Jean-Eudes Ayilo

Mostafa Sadeghi

Romain Serizel

DIFFUSION-BASED SPEECH ENHANCEMENT WITH A WEIGHTED GENERATIVE-SUPERVISED LEARNING LOSS

Keywords: Speech enhancement, diffusion models, generative modeling, supervised learning

Diffusion-based generative models have recently gained attention in speech enhancement (SE), providing an alternative to conventional supervised methods. These models transform clean speech training samples into Gaussian noise, usually centered on noisy speech, and subsequently learn a parameterized model to reverse this process, conditionally on noisy speech. Unlike supervised methods, generative-based SE approaches often rely solely on an unsupervised loss, which may result in less efficient incorporation of conditioned noisy speech. To address this issue, we propose augmenting the original diffusion training objective with an ℓ 2 loss, measuring the discrepancy between ground-truth clean speech and its estimation at each diffusion time-step. Experimental results demonstrate the effectiveness of our proposed methodology.

INTRODUCTION

Diffusion models are a recent class of generative models that have brought significant improvements in image and audio synthesis [START_REF] Song | Score-based generative modeling through stochastic differential equations[END_REF][START_REF] Kong | Diffwave: A versatile diffusion model for audio synthesis[END_REF]. Their underlying mechanism is to gradually turn training samples into noise, and then learn a parameterized model to revert this process, thus enabling data generation from pure noise. These models are also gaining increasing interest in the speech enhancement (SE) task, whose goal is to recover a clean speech signal recorded in adverse acoustic environments. In this context, diffusion models aim at learning the distribution of speech data, encoding their temporal-spectral characteristics, in order to infer clean speech from noisy observation [START_REF] Lu | Conditional diffusion probabilistic model for speech enhancement[END_REF]. Additionally, this distribution is learned conditionally on the associated noisy speech data to guide the data generation process [START_REF] Lu | Conditional diffusion probabilistic model for speech enhancement[END_REF][START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF][START_REF] Hu | Noiseaware speech enhancement using diffusion probabilistic model[END_REF][START_REF] Yen | Cold diffusion for speech enhancement[END_REF].

This generative-based SE approach is systematically different from the prevailing supervised counterpart [START_REF] Xu | A regression approach to speech enhancement based on deep neural networks[END_REF][START_REF] Wang | Supervised speech separation based on deep learning: An overview[END_REF], which learns a deep neural network (DNN) to directly estimate clean speech or a time-frequency mask from noisy input by minimizing a supervised loss, e.g., mean squared error (MSE). In contrast, diffusion-based SE usually follows the standard unsupervised (generative-based) loss used in diffusion models, with the difference that noisy speech is provided as an additional input. While this approach could potentially be advantageous, as it models the intrinsic properties of clean speech contrary to supervised methods, we hypothesize that it might not efficiently incorporate a measure of the goodness of the estimated clean speech.

The current study aims at addressing this issue, and bridging the performance gap between the supervised and diffusion-based approaches by combining the best of the two worlds. To this end, we propose to add an ℓ 2 loss to the original generative-based diffusion loss. This extra supervised loss measures the distance between ground-truth clean speech and its estimation at each diffusion time-step. In doing so, we hope to combine the effectiveness of diffusion models in unseen noise conditions and the strength of supervised methods in seen noise conditions. Experiments are performed to evaluate the effectiveness of the proposed approach against both supervised and standard diffusion-based approaches. The results show promising performance of the proposed training methodology.

In the rest of the paper, we present an overview of diffusion-based SE methods in Section 2, followed by a review of [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF]. Our proposed methodology is discussed in Section 3. Next, Section 4 presents the experiments and results, and Section 5 concludes the paper.

RELATED WORK

Diffusion-based speech enhancement

The fundamental concept behind diffusion models involves two main phases. Initially, within a forward process, clean data are progressively distorted by adding (usually Gaussian) noise, eventually resulting in entirely noise data following a tractable distribution like a standard Gaussian. Subsequently, through a reverse process, a DNN is trained to sequentially produce clean data, beginning from random noise sampled from the prior distribution.

Many recent studies have already used the diffusion principle for speech enhancement. [START_REF] Lu | Conditional diffusion probabilistic model for speech enhancement[END_REF] used Gaussian Markov chains to model the forward and reverse processes, where the mean of the forward Gaussian Markov chain is a linear interpolation between the clean speech and the associated noisy one. The training objective is obtained by minimizing the Kullback-Leibler (KL) divergence between the forward and reverse Markov chains, which leads to an objective where the trained network learns to predict both Gaussian noise and non-Gaussian noise. [START_REF] Hu | Noiseaware speech enhancement using diffusion probabilistic model[END_REF] added an auxiliary classification loss to the loss function of [START_REF] Lu | Conditional diffusion probabilistic model for speech enhancement[END_REF] to perform noise classification and help the model better use the noise information, which resulted in improved performance. [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF] leveraged stochastic differential equation (SDE) [START_REF] Song | Score-based generative modeling through stochastic differential equations[END_REF] to model the forward and reverse processes, and used denoising score matching [START_REF] Vincent | A connection between score matching and denoising autoencoders[END_REF] as the training objective.

All these works condition the diffusion process on the noisy speech to take into account the non-Gaussian nature of environmental noise. More precisely, to estimate clean speech, the reverse process is performed starting from a Gaussian noise centred on the noisy speech, which allows for iteratively recovering an enhanced version. [START_REF] Yen | Cold diffusion for speech enhancement[END_REF] proposed to extend the forward process by incorporating a deterministic, progressive degradation of the clean speech through linear interpolation between the clean speech and the noisy speech. The training objective here consists in minimizing an ℓ 1 loss between the clean speech and the reconstruction of the clean speech at a given step of the forward degradation.

The recent work [START_REF] Tai | Revisiting denoising diffusion probabilistic models for speech enhancement: Condition collapse, efficiency and refinement[END_REF] identified the condition collapse problem, and proposed an auxiliary conditional generation network for generating reliable condition representations as well as a dual-path parallel network architecture to provide fine-grained condition guidance for the diffusion model. Additionally, a refinement network is trained in a supervised way that takes the enhanced speech returned by the reverse sampling and outputs a refined version. We adopt a strategy different from these approaches by including an ℓ 2 supervision loss in the training objective function.

Score-based generative model for SE (SGMSE)

In this section, we review the score-based diffusion model proposed in [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF], as a closely related approach to our work. Let us consider the flattened short-time Fourier transform (STFT) representations of clean speech, noisy speech, and noise: x 0 , y, n ∈ C d , where d denotes the total number of complexvalued time-frequency (TF) bins. We assume that the noisy speech STFT is formed by the following mixture model: y = x 0 + n. The objective of SE is then to recover x 0 given y.

As previously mentioned, the forward process of diffusion involves gradually introducing Gaussian noise to the clean speech. This process is modeled by an SDE, and its solution is represented as the stochastic process {x t } t [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF]:

dx t = γ (y -x t ) :=f (xt,y) dt+ σ min σ max σ min t 2 log σ max σ min :=g(t)
dw

(1) where x t denotes the process state at time t ∈ (0, T ], γ ∈ R controls the transition from x 0 to y, and g(t) ∈ R, with fixed parameters σ min and σ max , is the diffusion coefficient that controls the amount of noise induced by a standard Wiener process w. Moreover, f (x t , y) is the drift term, which makes the forward process conditioned on the noisy speech. For numerical stability, the forward process starts at t ε ̸ = 0.

The final SE objective is to reverse the above forward process in order to estimate the clean speech. To this end, one needs to find the solution to the following associated reverse process SDE [START_REF] Anderson | Reverse-time diffusion equation models[END_REF]:

dx t = -f (x t , y) + g(t) 2 ∇ xt log p t (x t |y) dt + g(t)dw
(2) where w denotes a standard Wiener process running backwards in time. In [START_REF] Kong | Diffwave: A versatile diffusion model for audio synthesis[END_REF], the term ∇ xt log p t (x t |y) refers to the conditional score function, which is approximated by a socalled (conditional) score model s θ (x t , y, t) with parameters denoted θ. The score model can be trained by minimizing a noise-prediction loss [START_REF] Kingma | Understanding diffusion objectives as the elbo with simple data augmentation[END_REF] as follows

min θ E t,(x0,y),z,xt|(x0,y) L θ (x t , y, t, z) , (3)
where 1L θ (x t , y, t, z) := ∥σ(t)s θ (x t , y, t) + z∥ 2

and z ∼ N C (z; 0, I), with N C denoting the circularlysymmetric complex normal distribution. As the drift term is linear w.r.t x t , the transition kernel p 0t (x t |x 0 , y) admits a closed-form expression [START_REF] Song | Score-based generative modeling through stochastic differential equations[END_REF]:

p 0t (x t |x 0 , y) = N C x t ; µ (x 0 , y, t) , σ(t) 2 I , (5)
where

µ(x 0 , y, t) = e -γt x 0 + 1 -e -γt y (6)
and

σ(t) 2 = σ 2 min (σ max /σ min ) 2t -e -2γt log(σ max /σ min ) γ + log(σ max /σ min ) . (7)
Once the score model is trained, it is plugged in (2), replacing the score function, and the resulting reverse SDE is solved by a Predictor-Corrector sampling procedure [START_REF] Song | Score-based generative modeling through stochastic differential equations[END_REF] to iteratively generate clean speech's estimates.

WEIGHTED GENERATIVE-SUPERVISED LEARNING LOSS

The training objective function described in (3) and ( 4) aims to train the score model by minimizing an ℓ 2 loss between the added Gaussian noise z and its estimation given by ẑ = -σ(t)s θ (x t , y, t). This suggests that, in its current form, the score model is not informed of the specific SE task it is meant to perform, as it does not explicitly take into account the environmental noise present in y. Instead, the training loss primarily resembles the generative (unsupervised) loss typically utilized in unconditional diffusion models.

