102
Big VGAN
0.9658
3.2391
0.1388
0,9340
Vocos
0,9307
3.2785
0.1369
0,9361
Co
0.8776
3.5220
0.1304
0.9384
Table 4: Subjective evaluation the MUSDB18-HQ.
Model
Vocals
Drums
Bass
Others
Mixture
Average
GT
4.31 = 0.11
4.25 + 0.12
4.26 = 0.12
4.29 + 0.11
4.37 + 0.11
4.29 + 0.11
HiFi-GAN
3,83 = 0.14
3.93 4 0.13
3.43
0.19
3.21 + 0.19
3.60 1600:..16
3.61 1 0.16
iSTFTNet
3.82 = 0.14
0.13
3.37 0.18
3.17
0.19
0.17
3.59 1 0.17
BiAVGAN
4.07 = 0.12
4.19 4.19
3.59 0.17
3,57 = 0.15
0.12
3.88 - 0.14
Vocos
4.04 = 0.12
4.10
3.58
3.52
0.13
3.82 - 0.14
Com Vo
4.05 0.12
4.14 0.12
3.60 0.17
3.68 - 0.16
3,98 + 0.13
3.89 + 0.14
open-source reim- plementation, while the other models are trained using official code with recommended settings. We evaluate using both subjective and objective metries. Subjective quality is assessed via opinion score (MOS), similarity MOS (SMOS), and comparison MOS (CMOS). Objective metrics include UTMOS (Sacki al. (022), PESQ (Rix al. 2001), multi-resolution STFT (MR-STFT) error Yamamoto et al. (2200), periodicity RMSE, and V/UV Fl score (Morrison et al. 2022). Detailed explanations are provided Appendix K and Appendix
(base) (Lee et al. 2023). and Vocos (Siuzdak 2024). For iSTFTNet,
5.2 COMPARATIVE EVALUATION
Table 2 reports results on LibriTTS: ComVo achieves the highest objective scores among the base- lines, and the corresponding MOS and CMOS comparable those of strong baseline systems. Tables 3 and report results on MUSDB 18-HQ (Ratii el al. 2019), an out-of-distribution audio dataset: Com Vo achieves higher scores across all objective measures than the other models, and the corresponding subjective evaluations are comparable to strong baselines. The SMOS evalua- tion shows that Com Vo delivers competitive perceptual quality across individual source stems and mixture tracks, with its average scores typically at near the top. Taken together, these results indi- cate that an iSTFT-based model with complex-valuee modeling consistently improves performance while maintaining the standard pipeline.
5.3
IMPACT OF COMPLEX-VALLEE VALUED MODELING
We assess the contribution of each discriminator component individually. The MPD and MRD pro- vide complementary forms of supervision: the MPD emphasizes periodic structure, while the MRD supplies multi-rea resolution spectral constraints. understand how each behaves on its own, we eval- uate MPD-only, MRD-only, and cMRD-only configurations. The MPD-only variant lacks spectral guidance and exhibits higher MR-STFT error. The MRD-only variant attains low STFT-based emrors but produces lower UTMOS score, indicating that spectral constraints alone do not fully capture

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GnDn
GcDR
GRDe
GeDe
ii
iii
Figure 3: Grad-CAM comparison across generator-discrmmnnaoo configurations. Each row corre- sponds to cMRD sub-iiscriminator operating at different STFT resolution (i, ii, iii).
perceptual quality. The cMRD-only model improves the MRD-only bascline across all ob- jective metrics, showing that the comple--valued discriminator provides a more effective constraint when used alone.
than its real-valued counterpart
We then extend the analysis to the full gemerator-discrmmee discriminator combinations: GRDR. Ge Dis GcD and Gc where GR and Gc denote real-valued and complex-valued generators, and DR and De denote real-valued and complex-valued discriminators. To isolate the effect of complex- valued modeling, the nuase-quantization layer is disabled for all configurations, and the MPD branch is kept active without modification.
Replacing only the generator (GRDR + DcDR) consistently improves all objective metrics. Re- placing only the discriminator (GRDR + GRDc) also yields measurable gains, particularly in MR-STFT error and PESQ. The best performance achieved when both the generator and discrim- inator operate in the complex domain (G)De), confirming the effectiveness of complex -omain modeling for iSTFT-based waveform generation.
For qualitative analysis, we visualize Grad-CAM (Selvarauu et al. 2017) activations of the dis- criminator in Figure Each row the figure corresponds to a sub-discriminator index (i, ii, iii), and each column corresponds to one of the generator-discriminator configurations. In the config- urations with a real-valued MRD (GRDR and DR), the attention maps are diffuse and poorly aligned with speech-relevant spectral structures. In contrast, in the configurations with cMRD DG and GcDc), the highlighted regions consistently trace structured spectral patterns across all sub-dis bibb-iserrimintors.. These results indicate that complex- complex-valued discriminators provide more pre- cise spectral feedback to the generator, helping better match perceptually important features and ultimately improving synthesis quality, also reflected in the ablation metrics.

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Table 5: Ablation study comparing real-valued and complex-valuee architectures.
Model
UTMOS 1
MR-SIFT 4
PESQ 1
Periodicity 4
V/UV FI
MPD only
3.6357
0.8522
3.7670
0.0942
0.9613
MRD only
2.8338
0.8442
3.9868
0.0870
0,9610
cMRD only
2.9285
0.8398
4.0149
0.0859
0.9635
GgDR
3.6025
0.8856
3.6266
0.1061
0,9522
GnDe
3.5930
0.8679
3.6399
0.1060
0,9497
GcDa 3.6452
0.8597
3.7375
0.0978
0,9567
GcDe 3.6646
0.8435
3.7756
0.0915
0.9625
Table 6: Ablation on phase quantization levels. N₂ denotes the number of quantization levels.
Ng Quantization
UTMOS
MR-STFT
PESQ
Periodicity
V/UV f
0 3.6646
0.8435
3.7756
0.0915
0.9625
128
3.6901
0.8439
3.8239
0.0903
0.9609
256
3.6423
0.8466
3.8127
0.0926
0.9597
512
3,6412
0.8489
3.8248
0.0896
0.9613
Table 7: Comparison of standard Py PyTorch and refined implementations.
Implementation
MR-STFT 4
GPU xRT
Training Time
Nodes (Gen cMRD)
Native PyTorch
0.8465
702.26
183 hrs
5686 / 4248 2547 / 1404
Block-matrix
0.8435
696.91
138 hrs
5.4 EFFECT OF PHASE QUANTIZATION
Table shows that adding a phase quantization layer yields clear benefits in perceptual quality, despite only minor trade-off in reconstruction fidelity. The model without phase quantization (N, 0) achieves the lowest MR-STFT error, but moderate quantization level (N, 128) smooths out phase fluctuations, resulting in higher UTMOS and PESQ scores and fewer periodicity artifacts, with only small increase in MR-STFT error. Using finer quantization (e.g.. .g 256, Ng 512) can further boost perceptual metrics, but with diminishing returns and slight degra- dation in reconstruction accuracy. Overall, phase quantization acts as an effective regularizer: enhances listening quality while only modestly affecting spectral fidelity, with N₂ = 128 providing the best trade-off in our setup.
5.5 BLOCK- MATRIX COMPUTATION SCHEME
In this section, evaluate the efficiency and graph-complexity benefits of our block-matrix com- putation scheme. Table 7 reports the comparative results. It shows that our block-matrix imple- mentation achieves performance comparable PyTorch's native complex operations in terms of MR-STFT reconstruction error. While PyTorch's optimized complex kernels yield slightly faster forward-pass throughput, our overall training time is substantially shorter. Specifically, we reduce the number of backward graph nodes in the generator by over 55% and in the discriminator's cMRD by nearly 67%, resulting in a 25% reduction training time. This improvement arises primarily from the backward pass: examining the gradient computation graphs reveals that method dra- matically lowers the node count compared to PyTorch's 's default approach of separately tracking real and imaginary components. By replacing four independent real-valued multiplications with sim- ple channel concatenation and single matrix multiplication, we eliminate redundant operations and significantly accelerate gradient computation, all without sacrificing model fidelity.
5.6 EVALUATION IN TEXT-TO-SPEECI PIPELINE
We further evaluate each model in a text-to-speech (TTS) pipeline by pairing it with an acoustic model. In particular,
Matcha-TTS (Mehta al. 2024) the acoustic model to generate Mel-spectrograms from text, then pass those spectrograms to each model. Matcha-TTi TTS is trained on LibriTTS, and each model is trained independently LibriTTS and connected the Matcha-
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Table 8: UTMOS, MOS, and CMOS com- parison in the TTS pipeline.
Table 9: Comparison of computational cost and inference latency.
Model
UTMOS
MOS
CMOS
Model
Param (M)
Memory (MB)
GPU xRT
HiFi-GAN
3,2233
3,85 1 0.05
0.22
HiFi-GAN
14.00
53,40
259.08
iSTFTNet
3,2951
3.89 + 0.05
-0.15
iSTFTNet
13.33
50.83
402.21
BigVGAN
3.3022
3,92 = 0.05
-0.06
BigVGAN
14.02
53.46
158.07
Vocos
3,4357
3.91 = 0.05
-0.06
Vocos
13.54
51.62
4657.65
Co
3,4403
3,92 = 0.05
0
ComVo
13.28
101.24
819.02
Table 10: Objective evaluation and cost comparison: complex modeling vs. parameter scaling.
Model
Params. (M)
Memory (MB)
UTMOS
MR-STFT
PESQ f
Periodicity +
VUV FI
GRDR
13.54 27.05
51.62
3.6025
0.8856
3,6266
0.1061
0.9522
GRDR x
103.19
3.6164
0.8622
3.6336
0.1055
0,9524
Go
13.28
101.24
3.6452
0.8597
3.7375
0.0978
0.9567
TTS outputs without additional fine-tuning, Table 8 reports the MOS, UTMOS, and CMOS for the TTS pipeline evaluation. Com Vo achieves a MOS that matches the top score among the compared models, and it attains the highest UTMOS. This indicates that Com Vo reliably converts the predicted spectrograms into high-quality waveforms within the TTS setting.
5.7 COMPUTATIONAL ANALYSIS
TabpeeJs the inference throughput and memory usage of each model under a common setup (batch size I, no hardware-specific optimizations). HiFi-GAN and BigVGAN are upsampling-basee models, whereas iSTFTNet, Vocos, and Com Vo synthesize via frame-level iSTFT. The upsampling- based models exhibit the lowest throughput (lower xRT, indicating slower generation), while the isTeT-based models run significantly faster. Among them, Vocos achieves the highest throughput. ComVo's throughput (xRT) lies within the range of the other 1STFT-based models. However, its memory footprint is higher than the real-valued iSTFT baselines: with complex type, each weight is stored as real-ine aall-imaiinary pair, the same precision the per-parameter memory is roughly doubled for a fixed parameter count.
To test whether the improvements stem merely from the larger memory footprint of complex types, we trained real-valued model with twice the parameter count to match the complex model's mem- ory and compared cost-quality trade-offs. The results are reported in Table IO We compare three settings: the baseline real-valued model (R), a widened real-valued model with roughly 2x parameters (denoted GRDD 2 2x), and complex-valued model Dn). The discriminator is iden- tical across all settings. Dr and GRDR 2x have comparable memory footprints. As expected, GRDR 2x improves objective metrics relative to GR In fact, GR exceeds the widened model across all metrics despite similar memory cost. Taken together, Tables 9 and TO indicate that modeling real-imaginary correlations with CVNNs provides larger quality gains than simply scaling real-valuce models.
6 LIMITATIONS
ComVo integrates complex-valued networks into an isseT-based vocoder. To keep the implemen- tation straightforward, we adopt split-style designs. Concretely, we apply component-wise hinge losses the real and imaginary outputs of cMRD, and we use split GELU within the Xo backbone. We will explore more advanced designs for these components in future work. The block- matrix formulation accelerates training, but computational overhead remains high because complex layers store and process paired real and imaginary values. Empirically, multi-GPU Distributed Data Parallel experiments showed under-optimized performance for complex parameters in our current training setup and occasional numerical issues; accordingly, report single GPU results. With bet- ter multi-GB GPU optimization and broader design exploration. larger-scale studies should be feasible and can further catalyze research CVNNs for speech generation.