To address this issue, we propose the inclusion of a supervised loss as a form of regularization or guidance. This additional loss explicitly reinforces the SE objective during the score model's training. To accomplish this objective, we need to have an estimate of the clean speech at each diffusion timestep t during training, denoted x0,t , to be compared against the ground-truth x 0 . Such an estimate could be provided using Tweedie's approach [START_REF] Efron | Tweedie's formula and selection bias[END_REF]. To this end, by combining ( 5) and ( 6), we can write

x t = e -γt x 0 + 1 -e -γt y + e t (8)
where e t ∼ N C (e t ; 0, σ(t) 2 I). If we apply Tweedie's formula independently to the real and imaginary parts of the variables, we obtain the following result: We then propose to add an ℓ 2 loss between the above expression and the associated ground-truth one, i.e., e -γt x 0 + (1 -e -γt )y, to the original score loss (3). This leads to the following weighted training objective for learning the conditional score model

e -γt
min θ E t,(x0,y),z,xt|(x0,y) [(1 -α t ) L θ (x t , y, t, z) + α t x t + σ(t) 2 2 s θ (x t , y, t) -(e -γt x 0 + (1 -e -γt )y) 2 ] (10)
which is a weighted loss between the original generativebased training loss in (4) and a supervised ℓ 2 loss. α t are time-dependent scalar weights taking values in [0, 1]. We propose to use the following expression for the weights

α t = σ(T ) -σ(t) σ(T ) -σ(t ε ) . (11)
In the proposed loss function [START_REF] Tai | Revisiting denoising diffusion probabilistic models for speech enhancement: Condition collapse, efficiency and refinement[END_REF], we aim to make a balance between two essential tasks: conditional score estimation and supervised estimation of clean speech at every time-step t.

The parameter α t is a decreasing function of time, contrary to σ(t). Consequently, at earlier stages of the forward diffusion process, i.e., when the noise variance is small, the network is expected to give higher weights to the supervised component of the loss, while in the later stages, the score network is tasked with assigning more importance to the loss associated with Gaussian noise estimation.

EXPERIMENTS

Baselines. We evaluate the performance of our proposed weighted generative-supervised learning loss for SE, comparing it to the reference approach referred to as SGMSE+ [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF].

Both methods utilize the same network architecture, based on the Noise Conditional Score Network (NCSN++), which consists of a multi-resolution U-Net design. Additionally, we compare these two methods with a purely supervised model that directly predicts the clean speech spectrogram from the noisy speech spectrogram input. For the supervised approach, we also use the NCSN++ network, which is trained by minimizing the MSE loss between the enhanced and clean spectrograms.

Evaluation metrics. To measure the quality of the enhanced speech signals, we use standard instrumental evaluation metrics, including the scale-invariant signal-to-distortion ratio (SI-SDR) in dB [START_REF] Le Roux | SDR-half-baked or well done?[END_REF], the extended short-time objective intelligibility (ESTOI) measure [START_REF] Jensen | An algorithm for predicting the intelligibility of speech masked by modulated noise maskers[END_REF] ([0, 1]), and the perceptual evaluation of speech quality (PESQ) score [START_REF] Rix | Perceptual evaluation of speech quality (PESQ)-a new method for speech quality assessment of telephone networks and codecs[END_REF] ([-0.5, 4.5]). In addition, we use the DNS-MOS, as a nonintrusive objective evaluation metric [START_REF] Reddy | DNSMOS P. 835: A non-intrusive perceptual objective speech quality metric to evaluate noise suppressors[END_REF], which provides three MOS scores: speech signal quality (SIG), background intrusiveness (BAK), and overall quality (OVRL). For all these metrics, the higher, the better.

Datasets.

For training and evaluation, we used the WSJ0-QUT [START_REF] Leglaive | A recurrent variational autoencoder for speech enhancement[END_REF] and NTCD-TIMIT datasets [START_REF] Abdelaziz | NTCD-TIMIT: A new database and baseline for noise-robust audio-visual speech recognition[END_REF], to allow for a cross-dataset evaluation. The WSJ0-QUT dataset combines the clean speech signals from the WSJ0 dataset [START_REF] Garofolo | CSR-I (WSJ0) Complete[END_REF] with noise signals from the QUT-NOISE dataset [START_REF] Dean | The QUT-NOISE-SRE protocol for the evaluation of noisy speaker recognition[END_REF]. The test subset of WSJ0-QUT comprise 651 synthetic mixtures (roughly 1.5 hours). It is created by taking clean speech signals from the 'si et 05' subset of WSJ0 (unseen speech samples) and adding noise signals sampled uniformly from the 'verification' set of the QUT-NOISE dataset with signal-to-noise ratio (SNR) values of -5, 0, and 5 dB. The NTCD-TIMIT dataset comprises 62 English speakers (with/without Irish accent), divided into train, test, and validation subsets, where each speaker utters 98 different sentences. Duration of each utterance is approximately 5 seconds. To create the training and validation datasets, we combined the clean speech signals from the NTCD-TIMIT dataset with various types of noise from the DEMAND dataset [START_REF] Thiemann | The diverse environments multi-channel acoustic noise database (DE-MAND): A database of multichannel environmental noise recordings[END_REF]. We applied different SNR values, including -10, -5, 0, 5, and 10 dB. Each utterance in the training and validation sets was mixed with three different combinations of DEMAND noises and SNRs, resulting in a total of 12,348 training and 2,352 Hyperparameters setting for SDE and STFT. Input data representations, SDE, and STFT representations follow the same settings as in [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF]. Specifically, the STFT of the speech data, with a sampling rate of 16 kHz, is computed with a window size of 512, a hop length of 128 (75% overlap) and a Hann window which gives F = 256 as the number of frequency bins. For the drift and diffusion coefficients of SDE in (1), the parameters are set as γ = 1.5, σ min = 0.05, σ max = 0.5. The minimum and maximum process times are set to t ε = 0.03 and T = 1, respectively.

Results.

In Table 1, we present the average speech enhancement metrics for all the cross-dataset configuration settings, along with the corresponding standard error of the mean. To clarify, we use the term "matched condition" when the model is trained and tested on the same dataset, while "mismatched condition" refers to cases where the model is evaluated on a test set from a different dataset than the one it was trained on.

From Table 1, we can draw several conclusions. First, in both matched and mismatched conditions, the supervised method consistently outperforms the two diffusion-based methods when considering the SI-SDR, ESTOI, and BAK-MOS metrics. Our proposed method ranks second in performance. Nevertheless, when trained and evaluated on the NTCD-TIMIT dataset, the ESTOI and BAK-MOS metrics show a different trend. In these cases, the supervised method underperforms our proposed method. For the PESQ, SIG-MOS, and OVR-MOS metrics, the proposed method consistently performs the best.

A noteworthy observation from these findings is that in the matched conditions, when the supervised method performs the best, the gap between the supervised and SGMSE+ tends to narrow when the supervision loss is added. This trend also holds for the mismatched conditions.

In summary, our proposed method appears to inherit some capabilities from the supervised method, striving to match its performance in terms of SI-SDR, ESTOI, and BAK-MOS. Simultaneously, it retains and even improves upon the strengths of the baseline SGMSE+ method, resulting in better performance in terms of ESTOI, PESQ, SIG-MOS, and OVR-MOS. This suggests that the supervision loss provides valuable feedback for score estimation. Code is available online. 2

CONCLUSION

In this paper, we addressed the problem of diffusion-based speech enhancement and introduced a supervised training loss component alongside the generative-based Gaussian noise prediction loss. This weighted loss balances the original noise prediction loss with an ℓ 2 -based supervision loss, enhancing the mapping between clean and noisy speech by incorporating clean speech estimates at every diffusion time-step. This additional loss aids in training a score model better optimized for speech enhancement. Our experiments, conducted in both matched and mismatched conditions, demonstrate that our approach combines the strengths of supervised methods and diffusion-based approaches, resulting in improved performance. Future research directions involve exploring alternative supervised loss functions to ℓ 2 loss and developing more efficient adaptive weighting mechanisms.

Table 1 :

 1 Speech enhancement results (mean ± standard error) for WSJ0-QUT and NTCD-TIMIT under both matched and mismatched conditions. The best average metric value is highlighted in bold, while the second best is italicized. ± 0.27 2.05 ± 0.02 0.45 ± 0.01 4.04 ± 0.01 4.05 ± 0.01 3.57 ± 0.02 Proposed 4.42 ± 0.23 2.08 ± 0.02 0.48 ± 0.01 4.16 ± 0.01 4.20 ± 0.01 3.76 ± 0.01 validation mixtures. For the test subset, we retained the same noisy speech signals as provided in the NTCD-TIMIT dataset. These noisy samples were generated by adding six different noise types, including Living Room, White, Cafe, Car, Babble, and Street, with SNRs of -5 dB, 0 dB, and 5 dB. The test set comprises 810 mixtures.

	Training set	Method	SI-SDR (dB)	PESQ	ESTOI	SIG-MOS	BAK-MOS OVR-MOS
		Input (WSJ0-QUT)	-2.60 ± 0.16 1.83 ± 0.02 0.50 ± 0.01 4.04 ± 0.01 2.93 ± 0.02 3.13 ± 0.01
	WSJ0-QUT	Supervised SGMSE+ [4]	12.91 ± 0.14 2.67 ± 0.02 0.84 ± 0.00 4.38 ± 0.01 4.81 ± 0.01 4.30 ± 0.01 10.21 ± 0.16 2.83 ± 0.02 0.81 ± 0.00 4.52 ± 0.01 4.70 ± 0.01 4.31 ± 0.01
		Proposed	10.40 ± 0.15 2.88 ± 0.02 0.83 ± 0.00 4.56 ± 0.01 4.73 ± 0.00 4.37 ± 0.01
		Supervised	9.27 ± 0.15 2.36 ± 0.02 0.75 ± 0.00 4.22 ± 0.01 4.68 ± 0.01 4.15 ± 0.02
	NTCD-TIMIT	SGMSE+ [4]	7.32 ± 0.15 2.51 ± 0.02 0.72 ± 0.01 4.47 ± 0.01 4.60 ± 0.01 4.24 ± 0.01
		Proposed	7.55 ± 0.14 2.61 ± 0.02 0.75 ± 0.00 4.55 ± 0.01 4.66 ± 0.00 4.34 ± 0.01
		Input (NTCD-TIMIT) -7.81 ± 0.22 1.77 ± 0.02 0.31 ± 0.00 3.51 ± 0.01 2.28 ± 0.02 2.69 ± 0.01
	NTCD-TIMIT	Supervised SGMSE+ [4]	8.57 ± 0.19 2.18 ± 0.02 0.54 ± 0.01 3.69 ± 0.01 4.26 ± 0.01 3.42 ± 0.02 6.21 ± 0.23 2.35 ± 0.02 0.53 ± 0.01 4.02 ± 0.01 4.30 ± 0.01 3.68 ± 0.01
		Proposed	7.97 ± 0.18 2.46 ± 0.02 0.57 ± 0.01 4.14 ± 0.01 4.37 ± 0.01 3.83 ± 0.01
		Supervised	5.98 ± 0.22 2.02 ± 0.02 0.50 ± 0.01 3.76 ± 0.01 4.19 ± 0.01 3.34 ± 0.02
	WSJ0-QUT	SGMSE+ [4]	1.28

This is the actual loss used in the implementation of[START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF]: https:// github.com/sp-uhh/sgmse.

https://github.com/jeaneudesAyilo/weighted_ generative_supervised_DiffSE

This research was supported by the French National Research Agency (ANR) under the project REAVISE (ANR-22-CE23-0026-01). Experiments presented in this paper were carried out using the Grid'5000 testbed, supported by a scientific interest group hosted by Inria, and including CNRS, RENATER, and several universities as well as other organizations (see https://www.grid5000.fr).


Berné Nortier

Mostafa Sadeghi

Romain Serizel

Unsupervised speech enhancement with diffusion-based generative models

Keywords: Unsupervised speech enhancement, diffusionbased models, expectation-maximisation, posterior sampling

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Over the past decade, the speech enhancement (SE) task has been extensively investigated, and numerous novel approaches have been proposed that greatly leverage the advancements and efficacy of deep neural network (DNN) architectures [START_REF] Wang | Supervised speech separation based on deep learning: An overview[END_REF]. The majority of these approaches are based on supervised (discriminative) learning of a DNN over training pairs of clean and noisy speech signals, covering different speakers, noise types, and signal-to-noise ratio (SNR) values. Such an approach depends heavily on the number and diversity of training samples and noise conditions, and thus generalisation to unseen (out-of-domain) environments cannot be guaranteed.

Unsupervised SE based on deep generative models presents an alternative approach with improved generalisation performance [START_REF] Bando | Statistical speech enhancement based on probabilistic integration of variational autoencoder and non-negative matrix factorization[END_REF][START_REF] Bie | Unsupervised speech enhancement using dynamical variational autoencoders[END_REF][START_REF] Bando | Adaptive neural speech enhancement with a denoising variational autoencoder[END_REF]. In contrast to purely supervised methods, the generative-based (unsupervised) framework learns the statistical distribution of clean speech signals and uses it as a prior distribution for inferring the target signal from its noisy observation. In these methods, VAE [START_REF] Kingma | Auto-encoding variational bayes[END_REF] has been commonly used as a generative clean speech prior, which This work was supported by the French National Research Agency (ANR) under the project REAVISE (ANR-22-CE23-0026-01). Experiments presented in this paper were carried out using the Grid'5000 testbed, supported by a scientific interest group hosted by Inria, and including CNRS, RENATER, and several universities as well as other organizations (see https://www.grid5000.fr).

is combined with a non-negative matrix factorization (NMF)-based observation model to estimate clean speech following a statistical EM framework.

Recently, diffusion-based generative models have emerged as a powerful and state-of-the-art framework to model complex data distributions [START_REF] Sohl-Dickstein | Deep unsupervised learning using nonequilibrium thermodynamics[END_REF][START_REF] Song | Score-based generative modeling through stochastic differential equations[END_REF]. These models learn an implicit distribution by estimating the score, i.e., the gradient of the log probability density (with respect to data). This is done by gradually diffusing data samples into noise and then learning a score approximating model that can reverse the noising process for different noise scales. The forward process of corrupting data is modelled as a stochastic differential equation (SDE), which can be reversed and yields a corresponding reverse SDE that depends only on the score of the perturbed data and may easily be solved numerically. Diffusion-based models have been widely applied to the SE task in a supervised way [START_REF] Lu | Conditional diffusion probabilistic model for speech enhancement[END_REF][START_REF] Serrà | Universal speech enhancement with score-based diffusion[END_REF][START_REF] Welker | Speech enhancement with score-based generative models in the complex STFT domain[END_REF][START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF][START_REF] Yen | Cold diffusion for speech enhancement[END_REF] by incorporating noisy speech signals in the diffusion process as conditioning information.

In this paper, we develop an unsupervised speech enhancement framework leveraging diffusion-based generative models as data-driven priors. Specifically, in a training step, the statistical characteristics of clean speech signals are learnt in the complex STFT domain through the use of a score-based diffusion model. At test time, we perform posterior sampling by combining the learnt implicit clean speech prior with a parametric statistical model for noise to infer the clean speech signal. The noise parameters are estimated alongside the clean speech signal by following an iterative EM-based approach. To our knowledge, this is the first work that proposes using diffusion-based generative models for unsupervised SE, and explores their potential. We conduct experiments comparing the proposed framework with a VAE-based unsupervised approach [START_REF] Bie | Unsupervised speech enhancement using dynamical variational autoencoders[END_REF] as well as a state-of-the-art diffusion-based supervised method [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF]. The results demonstrate the effectiveness and promising performance of the proposed diffusion-based unsupervised approach, paving the path for future research in this direction.

The rest of the paper is organised as follows: Section 2 reviews score-based diffusion modelling and VAE-based SE as two closely related problems to our work. The proposed speech generative modelling and enhancement frameworks are detailed in Section 3. Experimental results are then presented in Section 4, followed by a conclusion and suggestions for future lines of work in Section 5.

BACKGROUND

Score-based diffusion models

Diffusion models are a state-of-the-art class of probabilistic generative models that have recently achieved remarkable performance in generating high-quality samples in different applications [START_REF] Song | Score-based generative modeling through stochastic differential equations[END_REF]. These models transform an unknown data distribution p0 to a tractable prior distribution, usually N (0, I), by gradually adding noise to training data in a forward process. Then, in a reverse process, a parameterised model is learnt to iteratively generate samples starting from noise and transform these into samples from the unknown data distribution. This action of smoothly injecting noise into training samples may be described by a SDE. Specifically, consider a diffusion process {st} t∈[0,1] , indexed by a continuous time-step variable t, which solves the following general linear SDE

dst = f (st)dt + g(t)dw, (1)
where w denotes a standard Wiener process, the vector-valued f is the drift coefficient term, and the scalar function g is the diffusion coefficient. Here, the forward process transforms a clean training sample s0 = s to a noise sample s1, whose distribution converges to p1 ∼ N (0, I). Under some light regularity conditions [START_REF] Anderson | Reverse-time diffusion equation models[END_REF], the SDE in (1) also has an associated reverse-time SDE:

dst = [f (st)dt -g(t) 2 ∇s t log pt(st)]dt + g(t)d w, (2)
where w is a standard Wiener process running backwards in time, dt is an infinitesimal negative time-step, and ∇s t log pt(st) is called the score function. In practice, the score is approximated by a timedependent neural network (NN) S θ * (st, t) ≈ ∇s t log pt(st), called the score model, where θ * denotes the learnt weights of the NN.

By plugging the score model in (2), we can solve the resulting SDE using a variety of solvers to sample from the unknown data distribution [START_REF] Song | Score-based generative modeling through stochastic differential equations[END_REF]. In this paper, we make use of the Predictor-Corrector (PC) sampler [START_REF] Song | Score-based generative modeling through stochastic differential equations[END_REF].

VAE-based unsupervised speech enhancement

Previous works on unsupervised SE use VAE to learn the prior distribution of clean speech signals, which is then combined with an observation model to estimate clean speech within a statistical framework. Specifically, in the STFT domain, a latent variable-based generative model is assumed as p θ (s, z) = p θ (s|z)p θ (z), where s denotes the STFT representation of clean speech and z represents the associated (latent) low-dimensional embedding. Some parameterised Gaussian forms for the generative distributions are usually assumed, whose parameters are learnt over clean speech data, following the evidence lower-bound optimisation principle [START_REF] Kingma | Auto-encoding variational bayes[END_REF].

For SE, it is assumed that x = s + n, where x, s, and n denote STFT representations of noisy (mixture) speech, clean speech, and background noise, respectively. The likelihood p ϕ (x|s) is usually a proper complex Gaussian distribution N C with mean s, whose variance is parameterised with a low-rank NMF factorisation. SE then amounts to inferring the latent variable z associated with s from x. This necessitates learning the NMF parameters, denoted ϕ, via an EM process formulated below

max ϕ E p ϕ (z|x) {log p ϕ (x|z)} . (3)
This could be solved using, e.g., the variational EM procedure developed in [START_REF] Bie | Unsupervised speech enhancement using dynamical variational autoencoders[END_REF][START_REF] Leglaive | A recurrent variational autoencoder for speech enhancement[END_REF], which approximates p ϕ (z|x).

PROPOSED FRAMEWORK

Diffusion-based speech generative modeling

Following [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF], we work with the complex-valued STFT representations of speech signals and apply an exponential amplitude transformation to balance the heavy-tailed distribution of STFT amplitudes.

Like VAE, the diffusion-based generative model is independently defined for each time-frequency (TF) bin. Therefore, as done in [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF], all the vector-valued variables st in boldface contain flattened TF representations of speech signals. For concrete instantiations of the forward and reverse SDE ((1) and ( 2) respectively), we use an alternative form of the well-known variance-exploding stochastic differential equation (VESDE) [START_REF] Song | Generative modeling by estimating gradients of the data distribution[END_REF] inspired by [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF], and adapt it to obtain the drift and diffusion coefficients as follows

f (st) = -γst, g(t) = σmin σmax σmin t 2 log σmax σmin , (4)
where γ is a constant parameter, and σmin and σmax are parameters defining the noise schedule of the Wiener process. The SDE in (1) then has the perturbation kernel defined below, which allows one to sample st directly given s0

p0t(st|s0) = N C (δts0, σ(t) 2 I), (5)
where δt = e -γt and the variance term is given by

σ(t) 2 = σ 2 min (σmax/σmin) 2t -δt 2 log(σmax/σmin) γ + log(σmax/σmin) . (6)
To learn the NN parameters θ, a weighted Fisher divergence [START_REF] Song | Generative modeling by estimating gradients of the data distribution[END_REF] between the true and approximated score is solved, which, after some mathematical manipulation, leads to the following training objective [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF] θ

* = argmin θ E t,s,ζ,s t |s ∥S θ (st, t) + ζ σ(t) ∥ 2 2 , (7)
where ζ ∼ N C (0, I), i.e., complex-valued Gaussian noise.