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7 CONCLUSION
We presented Com Vo, a vocoder that integrates CVNNs into both the generator and the discrimina- tor, establishing a complex-doman adversarial framework for iSTFT-based waveform generation. By modeling the real and imaginary components jointly, our method addresses the structural mis- matches conventional real-valued processing complex spectrograms. We also introduced a phase quantization layer as inductive bias and block-matrix formulation that simplifies compu- tation graphs and accelerates training. Com delivered higher synthesis quality than comparable real-valued basclines. In addition, the block-matris formulation reduced training time by approx- imately 25%. Future work will extend this framework beyond adversarial training to other gener- ative paradigms (e.g., diffusion flow-matching) and explore richer complex-domain activations and losses.
ACKNOWLEDGGENNT
This work was partly supported by Institute of Information & communications Technology Plan- ning & Evaluation(IIITP) grant funded by the Korea goVerament(MSIT) (No. RS-2019-11190079, Artificial Intelligence Graduate School Program (Korea University), IITP LTTP-2026-RS S-------2255022-0- Artificial Intelligence Star Fellowship Support Program to nurture the best talents and No. RS-2024- 00457882, AI Research Hub Project).
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A OVERVIEW OF COMPLEX- VALUED NEURAL NETWORKS
This section reviews the core building blocks of CVNNs complexX convolutions, activation func- tions, normalization, and optimization via Wirtinger calculus (Wirtinger 1927). CVNNs extend real-valued networks by jointly modeling the real and imaginary components (Trbellsi et al. 2018). By preserving cospoonttnnntt structure in the complex domain, they yield more coherent repre- sentations than split-channel parameterizations.
Complex Convolutions: A CVNN performs convolutions directly in the complex domain, jointly processing the real and imaginary parts. For an input complex feature : = x + iy and a complex filter h a + ib, the output z' of complex convolution is:
z' (x y 6) + (x bb++++++)
(5)
where y are the real and imaginary components of z, and a, b are the corresponding components of h. Here, denotes the convolution operation applied to each channel pair before recombining.
Complex-valued networks require activation functions that handle both mag- nitude and phase coherent way. Let fRe: fim, SMag R - IR be real-valued nonlinearities. A simple split activation applies fRe and fim separately to the real and imaginary components:
Activation Functions:
f(z) = (x) + i fim (v),
but this approach ignores the natural coupling between magnitude and phase. A more phase-aware alternative applies SMag the magnitude and then reattaches the original phase:
f(z) = fMag (1=1) e
thereby preserving all phase information while still introducing the desired nonlinearity. (Here | the magnitude and 0 the phase of re"
Normalization: Normalization in CVNNs accounts for the joint distribution of real and imaginary components. A general form of complex normalization is:
P
Borm
(8)
where and a are the mean and standard deviation of the complex input. To capture correlations between the real and imaginary parts, this basic normalization is extended using the covariance matrix:
Jaz
==
Ozy
Tyz
Tyv.
where dur and Byy denote the variances of the real and imaginary components, respectively, and Ozy = Fyz represents their cross-covariance. Using the estimated covariance, the input is normal- ized by centering and decorrelating:
bom (z - p),,
(10)
and an affine transformation is then applied to restore the network's ability to shift and scale the normalized features:
7 + 3,
(II)
where y and 3 are learnable complex-valus valuue parameters. This formulation can be applied to various normalizations (e.g., layer or instance normalization) while preserving the complex structure.
Gradient Optimization: Gradient computation in CVNNs requires special care due to the non- holomorphic nature of most complex-value functions. To handle this, CVNNs employ Wirtinger calculus [Wiriinger 1927). which defines the gradient of real-valued loss L(z) with respect to complex variable x iy as:
aL 1 1OL dx
OL By
OL
1(
OOL dr
OL Dy
(12)
dz 2
0=
For real-valued objectives, only the conjugate gradient ensures descent the loss landscape:
AL 85
is used for parameter updates, which
ah
(1+1) 2(t)
(13)
0z
where 77 is the learning rate.
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Table 11: Architecture used for both the CVNN and RVNN generators and discriminators. The two networks share the same layer structure and differ only in how complex variables are represented.
Component
CVNN
RVNN
Input
I2
Hidden dimension
128
256
Depth
Layer type Activation
Complex Linear Complex LeakyRel.U
Linear LeakyRel.U I2
Output
B INVESTIGATING REAL AND COMPLEX MODELS FOR COMPLEX-DOMANE GENERATION
To investigate how real-valued and complex-valued neural networks differ in learning distributions with coupled real-imaginary structure, we conduct minimal generative modeling experiment based on two-dimensional target density defined in the complex plane. This setting removes the influ- ence of architectural factors specific to waveform generation and isolates the effect of the underly- ing parameterization. The target distribution contains a nontrivial correlation between its real and imaginary components, providing simple but informative test case for comparing representational behavior.
The complex-valued models operate directly in C, receiving a one-dimensional complex latent vari- able and propagating through a stack of complex linear layers with complex activations. The real-valued models use equivalent architecture in depth but operate entirely in I, starting from a two-dimensional latent input and producing two real outputs that are interpreted as the real and imaginary components of sample. To match representational width between the two model fami- lies, the hidden dimension of the RVNN layers doubled relative to the CVNN. A concise summary of these architectural differences is provided in Table 11 All networks are trained using the standard GAN objective with binary cross-entropy loss and identical optimization hyperparameters.
For each random seed, examine three aspects of the learned distribution: (i) the scatter plot of generated samples, (ii) the magnitude histogram, and (iii) the phase histogram. These visualizations allow us assess how consistently each model reproduces the target structure across independent runs. Examples are shown in Figure While both models are capable of approximating the global geometry of the target, the complex-ves tude-phase statistics with reduced run-to-run variability. This experimental design does not aim to assert broad conclusions beyond this setting, but provides a controlled example which complex- valued parameterization can yield advantages when the modeled data are inherently expressed in the complex domain.
generator often produces more stable spirals and magni-
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Figure 4: Visualizations over multiple training seeds. Each row corresponds to one run and contains five subplots: ground-truth samples, RVNN outputs, CVNN outputs, and the corresponding magni- tude and phase distributions. This layout enables run-to run-to-run comparison of distributional behavior across the two models.
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C DETAILS OF TRAINING OBJECTIVE
The VomVo training objective integrates adversarial, reconstruction, and feature-matching losses from both the MPD and the eMRD.
C.I DISCRIMINATOR Loss
We use adversarial losses to push real samples above and generated samples below the decision boundary.
MPD Loss: Let D denote the k-th sub-discriminator operating raw waveforms. For each period Pk. the input segment is reshaped to (Pk. P/P;) to expose the periodic structure. We use hinge loss on the real-valued outputs:
CMPD Σ (max((0 max(0, DMPD (y)))
(14)
+ (mx((( + D MPD (1))
where y and y are ground toound-truth and generated waveform segments, respectively.
cMRD Loss: We apply hinge losses independently to the real and imaginary components to retain compatibility with standard real-valued GAN objectives, while allowing the discriminator to operate directly the complex domain. For any complex quantity let [u] and [u], denote its real and imaginary parts, respectively (these are operators on a single complex output, not separate networks). With DRMRD the k-th sub-discrii aab-isscrimintoor.
K
CD CeMRD
E. max(0, - [DE MRD (2)]R) + max(0, 1 [D MRD (z)]r)
(15)
Eg max(0, 1 (RM))(()))))) + max(0, 1 1 DD MRD (:)]
C.2 GENERATOR LOSS
The generator objective includes reconstruction, adversarial, and Seature-matching terms.
Mel-spectrogram Loss: We L1 loss log-scaled Mel-specttograms:
Lmel E M(y) M(i)l
(16)
where y and ý denote ground-trutt and generated waveforms, and M.)) is the log-Mel transform.
Adversarial Generator Loss: For the MPD operating on waveform segments ÿ:
K
MPD
Eg max(00 1 MPMP (i))).
(17)
For the cMRD operating generated spectrograms å, let [-]n and [-], denote the real and imagi- nary parts a complex output. We apply hinge losses to both components:
K
CcMRD
Eg max((, 1 [D MRD (2)] + max(0, 1 DDE MRD (3)]))-
(18)
Feature Matching Loss: We match intermediate representations in both discriminators.
For MPD (waveform segments y and g), we use & loss on feature maps:
L
CPM
Σ
MPD k.l
(v)
D
DMPP()))
MPD
(19)
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where D is the I-th layer feature of the k-th MPD -ubddiscrimintoo.
For cMRD (complex spectrograms and 2), let I and [-] denote the real and imaginary parts of a complex feature, respectively. We match the components separately:
K
Lx
FPM
ΣΣΣ
1
(z)] (x))a
[D (3)]a]h
(20)
(z)]r [D MDDD (8)).ll)
Total Generator Loss: The generator objective combines reconstruction, adversarial, and feature-
matching terms:
(MPD) Lgen AMdel LMel + AMPD ( (CMPD + FM
(21)
+ AMMDD (
PM
Here, AMel. AMPD and AeMRD weight the Mel, MPD, and eMRD terms, respectively. Detailed hyperparameters provided in Table 20
D PROOF OF EQUIVALENCE BETWEEN THE BILOCK-MATRIX COMPUTATION SCHEME AND STANDARD COMPLEX-VALUEI OPERATIONS
We now verify detail that applying the block-matrix operator
W - W2
W W
to the stacked real vector [x; y] reproduces exactly the real and imaginary components of the com- plex product Wz with Wr + WW
D.I FORWARD COMPUTATION
Let
2 = W W, WW,, where v, We, W; real-valued. Then the complex linear transformation be written
2 = (W, + (x)
W, + iW, + wi, 2W Way
(W, Way + (W,r + Wyy).
Thus
Ro(2') W,r W.y Im(z) Wa + Wyy.
On the other hand, the block-matrix product gives
x y/
W, W.y
Re(=')]
A
=
Wir W.y
Im(2'))
D.2 BACKWARD COMPUTATION
Let scalar loss be L, and denote
aL
aL Alm(z')
2Re(z')
In the complex formulation, the gradient with respect
aL W (gr + +i)))= = (W,J gr + W. 9.) + (-WWWW W. gr + W,T.T 9t).
Oz
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Define
9x = W, 9r + W.T.O 9b:
99 = W; gr + W, 9s.
Stacking these gives
9x
gr
W
- W;
gr
A
(22)
g4
99
W
W
9
which is precisely the transpose of the forward block-matrix. For convolutional layers, each trans- pose block corresponds the appropriate transposed-convllee convolutinnnoittionn operator.
Table 12: Average GPU execution times for generator (Gen) and discriminator (Dise) forward and backward passes.
Implementation
Gen Forward (s)
Gen Backward (s)
Disc Forward (s)
Dise Backward (s)
Native PyTorch
0.0055888
0.234591
0.0790773
O) 19190067
Gaussian trick
0.005 160
0.231555
0.074103
0.184894
Block-matrix
0.005786
0.18122 8828
0.050389
0.139159
E SPEED COMPARISON OF GENERATOR AND DISCRIMINATOR OPERATIONS
To isolate the effect of block-matrix fusion, we benchmark only the generator and the cMRD, ex- cluding the MPD and reusing the same pretrained hyperparameters across all implementations.
In addition to the native PyTorch implementation and our block-mane matrix formulation, we also evalu- ated Gauss' multiplication trick, implemented using the comp extorch library
Gauss' multiplication trick rewrites complex product using three real-valued convolutions instead of four, and a common arithmetic reduction technique for complex operations.
Table 12 reports the average GPU execution times for the forward and backward passes of both generator and the cMRD over runs with a batch size of 16. For generator, the forward time shows minimal variation across implementations, indicating that fusing real and imaginary components introduces little overhead this part the computation. In contrast, the block-matre formulation substantially reduces the generator's 's backward time and provides clear improvements both the forward and backward passes of the cMRD, leading to a noticeably faster end-to-end training step. Overall, these results indicate that the block-matrix formulation provide practical efficiency gains in our training setup.