Diffusion-based unsupervised speech enhancement

We now describe the unsupervised SE framework based on diffusionbased generative models. The prior clean speech distribution p = p(s) is unknown, but can be obtained by training a diffusion model as described in Section 3.1, yielding an implicit prior, as opposed to the explicit VAE-based speech prior modelling framework. This implicit diffusion-based speech prior only allows for iterative sampling, without an explicit density form. As such, the SE procedure adopted in VAE-based modelling cannot directly be used for diffusion-based learnt speech priors. Assuming the same observation model as before, i.e., x = s + n, and NMF-based likelihood parameterisation, we here propose to sample from the following intractable posterior distribution to estimate the clean speech s directly

p ϕ (s|x) ∝ p ϕ (x|s)p θ * (s), (8)
An overview of the proposed Unsupervised Diffusion-Based Speech Enhancement (UDiffSE) approach is provided in Algorithm 1. The following sections detail the E-step and M-step.

where again the score function can be approximated by S θ * (st, t).

However, the conditional score function ∇s t log p ϕ (x|st) is, in fact, intractable to compute in closed form, due to its dependence on time. That is,

p ϕ (x|st) = p ϕ (x|s0)pt0(s0|st)ds0, (11)
where pt0(s0|st) ∝ p0t(st|s0)p(s0) is intractable. As an approximation, we follow [START_REF] Meng | Diffusion model based posterior sampling for noisy linear inverse problems[END_REF] and assume an uninformative prior p(s0), which, along with (5), results in

pt0(s0|st) = N C st δt , σ(t) 2 δ 2 t I . (12)
Plugging this approximation in [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF] gives us the following noiseperturbed pseudo-likelihood

pϕ (x|st) ∼ N C st δt , σ(t) 2 δ 2 t I + diag(v ϕ ) . (13)
The conditional reverse process is then approximated as

dst = f (st)dt -g(t) 2 S θ * (st, t) dt + g(t)d w -g(t) 2 ∇s t log pϕ (x|st)dt. (14)
This is exactly the unconditional reverse process (2) for sampling clean speech, plus an additional term which imposes data consistency. We use the change of variables formula and take the gradient to compute ∇s t log pϕ (x|st), the noise-perturbed pseudo-likelihood score, as

∇s t log pϕ (x|st) = 1 δ t σ(t) 2 δ 2 t I + diag(v ϕ ) -1 ( st δt -x). (15)
Lastly, we introduce an additional weighting parameter λ to the pseudo-likelihood as in [START_REF] Meng | Diffusion model based posterior sampling for noisy linear inverse problems[END_REF] to balance the effect of the mixture signal on the estimated sample. We experimentally observed that performing the full posterior reverse step at each iteration enforces if i ≡ 0 (mod ℓ) then ▷ (Posterior)

10: end if 13: end for 14: return ŝ = s0 strongly the data consistency condition, causing the sample to converge to the mixture signal. To prevent this, we only perform the posterior step every ℓ iterations. We solve the reverse SDE using a PC sampler [START_REF] Song | Score-based generative modeling through stochastic differential equations[END_REF] -a numeric sampler consisting of a discretisation of ( 2) -the predictor -followed by a Langevin sampling step -the corrector -to 'correct' the marginal at time t. The overall E-step is summarised in Algorithm 2. The variable τ denotes discrete timestep in [0, 1]. For simplicity, we employ the shorthand στ , fτ , gτ for σ(τ ), f (τ ), g(τ ), respectively.

∇s τ log pϕ (x|sτ ) ← 1 δτ σ 2 τ δ 2 τ I + diag(v ϕ ) -1 ( sτ δτ -x)

M-Step

Having obtained a clean speech estimate ŝ in the E-step, we now consider updating the noise parameters ϕ = {W, H} via (9), replacing the expectation with a Monte-Carlo estimate as follows:

ϕ ← argmax {W,H}≥0 log p ϕ (x|ŝ) = argmin {W,H}≥0 i (x -ŝ) * i (x -ŝ)i v ϕ (i) + log(v ϕ (i)), (16)
where (•) * denotes the conjugate operation. The above problem can be solved using a variety of algorithms, e.g., the multiplicative update rules [START_REF] Févotte | Nonnegative matrix factorization with the itakura-saito divergence: With application to music analysis[END_REF][START_REF] Sadeghi | Robust unsupervised audio-visual speech enhancement using a mixture of variational autoencoders[END_REF].

EXPERIMENTS

In this section, we provide a performance evaluation of our proposed UDiffSE framework as compared against an unsupervised speech enhancement approach based on recurrent VAE (RVAE)1 [START_REF] Bie | Unsupervised speech enhancement using dynamical variational autoencoders[END_REF][START_REF] Leglaive | A recurrent variational autoencoder for speech enhancement[END_REF], as well as a state-of-the-art diffusion-based supervised SE method, called score-based generative model for speech enhancement (SGMSE+)2 [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF]. Evaluation Metrics. To measure the quality of the enhanced speech signals, we use standard instrumental evaluation metrics, including the scale-invariant signal-to-distortion ratio (SI-SDR) in dB [START_REF] Le Roux | SDRhalf-baked or well done?[END_REF], the extended short-time objective intelligibility (ESTOI) measure [START_REF] Jensen | An algorithm for predicting the intelligibility of speech masked by modulated noise maskers[END_REF] ranging in [0, 1], and the perceptual evaluation of speech quality (PESQ) score [START_REF] Rix | Perceptual evaluation of speech quality (PESQ)-a new method for speech quality assessment of telephone networks and codecs[END_REF] ranging in [-0.5, 4.5]. We also use the For performance evaluation, we use the WSJ0-QUT dataset created by [START_REF] Leglaive | A recurrent variational autoencoder for speech enhancement[END_REF], comprising 651 synthetic mixtures (about 1.5 hours of noisy speech data) which uses clean speech signals from the 'si et 05' subset of WSJ dataset and noise signals from the QUT-NOISE corpus [START_REF] Dean | The QUT-NOISE-SRE protocol for the evaluation of noisy speaker recognition[END_REF]. These include Café, Home, Street, and Car and have SNR values of -5 dB, 0 dB, and 5 dB. We also evaluate generalisation capability of different methods in mismatched conditions by using pre-computed noisy versions of the TCD-TIMIT data presented in [START_REF] Abdelaziz | NTCD-TIMIT: A new database and baseline for noise-robust audio-visual speech recognition[END_REF]. This set contains noise types Living Room (from the second CHiME challenge [START_REF] Vincent | The second 'CHiME' speech separation and recognition challenge: Datasets, tasks and baselines[END_REF]), White, Car, and Babble (from the RSG-10 corpus [START_REF] Steeneken | Description of the RSG-10 noise database[END_REF]) with SNR values of -5 dB, 0 dB, and 5 dB and. This yields 540 test speech signals (or approximately 45 minutes). Stochastic Differential Equation. The SDE in (4) has parameter values γ = 1.5, σmin = 0.05, σmax = 0.5. To avoid instabilities around 0, we adopt standard practice and set a minimum process time with tmin = 0.03. Models architecture. We adapt the SGMSE+ architecture developed in [START_REF] Welker | Speech enhancement with score-based generative models in the complex STFT domain[END_REF], which is based on a multi-resolution U-Net structure, by removing their x term and adapting the number of channels. RVAE consists of an encoder-decoder architecture composing bidirectional long short-term memory (BLSTM) networks. For both RVAE and SGMSE+, we use the pretrained models that are available in their associated public code repositories. Training setup. We train the score model S θ * for 220 epochs using an Adam optimiser with a learning rate of 0.0001 and a batch size of 16. Our loss is an exponential moving average of the network's weights, initialised with a decay of 0.999. EM settings. The reverse process in ( 14) is solved using a PC sampler with step size ϵτ := (στ /2) 2 , i.e., r = 0.5. The number of reverse sampling steps is set to N = 30. The posterior update step is performed every ℓ = 2 steps, and the NMF matrices possess a rank of 4. For each sample, we perform 5 EM iterations. We observe that performing the same denoising procedure over b samples in parallel and then averaging the result yields much better performance; we thus set the batch size to b = 4. The weighting parameter λ is set to 1.5. These parameter choices are motivated by an extensive set of experimental studies provided in the supplementary material. Results. We report our SE results in Table 1. Competing methods are evaluated in the matched and mismatched cases. Inspecting the results, we can make a number of conclusions: As may be expected, the supervised framework outperforms its unsupervised counterpart in the matched case, but at the cost of utilising labelled data. Our UDiffSE framework outperforms the alternative unsupervised RVAE on almost all metrics under both matched and mismatched conditions. In particular, it achieves much higher ESTOI, SIG-MOS, and OVR-MOS scores than RVAE, which is more noticeable in the mismatched condition.

Furthermore, the proposed UDiffSE method outperforms the supervised SGMSE+ framework for both the ESTOI and SIG-MOS metrics in the mismatched condition, with a comparable OVR-MOS score. While all three frameworks have very similar PESQ results in the mismatched case, the unsupervised methods significantly outperform SGMSE+ in terms of SI-SDR (by more than 4 dB). The performance of UDiffSE on the TCD-TIMIT dataset showcases its capacity to generalise to unseen data, which could possibly imply that it has learnt a good representation of general clean speech as the underlying prior. Supplementary material, including audio samples, is available online. 3

CONCLUSION

In this paper, we introduce UDiffSE, an unsupervised generativebased framework to solve the SE task by learning an implicit prior distribution over clean speech data. We do this by defining a continuous diffusion process in the STFT domain in the form of a conditional SDE, and imposing an NMF-based parameterised additive noise model. An EM approach is developed to simultaneously generate clean speech and learn the noise parameters. An approximation of the likelihood term in the E-step then yields a tractable posterior sampling procedure. This method outperforms an unsupervised VAE-based approach to SE for almost all metrics in matched and mismatched test conditions, while showcasing better generalisation performance than a state-of-the-art diffusion-based supervised method. UDiffSE does, however, have the disadvantage of being time-consuming, which originates from the complexity of the reverse diffusion process. Future works include speeding up the reverse process, utilising the recent advancements in diffusion-based image generation, and developing more efficient noise models.

  where θ * denotes the pretrained, and thus fixed, diffusion model's parameters. We model the noise by n ∼ N C (0, diag(vec(WH))) where W, H are low-rank matrices with non-negative entries and rank r and vec(WH) denotes the vectorised form of WH. The likelihood p ϕ (x|s) then writes as p ϕ (x|s) = N C (s, diag(v ϕ )), where v ϕ = vec(WH). Learning the NMF parameters, i.e., ϕ = {W, H}, is done by solving max ϕ E p ϕ (s|x) {log p ϕ (x|s)} .