Component-level differences in intermediate values and parameter gradients between native and refined implementations.
Table 13:
Metric
Convld
Conv2d
Linear
Input gradient
2e-09
6e-09
8e-10
Forward output
4e-09
1e-08
7e-09
Forward output gradient
9e-09
2e-08
le-08
Weight
0e+00
0e+00
0e+00
Weight gradient
1e-07
4e-07
4e-08
Bias
Oc+00
0c+00
Oc+00
Bias gradient
6c-08
4c-07
3e-08
F NUMERICAL CONSISTENCY VERIFICATION
To confirm that our block-matrix computation scheme maintains numerical fidelity, we compare forward outputs and gradients for each module against the native PyTorch implementation. Table B
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Table 14: Model-level differences in outputs, losses, and gradient magnitudes between native and refined implementations.
Metric
Generator
Discriminator
Forward output
7e-06
5e-06
Loss
5e-07
2e-07
Gradient
5e-06
le-06
reports mean absolute differences at the layer level for convolutional and linear modules-all within typical floating-poit point tolerances (~ 10 10-7). Table 14 summarizes end-to-end deviations in generator and discriminator outputs. losses, and gradient norms, all below 10 10-5. These results verify that, despite the structural optimizations, our block-matrix approach preserves numerical consistency and does not affect training dynamics.
G BACKWARD GRAPH VISUALIZATION
10 and show the backward computation graphs of the generator using (i) the native Py PyTorch complex implementation, (ii) Gauss' multiplication trick, and (iii) the block-matrix formu- lation, respectively. Figures 12 13 and 14 present the corresponding graphs for the cMRD. For clarity, both models are simplified by using a single -ce-speetr loss and reducing the number of layers and channels.
Figures
Across all configurations, the block-matrix formulation (Figures and yields the most com- pact backward graph. Compared to the native (Figures 12 and Gauss-based implementations (Figures 10, 13p. it avoids redundant branches and reduces the number of elementwise operations, resulting significantly simpler and more efficient gradient flow.
Avg cost by duration bin (bin=1.0s)
HiFiGAN
0.12
ISTFTNet
HigVGAN
Vocos
0.10
ComVo
0.08
Avg cost
0.06
0.04
0.02
0.00
10
15
20
25
30
35
Duration bin center
Figure 5: Average inference cost as function of utterance duration.
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H RUNTIME AS A FUNCTION OF UTTERANCE LENGTH
Figure 5 plots average inference cost versus utterance duration using 1-second bins and consis- lency under the same setup as Table points indicate bin means and vertical bars show variability. Upsampling-bane based vocoders increase approximately in proportion to duration with clear positive slope. whereas iSTFT-based vocoders exhibit a flatter, near-constant profile over the plotted range. The proposed method follows the iSTFT family: its curve lies above Vocos but remains below iSTFTNet and the upsampling-hase based systems across bins. Although CVNNs introduce computa- tional overhead, Com Vo maintains competitive runtime characteristics within the iSTFT class.
Table 15: Ablation comparing real Conv1 producing two-channel (Re, Im) feature, Complex Conv w/o PQ, and Complex Conv w/ PQ
Model
UTMOS
MR-STFT
PESQ
Periodicity +
VUV Fl
Real Cony
3.6337
0,8610
3.7774
0.0980
0,9574
Complex Conv w/o PQ
3,6646
0.8435
3.7756
0.0915
0.9625
Complex Conv w/ PQ
3.6901
0.8439
3,8239
0.0903
0.9609
I ANALYSIS OF PHASE QUANTIZATION
The generator receives real-valued inputs, and the imaginary component of the initial complex rep- resentation must therefore be synthesized internally by the network. At this early stage, the phase can vary freely, as there is no signal-driven constraint guiding how the initial complex feature should be formed. Prior work in speech coding has also observed that unconstrained phase can introduce instability or unnecessary variability during optimization (Yu & Chan 2002] Kim 2003). Motivated by these considerations, we insert phase quantization (PQ) step immediately after the first com- plex Conv ID layer to lightly regularize the formation of the initial complex features while allowing subsequent layers to operate without explicit phase constraints.
To examine whether the effect of PQ is tied to its interaction with the first complex layer, we trained a variant where the first complex C was replaced with a real Conv that outputs two channels, which are then interpreted as the real and imaginary components of a complex feature. Aside from this modification, the architecture remains unchanged. This variant trains properly and produces results similar to the version without PQ, whereas the original configuration with complex Conv ID followed by PQ achieves higher scores across all metrics (Table 15). This comparison indicates that the benefit of PQ is associated with its placement at the point where complex features first emerge.
Table 16: Comparison with amplitude-phart prediction vocoders
Model
UTMOS t
MR-STFT 4
PESQ t
Periodicity +
V/UV
PPNet
2,4015
1.3375
2.8457
0.1582
0.9185
APNe12
2.7379
1.1582
2.7748
0.1448
0.9243
FreeV
2.6971
1.1782
2.7960
0.1581
0.9105
Com Vo
3.6901
0.8439
3.8239
0.0903
0.9609
J COMPARISON WITH AMPLITUDE-PHAST PREDICTION VOCODERS
In addition to comparing against GAN-based vocoders, we also consider amplitude-phas predic- tion methods, in which magnitude and phase are modeled separately using real-valued networks. Representative examples include APNet (Ai & Ling, 2023). APNet2 (Du et al. 2024), and FreeV (Lv et al. 2024), all of which treat the two components independent regression targets.
formulation relative to this family of methods, we trained APNet, APNet2, and FreeV with their official implementations under the same data and training settings used in our system. This provides a controlled comparison between explicit amplitude- -hase esti- mation and directly modeling complex STFT coefficients.
To position our complex-dome
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Table 16 presents the results. Across all metrics, the proposed model achieves higher quality, sug- gesting that learning in the complex domain an effective parameterization for iSTFT-basee gener- ation compared to treating magnitude and phase separate prediction targets.
Table 17: Baseline model implementations and sources.
Model
Implementation Source
iSTF
HiFi-GAN
Big VGAN
Vocos
APNet
APNe12
Free'V
K BASELINE MODEL IMPLEMENTATIONS
We evaluate our proposed method against several representative neural vocoders, each with distinct architectural designs:
HiFi-GAN (v1) (Kong et al. 2020): A GAN-based vocoder that uses multiple discriminators (MPD and MRD) with transposed convolutional generator. It emphasizes high-fidelity waveform gener- ation with fast inference.
TeeFTNet (Kaneko al. 2022 ; lightweight vocoder that replaces upsampling layers with iSTFT
to reduce redundant computations. It directly predicts complex-valued spectrograms, simplifying the overall architecture.
BiAVGAN (base) (Lee et al. 2023): : An improved HiFi-GAN variant that introduces the Snake func- tion (Ziyin et al. 2020) for better modeling of periodicity and high geehffreuency details. It also adopts a scaled discriminator design, contributing to more stable GAN training and enhanced performance on challenging inputs.
Vocos (Siuzdak) 2024): An iSTFT-based vocoder built on ConvNeXt (Liu et al. 2022) architec- ture that predicts Fourier spectral coefficients waveform reconstruction. It achieves high-quality synthesis with low latency.
APNet (Ai & Ling. 2023): vocoder that separately predicts amplitude and phase spectra using
independent real-valued branches. Phase is modeled explicitly through parallel estimation module with anti-wrapping losses, and the waveform is reconstructed via iSTFT.
APNet2 (Du al. 2024): An improved version of APNet that adopts a ConvNeXt v2 backbone and multi-resolution discriminators. retains the separate amplitude-phase prediction design while offering higher fidelity and greater training stability.
V (L et al., 2024): A lightweight amplitude- vocoder derived from APNet2 that incor- porates gegnal-processing priors. It obtains approximate amplitude spectrum via pscudo-inverse mel filtering, reducing ASP complexity while maintaining quality.
We use the official implementations provided by the authors whenever available, except for iSTFT- Net, which lacks an official repository. For iSTFTNet, we adopt a publicly available open-source implementation instead. Implementation sources are summarized in Table
Table 18: Implementation sources for objective evaluation metrics.
Model
Implementation Source
UTMOS
MR-STFT
PESQ
Periodicity RMSE V/UV Fl score
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L EVALUATION METRICS
L.1 SUBJECTIVE EVALUATION
We conducted mean opinion score (MOS) listening tests on Mechanical Turk with 20 U.S.-based native English speakers, each evaluating 50 samples. We also ran similarity mean opinion score (SMOS) tests under the same conditions. In MOS, listeners rated naturalness on 1-5 scale; in SMOS, they rated similarity between synthesized and reference audio on 1-5 scale. In addition, we conducted comparison MOS (CMOS) using 7-point scale. For reporting, we use pairwise comparisons against our system the reference; thus the reference row is centered 0 and other systems' scores reflect average preference relative to it. To filter inattentive participants, we inserted fake samples and instructed listeners to mark them "X"; any listener who missed these was excluded. Figure o shows the MOS interface, Figure 7 shows the SMOS interface and Figure shows the CMOS interface.
L.2 OBJECTIVE EVALUATION
We measure performance using five objective metrics: UTMOS (Saeki et al. 2022), multi-resolution short-time Fourier transform error (MR-STFT) (Yamamoto et al., 2020), perceptual evaluation of speech quality (PESQ) (Rix et al., 2001), periodicity RMSE, and voiced/unvoiced (V/UV) Fl score (Morison et al. 2022). Implementation sources are listed in Table [8
UTMOS: We use the open-source UTMOS model to predict MOS scores for evaluating speech naturalness.
MR-STFT: We the multi-resolution STFT loss implementation from Auraloss (Steinmetz & Reiss 2020) to measure spectral distortion between the generated and ground-truth audio.
PESQ: We use the wideband version of PESQ with audio resampled to 16 to assess perceptual quality.
Periodicity and V/UV F1: Periodicity RMSE is used to quantiffy periodic artifacts, while the V/UV Fl score measures the accuracy of voiced/unvoiced classification.
Table 19: Comparison of large-scale models
Model
Params. (M)
UTMOS
MR-STFT +
PESQ f
Periodicity +
VUVV F1
GT
3.8712
BigVGAN (large)
112.41
3.5489
0.8644
3.8197
0.0888
0.9607
Vocos (large)
114.51
3.6923
0.8625
3,8362
0.0933
0.9596
Com (large)
114.56
3.7337
0.8443
3,8831
0.0871
0.9629
M EXTENDED EXPERIMENTS WITH LARGE-SCALE CONFIGURATIONS
To test whether the benefits of complex-valued modeling persist higher capacity, conducted a scaling study with large variants the baselines and our model. All systems were trained on the same LibriTTS splits in the base-scale experiments. For BigVGAN, used the authors' official large configuration; for Vocos and Com Vo, we set configurations to match the BiAN large model's parameter budget closely as possible while keeping architectures comparable. All runs were trained for 1M optimization steps on a single GPU. Table 19 summarizes the large-scale results. In this setting, Com Vo scaled effectively, showing clear quality gains across evaluation metrics. Overall, the complex-values approach scales well, and increasing capacity yields consistent quality gains.
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Table 20: Training hyperparameters.
Mel-spectrogram
Sampling rate
24,000
FFT size
1024
Hop length
256
Window size
1024
Mel bins
100
Generator
Base
Large
Input channels
100
100
Model dimension
512
1536
Intermediate dimension
1536
4608
Number layers
8
Phase quantization levels
128
128
MPD
Periods Pk
[2, 3, 5,
7, 11]
MRD cMRD
FFT sizes
[512, 1024, 2048]
Hop sizes
[128, 256, 512]
Window sizes
[512, 1024, 2048)
Bands ratio
[0, 0.1, 0.25, 0.5, 0.75, 1.0]
Training
Base
Large
Batch size
16
32
Steps
1M
IM
Segment size
16,384
16,384
Initial learning rate
2e-4
2c-4
Scheduler
cosine
cosine
Optimizer
AdamW
AdamW W
31,32
(0.8, 0.9)
(0.8, 0.9)
AMol
45
45
AMPD
1.0
1.0
AeMRD
0.1
0.1
Hardware
GPU CPU
Ix NVIDIA A6000
Intel Xeon Gold 6148 @ 2.40 GHz
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highly recommend
with headphone the environment
background.