Algorithm 2 1 N 2 : 6 :√ 2ϵτ ζc 7 : 8 :

 212678 Posterior sampling (E-step) of UDiffSERequire: x, N, ℓ, λ, r(signal-to-noise ratio) 1: s1 ∼ N C (x, I), ∆τ ← for i = N, . . . , sτ ← sτ + ϵτ S θ * (sτ , τ ) + ζp ∼ N C (0, I) ▷ (Predictor) sτ ← sτfτ ∆τ + g 2 τ S θ * (sτ , τ )∆τ + gτ √ ∆τ ζp 9:

  Given a current estimate of ϕ, the E-step (posterior sampling) entails the generation of speech samples from the posterior distribution p ϕ (s|x) to approximate the expectation in[START_REF] Serrà | Universal speech enhancement with score-based diffusion[END_REF]. This is done via the construction of a stochastic process {st|x} t∈[0,1] by conditioning the original process {st} t∈[0,1] on the observation x to obtain an estimate ŝ ∼ p ϕ (s|x). To this end, we modify the reverse SDE (2) as follows

	Algorithm 1 UDiffSE
	1: ϕ0 = {W0, H0}
	2: for k = 1, . . . , K do
	3:	ŝ ∼ p ϕ k-1 (s|x)	▷ (E-Step)
	4:	ϕ k ← argmax ϕ log p ϕ (x|ŝ)	▷ (M-Step)
	5: end for
	6: return	ŝ
	3.2.1. E-Step

dst = f (st)dt -g(t) 2 ∇s t log pt(st|x) dt + g(t)d w = f (st)dt -g(t)

2 (∇s t log p ϕ (x|st)+∇s t log pt(st)) dt + g(t)d w

Table 1 :

 1 Speech enhancement results under both matched and mismatched conditions. 'S': supervised, 'U': unsupervised. Bold and italicised indicate the best and second best performances, respectively. To learn the clean speech prior model, we train on the 'si tr s' subset of the Wall Street Journal (WSJ) corpus[START_REF] Garofolo | CSR-I (WSJ0) complete LDC93S6B[END_REF], which amounts to roughly 25 hours of data. The STFT is computed using a window size of 510, a hop-length of 128 (≈ 75% overlap), and a Hann window, which gives F = 256 frequency bins. All signals have a sampling rate of 16 kHz. To ensure similarity across samples of different lengths during training, subsamples are randomly selected from STFT data so that we get T = 256 time frames with start and end positions randomly generated.

	Method	Type SI-SDR (dB)	PESQ	ESTOI	SIG-MOS	BAK-MOS	OVR-MOS
	Input (WSJ0-QUT)	-	-2.60 ± 0.17 1.83 ± 0.02 0.50 ± 0.01 4.04 ± 0.01 2.93 ± 0.02 3.13 ± 0.01
	RVAE [3, 14]	U	4.39 ± 0.21	2.20 ± 0.02 0.59 ± 0.01 3.88 ± 0.02 3.32 ± 0.02 3.13 ± 0.02
	UDiffSE (Ours)	U	4.80 ± 0.23	2.21 ± 0.02 0.63 ± 0.01 4.33 ± 0.01 3.74 ± 0.02 3.74 ± 0.02
	SGMSE+ [11]	S	9.41 ± 0.18	2.66 ± 0.02 0.77 ± 0.01 4.48 ± 0.01 4.51 ± 0.01 4.19 ± 0.01
	Input (TCD-TIMIT)	-	-8.74 ± 0.29 1.84 ± 0.02 0.35 ± 0.01 3.52 ± 0.02 2.22 ± 0.03 2.68 ± 0.01
	RVAE [3, 14]	U	1.44 ± 0.31	2.02 ± 0.02 0.35 ± 0.01 3.08 ± 0.03 3.18 ± 0.02 2.61 ± 0.02
	UDiffSE (Ours)	U	0.37 ± 0.25	2.01 ± 0.02 0.41 ± 0.01 3.91 ± 0.01 2.88 ± 0.03 3.08 ± 0.02
	SGMSE+ [11]	S	-3.97 ± 0.41 2.04 ± 0.03 0.38 ± 0.01 3.79 ± 0.02 3.43 ± 0.02 3.13 ± 0.02
	DNS-MOS [22], a non-intrusive speech quality metric, which pro-
	vides scores for the speech quality (SIG), background noise quality
	(BAK), and overall quality (OVRL) of speech. For all the metrics,
	higher values indicate improved performance.
	Datasets.

https://github.com/XiaoyuBIE1994/DVAE_SE/

https://github.com/sp-uhh/sgmse

https://team.inria.fr/multispeech/demos/udiffse/


Mostafa Sadeghi

Romain Serizel

POSTERIOR SAMPLING ALGORITHMS FOR UNSUPERVISED SPEECH ENHANCEMENT WITH RECURRENT VARIATIONAL AUTOENCODER

Keywords: Unsupervised speech enhancement, deep generative models, recurrent variational autoencoder, posterior sampling

In this paper, we address the unsupervised speech enhancement problem based on recurrent variational autoencoder (RVAE). This approach offers promising generalization performance over the supervised counterpart. Nevertheless, the involved iterative variational expectation-maximization (VEM) process at test time, which relies on a variational inference method, results in high computational complexity. To tackle this issue, we present efficient sampling techniques based on Langevin dynamics and Metropolis-Hasting algorithms, adapted to the EM-based speech enhancement with RVAE. By directly sampling from the intractable posterior distribution within the EM process, we circumvent the intricacies of variational inference. We conduct a series of experiments, comparing the proposed methods with VEM and a state-of-the-art supervised speech enhancement approach based on diffusion models. The results reveal that our sampling-based algorithms significantly outperform VEM, not only in terms of computational efficiency but also in overall performance. Furthermore, when compared to the supervised baseline, our methods showcase robust generalization performance in mismatched test conditions.

INTRODUCTION

Speech enhancement is a fundamental signal processing technique, aiming to improve the quality and intelligibility of a noisy speech signal corrupted by acoustic noise [START_REF] Vincent | Audio source separation and speech enhancement[END_REF]. Over the past few years, and with the unprecedented success of deep learning, speech enhancement approaches have shifted from traditional statistical methods to data-driven approaches based on deep neural networks (DNNs) [START_REF] Wang | Supervised speech separation based on deep learning: An overview[END_REF][START_REF] Xu | A regression approach to speech enhancement based on deep neural networks[END_REF][START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF][START_REF] Yen | Cold diffusion for speech enhancement[END_REF][START_REF] Lu | Conditional diffusion probabilistic model for speech enhancement[END_REF]. Predominantly, current DNN-based speech enhancement techniques adopt a supervised (discriminative) paradigm, wherein a DNN is trained to map noisy speech inputs to their corresponding clean counterparts, leading to state-of-theart performance. However, a notable challenge pervasive in these methods concerns generalization to test conditions not encountered during training, such as distinct noise types and noise levels that deviate from training conditions.

In contrast, unsupervised speech enhancement methods based on deep generative models do not learn noise characteristics during the training process [START_REF] Bando | Statistical speech enhancement based on probabilistic integration of variational autoencoder and non-negative matrix factorization[END_REF][START_REF] Bie | Unsupervised speech enhancement using dynamical variational autoencoders[END_REF][START_REF] Carbajal | Guided variational autoencoder for speech enhancement with a supervised classifier[END_REF][START_REF] Bando | Adaptive neural speech enhancement with a denoising variational autoencoder[END_REF]. Specifically, a deep generative model, most commonly based on the variational auto-encoder (VAE) [START_REF] Kingma | Auto-encoding variational Bayes[END_REF], is trained solely on clean speech signals. This trained model then serves as a prior distribution for estimating clean speech from noisy input using an expectation-maximization (EM) approach. This gives them a generalization advantage over discriminative approaches. However, unsupervised methods remain significantly less explored than their supervised counterparts and suffer from some challenges, including their notably high computational complexity. This complexity originates from the iterative EM process during inference, which requires sampling from an intractable posterior distribution. In particular, the current state-of-the-art method for unsupervised speech enhancement relies on recurrent VAE (RVAE) [START_REF] Bie | Unsupervised speech enhancement using dynamical variational autoencoders[END_REF][START_REF] Leglaive | A recurrent variational autoencoder for speech enhancement[END_REF], as a dynamical and more efficient version of the standard VAE. This approach adopts a variational EM (VEM) strategy, involving the fine-tuning of the trained encoder at each EM iteration on the input noisy speech. Its computational complexity thus grows with the complexity (number of parameters) of the encoder.

To address this issue, we propose alternative, more efficient posterior sampling-based methods for speech enhancement with RVAE. The first approach extends the Langevin dynamics EM (LDEM) method for standard, non-dynamical VAE presented in [START_REF] Sadeghi | Fast and efficient speech enhancement with variational autoencoders[END_REF] to RVAE. This technique involves sampling from the intractable posterior using gradient descent steps combined with Gaussian noise injection. Additionally, we develop a Metropolis-Hastings (MH) sampling technique [START_REF] Bishop | Pattern recognition and machine learning[END_REF], relying on a proposal and acceptance/rejection mechanism, to generate a sequence of samples. Lastly, a Metropolisadjusted Langevin algorithm (MALA) [START_REF] Roberts | Langevin diffusions and metropolis-hastings algorithms[END_REF] is proposed, combining the strengths of both LDEM and MH methods. We assess the effectiveness of these algorithms for RVAE-based speech enhancement by comparing them to the VEM method and a state-of-the-art supervised speech enhancement approach based on diffusion models [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF], in both matched and mismatched test conditions. The results demonstrate that our proposed speech enhancement algorithms outperform VEM significantly in terms of performance and computational efficiency. Furthermore, they exhibit more robust generalization performance when compared to the supervised baseline method.

The paper is organized as follows: In Section 2, we present an overview of unsupervised speech enhancement based on RVAE. Section 3 introduces the proposed posterior sampling methods. Our experimental results are detailed in Section 4. Lastly, Section 5 provides some conclusions.