Evellute naturainess audio samples.
This score should reflect your opinion of how natural the audio sounded.
Note that should not judge the grammer content the sudio, just how sounds and pronounces.
evalustion.
Warning: Noise samples are included. If you rate noise sample with score other than X, your evaluation will be rejected.
Example oudio (expected 'Excellent- -F)
Example (expected "Good-
63t roze
Example of audio (expected "Fair FFir-37
instrucion
you
030/ E18
Excelent Compant FOLLM speech-
Fal- Cqual THUSS Lenutaaa speecs
Pook MOSH STORRE speech 2
deert roo
surph
Figure 6: MOS evaluation interface.
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Please wear earbuds or headphones before start the lask
Instructions
Evaluate audio similarity audio pair.
Please listen to the two audio samples and rate how similar they are. Your rating
reflect evaluation
the
two samples sound.
Please listen to each the audio files carefully during evaluation.
reliability of your evaluation is less than or the total evaluation shorter than the total length of the audio files, we will reject your review.
We put some samples. So, your evaluation on fake samples looks doubtful, we reject your review.
Example audio pairs (expected "Excellent- very similar 5)
$90 1o04
$90 Fooe
Example audio pairs (expected "Good similar
#82/s04
880
Example sudio peirs (expected "Poor - dissimilar - 2n)
280 /004
$30
Example audio peirs 4 (expected "Bad very dissimilar 1*)
840 FCB5
[*] Before you start, please instructions
answer one carefully, thanks!!!
option
Plcasr heuelghrve
Excellen- Comprstoy sentar 5
can colema devics
0:00 1936
Good Mosty sendal speech -4
Equally
Be serole hoross
Feer Mostly unsimi speech
Dad: Compleelyy unsimitrr speech
contio
Fake sample
Figure 7: SMOS evaluation interface.
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Instructions
Compare
rate relative
The score should refect the audio quality the second sample compared to the first.
Do not consider pronunctation, prosody, grammar, or content. Base your judgment only on how the audio sounds.
Please
evaluation.
reliability evalustion than 50% the evaluation time shorter the local audio
We some fake samples So, your evaluation take samples looks doubiful, we will nevile
Example of audio pairs (expected "3 Much bettert)
0/00 0:04
Example audio pairs 2 (expected 10 About the same")
0:00 0:04
0:00 6004
Example audio (expected --3 Much worse")
0:00 0:04
0100 0004
Please read the Instruction next to each task before startill
Instructions
Shorizats
does secoed recording sound tarn autoo qualtty companed
Select an option
0'00
Much better
Set audio confortatle level
0:00
1 Slightl better
Listen each audio sample casetulyy We ixcommeed liserin stlaas trice.
-1 Slightty worse
second
2 Worse
companed the
Select Take campplr the audio is Dite samale
Fake sample
Subeet buttoe to
More Instrcciions
Figure 8: CMOS evaluation interface.
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Figure 9: Backward computation graph of the generator using the native PyTorch complex imple- mentation.
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Figure 10: Backward computation graph of the generator using Gauss' multiplication trick.
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Figure 11: Backward computation graph of the generator using the block-matrix operation.
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Figure 12: Backward computation graph of the cMRD using the native PyTorch complex imple- mentation.
Figure 13: Backward computation graph of the cMRD using Gauss' multiplication trick.
Figure 14: Backward computation graph of the cMRD using the block-matrix operation.
32


Reviews:
[{"review_id": "HgaZ4k7OlV", "reviewer": "Reviewer_eE9R", "summary": "This paper introduces ComVo, a neural vocoder for waveform generation that operates within the complex domain. The core idea is to leverage complex-valued neural networks (CVNNs) for both the generator and a multi-resolution discriminator (cMRD), arguing that this allows the model to better capture the intrinsic structure of complex spectrograms compared to conventional real-valued networks that process real and imaginary components independently. The authors also introduce two technical refinements: a phase quantization layer to act as a regularizer, and a block-matrix computation scheme to improve training efficiency. The paper presents a series of experiments and ablation studies showing that ComVo achieves competitive or superior performance on objective and subjective metrics against several strong real-valued vocoder baselines.", "questions": "I hope the authors can address the following points to strengthen their claims and clarify the contributions of their work:\n1. The core premise of the paper needs a stronger defense. Given that the ultimate target is a real-valued signal, which implies its spectrogram must satisfy time-frequency consistency, why is it advantageous for the hidden layers to operate under general complex arithmetic rather than in a way that respects this physical constraint from the outset? Could the unconstrained complex modeling actually be a less efficient path to the desired solution space?\n2. The central claim is that the model better \"captures intrinsic dependencies\" between real and imaginary parts. Beyond the final performance metrics, is there any more direct analysis you can provide to support this? For instance, have you analyzed the learned phase-magnitude relationships or the structure of the internal representations to show they are more coherent than in real-valued models?\n3. The ablation study in Table 4 is insightful, but the powerful real-valued MPD remains a constant across all configurations. How can we be confident that the gains attributed to the complex-valued cMRD are not simply an artifact of having two strong discriminators, with the MPD still being the primary driver of quality? Have you considered an ablation where only the cMRD is used, to truly isolate its effectiveness?\n4. Other iSTFT-based models like Vocos and iSTFTNet also operate on complex spectrograms, but their motivation seems more direct—it's a computationally efficient and direct target for the network. Your paper claims a more fundamental modeling advantage. Could you elaborate on why your motivation leads to a better vocoder than one motivated purely by computational efficiency?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The exploration of complex-valued networks for generative audio tasks is a compelling research direction, and the authors present a well-executed implementation. The paper is technically solid; the proposed phase quantization is an interesting inductive bias for stabilizing phase prediction, and the block-matrix formulation for accelerating training is a valuable engineering contribution that demonstrably reduces training time by 25%. The experimental evaluation is thorough, with comparisons against strong, widely-used baselines like HiFi-GAN and Vocos on standard datasets. The ablation studies in Table 4 are particularly useful for dissecting the contributions of the complex-valued generator and discriminator.", "weaknesses": "Despite the positive results, I have fundamental reservations about the paper's central motivation. The primary claim is that CVNNs are superior because they \"capture the intrinsic dependencies between the real and imaginary components.