BACKGROUND

RVAE as a deep speech prior

We denote by s ≜ s1:T = {s1, . . . , sT } a sequence of clean speech time-frequency representations computed using short-time Fourier transform (STFT), where st = [s f t ] F f =1 ∈ C F . RVAE [START_REF] Leglaive | A recurrent variational autoencoder for speech enhancement[END_REF], as a latent variable-based model, considers the following generative model for the speech time frames s:

p θ (s, z) = T t=1 p θ (st|z)p(zt), (1)
where z = {z1, . . . , zT }, zt ∈ R L (L ≪ F ), are low-dimensional latent variables associated with s. Moreover,

p θ (st|z) = Nc(0, diag(v θ,t (z))) (2)
is a circularly-symmetric complex Gaussian distribution, with a diagonal covariance matrix whose entries, given by v θ,t (z), are modeled by a recurrent neural network (RNN), called decoder. Here, v θ,t (z) refers to the output at time frame t of the RNN with z as the input. This dynamical modeling makes RVAE more efficient than the standard VAE. Similar to VAE, the prior p(zt) is set to a standard Gaussian distribution.

Training the generative model (1) involves learning the RNN parameters θ following an EM procedure. The intractable posterior p θ (z|s) is approximated with a parametric Gaussian distribution as follows

q ϕ (z|s) = T t=1 q ϕ (zt|z1:t-1, st:T ), (3)
where similarly as in (2), the mean and variance are modeled via an RNN, called encoder, with parameters denoted ϕ. The encoder and decoder parameters, i.e., {θ, ϕ}, are then jointly learned by optimizing an evidence lower-bound [START_REF] Kingma | Auto-encoding variational Bayes[END_REF].

Variational EM for speech enhancement

The observation model for speech enhancement is assumed to be xt = st + bt, with bt corresponding to noise. As a statistical model for clean speech st, the pretrained RVAE model, i.e., p θ (s, z) is used. Moreover, noise is modeled based on a non-negative matrix factorization (NMF) framework [START_REF] Bando | Statistical speech enhancement based on probabilistic integration of variational autoencoder and non-negative matrix factorization[END_REF], where a circularly symmetric Gaussian form p ψ (bt) ∼ Nc(0, diag([WH]t)) is considered. The non-negative matrices W, H form the noise parameters ψ to be learned from x. This is done following an EM approach, that is

ψ * = argmax ψ E p ψ (z|x) {log p ψ (x, z)} , (4)
where [START_REF] Leglaive | A recurrent variational autoencoder for speech enhancement[END_REF]. Here, the posterior p ψ (z|x) in ( 4), needed for the E-step, is intractable to compute. The variational EM (VEM) approach proposed in [START_REF] Leglaive | A recurrent variational autoencoder for speech enhancement[END_REF] fine-tunes the pretrained encoder q ϕ (z|s) on x at each E-step, to serve as an approximation of p ψ (z|x). This approach aligns with the principles of standard variational inference methods. Then, at the M-step, using latent variables sampled from the approximate posterior, the NMF parameters are updated by optimizing (4). Once the EM steps converge, the speech signal is estimated as ŝ = E p ψ * (s|x) {s}.

p ψ (x, z) = t p ψ (xt|z)p(zt), with likelihood computed as p ψ (xt|z) = Nc(0, diag(v θ,t (z) + [WH]t))

POSTERIOR SAMPLING ALGORITHMS

In this section, we present our EM-based speech enhancement frameworks, utilizing RVAE as a deep speech prior. These frameworks share a common structure but vary in the E-step, where each employs a distinct strategy to draw samples from the intractable posterior p ψ (z|x). We provide a concise summary of the overall

= v θ,t (z) v θ,t (z)+[WH] t ⊙ xt T t=1
speech enhancement process in Algorithm 1. Specifically, the first approach extends the LDEM method, as proposed in [START_REF] Sadeghi | Fast and efficient speech enhancement with variational autoencoders[END_REF], to RVAE. The second approach utilizes the Metropolis-Hastings sampling algorithm, while the third algorithm is a Metropolis-adjusted version of LDEM.

Langevin dynamics (LD)

In the conventional VAE-based speech enhancement method described in [START_REF] Sadeghi | Fast and efficient speech enhancement with variational autoencoders[END_REF], the process of sampling from the posterior distribution is carried out independently for each latent variable. To capture temporal dependencies, a total variation (TV) regularization term is introduced. However, in the context of RVAE, latent variables are naturally interconnected through an RNN model, making the TV regularization term less effective.

Langevin dynamics enables the generation of a sequence of samples from the posterior distribution p ψ (z|x) solely using its score function, defined as follows:

f ψ (z) = ∇z log p ψ (z|x) = ∇z log p ψ (x|z) + log p(z) = ∇z T t=1 log p ψ (xt|z) + log p(zt) . (5)
In contrast to VAE, this score function cannot be decomposed over individual latent variables, meaning that f ψ (z) ̸ = t f ψ (zt). Consequently, each zt must be sampled individually, akin to the sequential Gibbs sampling procedure [START_REF] Bishop | Pattern recognition and machine learning[END_REF]. This sequential approach would significantly increase complexity. Instead, we adopt a parallel sampling strategy, wherein all latent variables are sampled simultaneously. Furthermore, following the methodology employed in LDEM for VAE, we generate multiple samples for each latent variable to obtain a more robust and efficient approximation of the expectation in (4). Therefore, starting from z = (z1, • • • , zT ), we initially draw M distinct samples (states) for each latent variable zt, denoted as

z(0) = z (0) t,i t,i
, with t = 1, . . . T and i = 1, . . . M , using a random walk approach by sampling from the following proposal distribution:

z (0) t,i |zt ∼ N (zt, σ 2 I), ∀t, i (6)
or z (0)

t,i = zt + σϵt,i, where ϵt,i ∼ N (0, I) and σ > 0. The next states are produced by sampling from the following distribution:

z (k) t,i |z (k-1) ∼ N (z (k-1) t,i + η 2 f ψ (z (k-1) ), ηI), (7)
or, equivalently

z (k) t,i = z (k-1) t,i + η 2 f ψ (z (k-1) ) + √ ηζ t,i , (8)
Algorithm 2 LD sampler

1: Inputs: z(0) = z (0) t,i t,i
, H (hyperparameters).

2: for k = 1, • • • , K do 3: ζ = ζ t,i t,i , with ζ t,i ∼ N (0, I) 4: z(k) = z(k-1) + η 2 f ψ (z (k-1) ) + √ ηζ, 5: end for 6: Output: z(K) = z (K) t,i t,i Algorithm 3 MH sampler 1: Require: z (0) = z (0) t t
, H (hyperparameters).

2:

for k = 1, • • • , K do 3: ζ = {ζ t } t , with ζ t ∼ N (0, I) 4: z(k) = z (k-1) + √ ηζ, 5:
Accept z(k) t (∀t) according to (10) 6: end for 7:

Output: z = z (k) k>kburn-in Algorithm 4 MALA sampler 1: Require: z (0) = z (0) t t
, H (hyperparameters).

2: for k = 1, • • • , K do 3: ζ = {ζ t } t , with ζ t ∼ N (0, I) 4: z(k) = z (k-1) + η 2 f ψ (z (k-1) ) + √ ηζ, 5: Accept z(k)
t (∀t) according to (12) 6: end for 7: Output: z = z (k) k>kburn-in where ζ t,i ∼ N (0, I), and η > 0 is a step size. The added noise introduces stochasticity, which enhances exploration within the highdensity regions of the posterior. The generated samples converge to the true posterior distribution under some regularity conditions [START_REF] Welling | Bayesian learning via stochastic gradient Langevin dynamics[END_REF]. The LD sampler is summarized in Algorithm 2.

Metropolis-Hastings (MH)

Metropolis-Hastings (MH) [START_REF] Robert | Monte Carlo statistical methods[END_REF] is a Markov chain Monte Carlo (MCMC) sampling technique for generating a sequence of samples from a probability distribution. It begins with initial states and iteratively proposes new states using a typically Gaussian proposal distribution. Candidate states are either accepted or rejected based on defined probabilities.

For RVAE, similarly as done in the LD sampler, the MH algorithm generates samples for all the latent variables in parallel. More precisely, starting from some initial states z

(0) = z (0) 1 , • • • , z (0) T
, at the k-th iteration, a sequence of candidate states, denoted z(k) , are sampled from the following proposal distribution

z(k) t |z (k-1) t ∼ N (z (k-1) t , σ 2 I), ∀t. (9)
Each candidate state z(k) t in the sequence z(k) is then accepted according to the following probability:

αt = min 1, p ψ (xt|z (k) )p(z (k) t ) p ψ (xt|z (k-1) )p(z (k-1) t ) (10)
Let ut be drawn from the continuous uniform distribution over [0, 1]. Then, if ut ≤ αt, the proposal is accepted and z

(k) t ← z(k) t . Other- wise, the current state is retained z (k) t ← z (k-1) t .
A key observation here is that the acceptance ratios, α1, . . . , αT , are computed in parallel, with the same current likelihood p ϕ (x|z (k-1) ) for all the samples. Once a sufficient number of iterations is performed, the initial samples corresponding to the so-called burn-in period are discarded. The overall MH sampler is summarized in Algorithm 3.

Metropolis-Adjusted Langevin Algorithm (MALA)

The Metropolis-Adjusted Langevin Algorithm (MALA) [START_REF] Roberts | Langevin diffusions and metropolis-hastings algorithms[END_REF] aims at combining the MH and LD mechanisms to achieve a more efficient exploration of the target distribution. MALA follows the same steps as MH with the difference that the candidate states are generated using a proposal distribution guided by LD. More precisely, the proposal distribution takes a similar form as [START_REF] Bando | Statistical speech enhancement based on probabilistic integration of variational autoencoder and non-negative matrix factorization[END_REF], except for the fact that here we do not generate multiple samples per latent variable:

z(k) t |z (k-1) t ∼ N (z (k-1) t + η 2 f ψ (z (k-1) t ), ηI), (11)
Nevertheless, in contrast to LD sampler, which always updates the states according to the update rule [START_REF] Bando | Statistical speech enhancement based on probabilistic integration of variational autoencoder and non-negative matrix factorization[END_REF], MALA considers the updated states as candidates, similar to MH, and accepts them according to the following probability

αt = min 1, p ψ (xt|z (k) )p(z (k) t )q(z (k) |z (k) ) p ψ (xt|z (k-1) )p(z (k-1) t )q(z (k) |z (k) ) (12)
where q(u|v) ∝ exp -

1 2η ∥u -v - η 2 f (v)∥ 2 (13)
is the transition probability density from v to u. Unlike the basic MH approach, MALA often suggests moves towards regions of higher probability, thus increasing the probability of their acceptance. The overall MALA sampler is described in Algorithm 4.

EXPERIMENTS

Baselines

This section presents and discusses the speech enhancement results of the proposed EM-based approaches 1 , i.e., LDEM, MHEM, and MALAEM, with RVAE [START_REF] Leglaive | A recurrent variational autoencoder for speech enhancement[END_REF] as the generative model. We compare against the VEM method2 [START_REF] Bie | Unsupervised speech enhancement using dynamical variational autoencoders[END_REF][START_REF] Leglaive | A recurrent variational autoencoder for speech enhancement[END_REF], and SGMSE+3 [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF], as a state-ofthe-art diffusion-based speech enhancement method.

Evaluation metrics

To evaluate the speech enhancement performance, we use standard metrics, including the extended short-time objective intelligibility (ESTOI) measure [START_REF] Jensen | An algorithm for predicting the intelligibility of speech masked by modulated noise maskers[END_REF], ranging in [0, 1], the perceptual evaluation of speech quality (PESQ) metric [START_REF] Rix | Perceptual evaluation of speech quality (PESQ)-a new method for speech quality assessment of telephone networks and codecs[END_REF], ranging in [-0.5, 4.5], and the scale-invariant signal-to-distortion ratio (SI-SDR) metric [START_REF] Le Roux | SDRhalf-baked or well done?[END_REF] in dB. For all these metrics, the higher the better. Moreover, as a rough measure of the computational complexity of different methods, we report the average real-time factor (RTF), which is defined as the time (in seconds) required to enhance one second of speech signal. Our experiments were conducted using a machine with an AMD EPYC 7351 CPU and an NVIDIA Tesla T4 GPU.

Model architectures

We utilized pretrained models from the respective code repositories for both RVAE and SGMSE+. In the RVAE architecture, the encoder and decoder employ bidirectional long short-term memory (BLSTM) networks with an internal state dimension of 128, and the latent space is of dimension L = 16. The input data consists of STFT power spectrograms with a dimension of F = 513. This model was trained on the training subset of the Wall Street Journal (WSJ0) corpus. The architecture of SGMSE+ is detailed in [START_REF] Richter | Speech enhancement and dereverberation with diffusion-based generative models[END_REF], and its core network is based on the Noise Conditional Score Network (NCSN++) [START_REF] Song | Score-based generative modeling through stochastic differential equations[END_REF], adapted for processing complex-valued STFT features. The overall model was trained using the same clean training utterances as RVAE, combined with noise signals from the CHiME3 dataset [START_REF] Barker | The third 'CHiME' speech separation and recognition challenge: Dataset, task and baselines[END_REF]. The input data for SGMSE+ comprises STFT representations with F = 256, and training was conducted on sequences with a length of T = 256, as opposed to T = 50 used for RVAE.

Parameter settings

For the inference parameters of SGMSE+, we adhered to their default values. In the case of RVAE-based methods, we conducted a total of J = 100 EM iterations. The learning rate for VEM, with the Adam optimizer, as well as for LDEM and MALAEM (denoted as η), was consistently set at 0.005. For LDEM and VEM, we empirically selected K = 1, while for MHEM and MALAEM, we opted for K = 10, and included a burn-in period of kburn-in = 5. Additionally, we set σ 2 = 0.02 in both ( 6) and (9).

Evaluation datasets

For performance evaluation, we used the test set of the WSJ0-QUT corpus [START_REF] Leglaive | A recurrent variational autoencoder for speech enhancement[END_REF], created by mixing clean speech signals from WSJ0 (distinct speakers from training) with noise signals from the QUT-NOISE corpus [START_REF] Dean | The QUT-NOISE-SRE protocol for the evaluation of noisy speaker recognition[END_REF]. It includes Café, Home, Street, and Car noise types, with signal-to-noise ratio (SNR) values of -5 dB, 0 dB, and 5 dB. The test set amounts to 651 noisy speech signals with a total duration of 1.5 hours. Furthermore, we evaluated the generalization performance of various methods under mismatched conditions by incorporating pre-computed noisy versions of the TCD-TIMIT data as introduced in [START_REF] Abdelaziz | NTCD-TIMIT: A new database and baseline for noise-robust audio-visual speech recognition[END_REF]. The noisy test set that we used includes Living Room (LR) (from the second CHiME challenge [START_REF] Vincent | The second 'CHiME' speech separation and recognition challenge: Datasets, tasks and baselines[END_REF]), White, Car, and Babble (from the RSG-10 corpus [START_REF] Steeneken | Description of the RSG-10 noise database[END_REF]), with SNR4 values of -5 dB, 0 dB, and 5 dB, yielding 540 test speech signals, with a total duration of about 0.75 hours.

Results

The average speech enhancement metrics, under both matched (WSJ0-QUT) and mismatched (TCD-TIMIT) conditions, are presented in Table 1, with the associated average RTF values reported in Table 2. From these results, we can make several observations. First of all, among the RVAE-based algorithms, the proposed posterior sampling methods outperform VEM with a significant margin. The performance gap is even more remarkable in the mismatched conditions, which demonstrates that VEM is not as generalizable as our proposed methods. This could be due to the fact that VEM relies on fine-tuning the trained encoder on the new data, which might not be efficient. For instance, in the mismatched condition, MALAEM outperforms VEM by around 3 dB in SI-SDR, 0.19 in PESQ, and 0.07 On the other hand, SGMSE+, as the supervised baseline, showcases remarkable performance in the matched condition, achieving much higher speech enhancement metrics than those of the unsupervised RVAE-based methods. Nevertheless, when tested in the mismatched condition, the performance of SGMSE+ drops significantly, under-performing our proposed methods with a large margin. This confirms the generalization dilemma of supervised methods.

In terms of computational complexity at inference time, all the proposed methods achieve much smaller RTF values than VEM, making them more applicable. In particular, the LDEM algorithm demonstrates a competitive RTF of 0.21 sec, as compared to 12.55 sec for VEM and 3.85 sec for SGMSE+, showcasing its high computational efficiency in enhancing speech signals.

CONCLUSIONS

In this paper, we present new posterior sampling techniques for EMbased unsupervised speech enhancement using RVAE. These methods serve as viable alternatives to the computationally intensive variational inference-based VEM approach. Our experimental results illustrate the efficiency of the proposed techniques-LDEM, MHEM, and MALAEM-which not only significantly reduce computational complexity but also outperform VEM. Notably, the LDEM algorithm demonstrates high efficiency and competitive enhancement outcomes. In contrast, the supervised baseline, SGMSE+, excels under matched conditions but faces challenges in mismatched scenarios, highlighting generalization limitation of supervised methods. In summary, our proposed methods offer a promising avenue for efficient and effective unsupervised speech enhancement.

  Initialize: z = {zt} T t=1 , ψ ={W, H}. 3: for j = 1, • • • , J do

	4:	E-step:	z ← Sampler ψ (z, H)
	5:	M-step:	ψ ← argmax ψ log p ψ (x|z)
	6: end for
	7: Clean speech estimation: ŝ

Algorithm 1 EM-based speech enhancement 1: Inputs: x = {xt} T t=1 (noisy STFT data), H (hyperparameters). 2:

Table 1 :

 1 Speech enhancement results under matched (WSJ0-QUT) and mismatched (TCD-TIMIT) test conditions.

	Metric		SI-SDR (dB)	PESQ	ESTOI
	Input (WSJ0-QUT)	-2.60 ± 0.16 1.83 ± 0.02 0.50 ± 0.01
		VEM [8]	4.5 ± 0.21	2.21 ± 0.02 0.60 ± 0.01
	RVAE	MHEM MALAEM	5.15 ± 0.20 5.52 ± 0.21	2.24 ± 0.02 0.62 ± 0.01 2.28 ± 0.02 0.62 ± 0.01
		LDEM	5.58 ± 0.20	2.32 ± 0.02	0.63 ±0.01
	SGMSE+ [4]	9.41 ± 0.18	2.66 ± 0.02 0.77 ± 0.01
	Input (TCD-TIMIT) -8.74 ± 0.29 1.84 ± 0.02 0.35 ± 0.01
		VEM [8]	1.44 ± 0.30	2.02 ± 0.02 0.35 ± 0.01
	RVAE	MHEM MALAEM	3.72 ± 0.27 4.49 ± 0.29	2.12 ± 0.02 0.42 ± 0.01 2.21 ± 0.02 0.42 ± 0.01
		LDEM	4.18 ± 0.29	2.21 ± 0.02 0.42 ± 0.01
	SGMSE+ [4]	-3.97 ± 0.41 2.04 ± 0.02 0.38 ± 0.01

Table 2 :

 2 RTF values (average processing time per 1-sec speech). Furthermore, the LDEM algorithm consistently stands out with the highest or second-highest scores in all three metrics under both test conditions. The observation that LDEM outperforms MALAEM could be because of their different sampling strategies. That is, LDEM creates multiple parallel sequences of samples at each E-step, whereas MALAEM has only one sequential sequence of samples, where the final retained samples are chosen based on a probabilistic acceptance strategy.

	VEM [8]	MHEM	MALAEM	LDEM	SGMSE+ [4]
	12.55 ± 0.01 0.92 ± 0.01 2.49 ± 0.01 0.21 ± 0.01	3.85 ± 0.01
	in ESTOI.

https://github.com/msaadeghii/ps-rvae-se

https://github.com/XiaoyuBIE1994/DVAE_SE/

https://github.com/sp-uhh/sgmse

Here, the protocol used to compute SNR is different than the one used in[START_REF] Leglaive | A recurrent variational autoencoder for speech enhancement[END_REF].

This work was supported by the French National Research Agency (ANR) under the project REAVISE (ANR-22-CE23-0026-01). Experiments presented in this paper were carried out using the Grid'5000 testbed, supported by a scientific interest group hosted by Inria, and including CNRS, RENATER, and several universities as well as other organizations (see https://www.grid5000.fr).