\" However, this central hypothesis is asserted rather than rigorously validated. The performance gains, while present, do not in themselves prove that this specific mechanism is the cause. My main conceptual issue is that for a spectrogram to be perfectly invertible back to a real-valued signal, it must satisfy time-frequency consistency. This means the space of valid spectrograms for real audio is a highly structured subspace within the broader domain of all possible complex spectrograms. By moving all computations into an unconstrained complex domain, the model may actually face the additional burden of learning to stay within this physically valid subspace, which might not be an advantage. The paper does not address this potential conflict.\nFurthermore, the overall architecture still relies on a standard Multi-Period Discriminator (MPD) operating on the real-valued waveform. The MPD is known to be a very powerful component in modern vocoders. Its presence makes it difficult to ascertain whether the observed quality improvements truly stem from the benefits of complex-domain feedback via the cMRD, or if the MPD is still doing the majority of the perceptual heavy lifting. Finally, from a structural standpoint, the detailed background in Section 2.1 on the fundamentals of CVNNs feels more appropriate for an appendix, as it disrupts the main narrative of the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "n7UaMLxvI1", "reviewer": "Reviewer_gyMx", "summary": "This paper introduces ComVo, a complex-valued neural vocoder that performs waveform generation entirely in the complex domain using a GAN-based architecture. The authors argue that existing iSTFT-based vocoders rely on real-valued networks that process real and imaginary parts independently, limiting their ability to capture the inherent structure of complex spectrograms. The proposed model employs complex-valued neural networks (CVNNs) to jointly model the real and imaginary components of spectrograms. The paper also introduces phase quantization as a regularization method and a block-matrix computation scheme to improve training efficiency. The experimental results show that ComVo outperforms existing real-valued vocoders in terms of synthesis quality and training time.", "questions": "See the above Weaknesses.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Originality: The introduction of complex-valued neural networks (CVNNs) for waveform generation is an interesting and novel approach that is not widely explored in the context of vocoders. The proposed method shows potential in capturing the structure of complex spectrograms by treating them as unified complex entities.\n\nQuality: The paper is well-written, and the experimental setup is clearly described. The proposed method shows promising results in terms of both objective and subjective evaluations.\n\nClarity: The paper is presented in a clear and structured manner. The explanations of the method, including the details of the generator, discriminator, phase quantization, and block-matrix computation scheme, are well-explained.", "weaknesses": "A major weakness of this paper is that it compares the proposed method only to real-valued vocoders and iSTFT-based methods. The paper does not include a comparison with vocoders that predict both amplitude and phase spectrograms (such as APNet and FreeV). These methods already integrate both real and imaginary parts in their amplitude and phase spectrogram predictions, which might address the issue the authors claim with real-valued networks. Without this comparison, it is difficult to conclusively prove that ComVo offers a significant advantage over existing methods. The authors should include such comparisons to strengthen the argument for their method's effectiveness.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oJZyoMr2xN", "reviewer": "Reviewer_GeoB", "summary": "This paper introduces ComVo, a complex-valued neural vocoder designed to generate high-fidelity audio within the iSTFT-based vocoder framework. ComVo leverages complex-valued neural networks (CVNNs) for both the generator and discriminators, enabling better modeling of signal characteristics and thereby enhancing audio reconstruction quality. In addition, two practical techniques are proposed: (1) Phase quantization - stabilizes training by mitigating abrupt phase drift, and (2) Block-matrix computation - improves computational efficiency by performing block-wise operations on real and imaginary components of complex-valued tensors. Experimental results show that ComVo outperforms real-valued vocoders with comparable parameter counts across multiple evaluation metrics (e.g., UTMOS, PESQ, MR-STFT). Further ablation studies demonstrate the effect of applying the phase quantization and block-matrix computation.", "questions": "- The phase quantization layer is applied after the first complex Conv1D in the generator. If the first Conv1D was not complex-valued, the phase quantization layer would not be required. What would happen if this sequence (complex Conv1D followed by phase quantization) was replaced with a single real-valued Conv1D layer? Such a comparison would help clarify the necessity and contribution of the phase quantization layer.\n- Why is the phase quantization layer not commonly employed in other complex-valued CNNs?\n- Could the authors provide a theoretical justification for why the phase quantization layer is beneficial or necessary for improved performance?\n- Why the inference latency of ComVo is noticeably slower than Vocos, despite adopting a similar architectural foundation?\n- A comparison of models with equivalent memory usage would be informative, since ComVo uses complex-valued tensors for its parameters, which actually consist of two float tensors. For example, it would be interesting to see whether a ComVo model with half the number of parameters still outperforms the real-valued baselines.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The use of CVNNs demonstrates clear performance gains over real-valued architectures.\n- The paper presents various ablation studies, providing insight into the role of each proposed component in the overall system. The architectural design is well-motivated and supported by empirical evidence.\n- ComVo scales effectively, showing competitive performance across both lightweight and large model configurations.\n- The advantage of ComVo is maintained even when integrated into a TTS pipeline, highlighting its versatility and robustness.", "weaknesses": "- For me, the proposed block-matrix computation seems more like an implementation-level optimization rather than a novel research contribution.\n- The ablation results for phase quantization appear relatively weak, offering limited evidence to justify its effectiveness.\n- Although Table 8 indicates that ComVo and the baselines have similar parameter counts, ComVo uses a different parameter datatype (e.g., ComVo’s parameters are stored in complex64 format, which effectively corresponds to two float32 tensors for real and imaginary parts). Therefore, for a fair comparison of model capacity and memory usage, the baseline parameter counts should be doubled.\n\nOverall, the ComVo model itself is competitive and valuable, and its performance improvements are noteworthy. However, given that ‘phase quantization’ and ‘block-matrix computation’ are addressed as major contributions in the main text, I think their novelty and impact should be more clarified and empirically supported to substantiate their significance as research contributions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)