Yoan Le Noane #

Evan Roué
email: 1evan.roue@univ-brest.fr

Norbert Parker #

Mihai Telescu #
email: 2mihai.telescu@univ-brest.fr

Vincent Laur #

Noël Tanguy

Uncertainty Quantification for a Microstrip Self-Biased Ku-band Circulator

Keywords: circulator, polynomial chaos expansion, selfbiased device, stochastic analysis

This paper focuses on uncertainty quantification for a Y-junction circulator. In the process of transitioning from prototyping to mass-production it is very important to understand how the uncertainties affecting input design parameters impact crucial features of the device behavior such as bandwidth or resonant frequency. This paper demonstrates the construction of a polynomial chaos expansion metamodel and its use in performing sensitivity analysis. The impact of the various sources of uncertainty affecting the circulator's behavior is quantified and the results are discussed.

I. INTRODUCTION

Circulators are commonly found in single antenna fullduplex systems or in RF front-ends to provide protection against impedance mismatches. The non-reciprocal behavior is obtained using a magnetically saturated ferrite whose permeability is thus anisotropic. Despite good electrical performance, these devices are known for their bulkiness. Indeed, a magnet is required to magnetize the soft ferrite. Furthermore, they are relatively expensive due to their complicated hybrid-manufacturing process (insertion of ferrite pucks in a stripline or microstrip structure).

One of the most promising approaches to mitigate these drawbacks is the use of a pre-oriented hexaferrite substrate [START_REF] Parker-Soues | Compact Self-Biased Q-Band Circulators[END_REF]. This ferrite type is able to maintain a strong remanent magnetization 𝑀 (without applying a static magnetic field) while also featuring a high anisotropy field 𝐻 . These properties allow for the development of self-biased circulators. In doing so, a thickness reduction of up to more than 90% can be achieved with respect to commercial circulators using a soft ferrite magnetized by a permanent magnet. However, precise characterization of the electromagnetic properties of hexaferrites is challenging, in particular the evaluation of the relative permittivity 𝜀 and the anisotropy field 𝐻 . Furthermore, sensitivity to the metallization dimensions has to be studied and quantified.

To better understand uncertainty propagation due to both material properties and device geometry, a stochastic approach was devised. Using data collected from a full-wave simulator, polynomial chaos metamodels were constructed and used to predict how the variability of design parameters impacts Sparameter response.

II. CIRCULATOR INDUSTRIALIZATION: PROBLEM OVERVIEW

The circulator topology used in this study and illustrated in Fig. 1 is a self-biased microstrip Y-junction resonator with quarter-wavelength impedance matching lines. The substrate is a hexaferrite sheet. Nominal values for material characteristics and design parameters cannot be disclosed. However, the tolerances are given in Table 1. Dimensions 𝑙 , 𝑅 𝑤 , 𝑤 are affected by the tolerances of the metallization; variation in substrate height ℎ is also considered. The uncertainties related to the hexaferrite mainly impact the anisotropy field 𝐻 and the relative permittivity 𝜀 . In the early stages of the design flow, the resonator radius (𝑅 ) and the quarter-wavelength line width (𝑤 ) of the circulator were determined using Bosma's theoretical model [START_REF] Bosma | On Stripline Y-Circulation at UHF[END_REF] for a central frequency of 16 GHz. The width of the input lines ( 𝑤 ) was computed in order to guarantee a characteristic impedance of 50 . This topology was subsequently optimized using the Ansys HFSS™ electromagnetic solver. The obtained S-parameter response is given in Fig. 2. Both adaptation and isolation achieve good levels with peaks below -20 𝑑𝐵 at the desired frequency of 16 GHz.

In order to qualitatively illustrate the effects of variability, 10 simulations, with the uncertain parameters varying randomly within the prescribed tolerances, were carried out and the Sparameter responses were plotted Fig. 3. The quantitative analysis is discussed in the following section.

III. STOCHASTIC METAMODELING

A. Polynomial chaos approach

Traditionally, uncertainty analysis was almost synonymous of Monte Carlo (MC) analysis: brute force stochastic investigation in which design parameters are varied randomly according to their respective tolerances. In order to properly observe quality metrics this process requires a very large number of simulations and entails a great computational cost, especially when it involves a full-wave electro-magnetic solver.

A solution allowing to circumvent the problem is the use of Polynomial Chaos Expansion (PCE). A popular tool implementing this approach is the UQLab toolbox [START_REF] Marelli | UQLab: A Framework for Uncertainty Quantification in Matlab[END_REF]. PCE considers the random vector 𝑿 ∈ 𝑅 which includes variables such as design parameters or material characteristics, and attempts to predict the behavior of random variable 𝑌 which is a quality metric, assumed to have finite variance by means of a metamodel ℳ defined as

𝑌 = ℳ(𝑿) = 𝑧 𝛹 (𝑿) ∈ℕ ( 1
)
where 𝑧 are the coefficients associated to the multivariate, orthonormal polynomials 𝛹 which form a basis of the Hilbert space. Index vector 𝜶 contains ordered lists of integers such that 𝜶 = (𝛼 , ⋯ , 𝛼 ) and is associated with 𝛹 through

𝛹 𝜶 (𝑿) = 𝜓 (𝑋 ) (2)
where 𝜓 are the univariate polynomials chosen according to the distribution of 𝑋 . If the input random variables follow uniform distributions, Legendre polynomials are usually chosen.

In practice, the space α is reduced to a set 𝒜 by limiting the total degree of a multivariate polynomial 𝛼 = ∑ 𝛼 to a degree 𝑝 (usually 3 to 5) according to one of the truncation schemes described in [START_REF] Sudret | Polynomial chaos expansions and stochastic finite element methods[END_REF].

The construction of the metamodel requires a training phase. Variable 𝑌 is computed for a reduced set of random values of 𝑿 . With this knowledge available, the computation of coefficients 𝑧 follows. Depending on the application, this operation may prove non-trivial and several approaches are possible. The example presented in this paper required the use of Bayesian compressive sensing [START_REF] Ji | Bayesian Compressive Sensing[END_REF].

B. Validation and numerical results

In the specific case of the device under study, vector 𝑿 contains the input variables listed in Table 1 and illustrated in Fig. 1. A total of 7 variables is thus considered: 𝑙 , 𝑅 𝑤 , 𝑤 , ℎ , 𝐻 and 𝜀 .

Four different output metrics 𝑌 were studied: the best isolation frequency, the best adaptation frequency, the bandwidth for isolation (absolute value measured at -12 dB) and the bandwidth for adaptation (also absolute value at -12 dB). These quantities are subsequently noted 𝑓 , 𝑓 , 𝐵 and 𝐵 , respectively.

In order to study uncertainty propagation, the authors constructed PCE metamodels from data obtained via HFSS™ simulation using 150 identification runs. Another 150 runs were then used for validation: probability density functions were plotted using the validation data directly and the prediction derived from the metamodels, they are shown in Fig. 4 and Fig. There is excellent agreement between the prediction and reference. The value of the metamodel comes from its Magnitude (dB) PDF PDF computational advantage. Indeed 150 runs take approximately 2 hour 35 minutes using HFSS with 6 parallel simulations. The evaluation of the PCE models in 150 points only takes 0.9 seconds. The hardware used is an AMD Ryzen 9 3900X, 64 GB of ram and a NVMe solid state drive.

It is important to understand the parametric nature of the polynomial chaos metamodel. It can be evaluated at great speed for any combination of the parameters illustrated in Fig. 1. Consequently, should the tolerances in Table 1 be reduced as industrialization progresses there is no use to build a new metamodel: a new statistical assessment can be carried-out simply by using the existing metamodel with new tolerance values. Moreover, the availability of the metamodel makes accurate variance analysis possible, the benefits of this investigation are shown in the following section.

IV. SENSITIVITY ANALYSIS The focus of this section is understanding and quantifying the impact of various input variables on an output metric. Sensitivity analysis usually comes in the form of variance analysis and entails the computation of the so-called Sobol' indices [START_REF] Sobol | Sensitivity analysis for non-linear mathematical models[END_REF].

Under a few reasonable assumptions, output variable Y can be expressed as 𝑌 = 𝑓(𝑋 , … , 𝑋 )

= 𝑓 + 𝑓 𝑋 (3)
+ 𝑓 𝑋 , 𝑋 + ⋯ + 𝑓 , ,…, (𝑋 , … , 𝑋 )

where the first term 𝑓 is a constant that is equal to the expected value of 𝑓(𝑿) and, furthermore

𝑉𝑎𝑟(𝑌) = 𝐷 + 𝐷 , + ⋯ + 𝐷 , ,…, (4)
where the first summation essentially designates the variance due to the effects of each unique variable 𝑋 .The second summation represents the residual influence due conjointly to 𝑋 and 𝑋 but not including 𝐷 and 𝐷 . Further orders can be considered but in practice they are negligible most of the time. First order Sobol' indices are defined as

𝑆 = 𝐷 𝑉𝑎𝑟(𝑌) (5)
Depending on the application, accurate computation of the indices may require tens of thousands of simulations. A detailed discussion of this topic is beyond the scope of the present paper, but it is important to note that in our case 10 simulations were needed. This operation required little computational effort with the metamodels, but could not have been carried-out in practice with a full-wave simulator.

Fig. 6 and Fig. 7 provide an important takeaway for circulator designers and manufacturers. Uncertainties affecting hexaferrite material properties such as permittivity or anisotropy field have far greater impact than those affecting the circuit geometry. It appears obvious that improving the accuracy of material characterization and closely controlling the production process of the hexaferrite has far greater importance than reducing the geometric tolerances of the metallization. This was expected and is inline with the theory behind circulator design, but the interest of the present paper is that it quantifies uncertainty effects. Furthermore, note that when investigating bandwidth, the results show that the anisotropy field is the single dominant source of variability. V. CONCLUSION In this paper the authors studied uncertainty propagation in a Y-junction circulator in preparation of the device's industrialization. The practicality of the polynomial chaos approach using Bayesian compressive sensing was demonstrated. The resulting metamodels were used to produce comprehensive statistical assessments at negligible computational cost. Variance analysis was conducted and the impact of the different sources of uncertainty was quantified.

Fig. 1 .

 1 Fig. 1. Analyzed microstrip Y-junction circulator topology

Fig. 2 .

 2 Fig. 2. Simulated S-parameters for the nominal circulator design (return loss in red/circles, isolation in cyan/squares, transmission in purple/triangles).

Fig. 3

 3 Fig.3The effects of variability on S-parameters (return loss in red, isolation in cyan, transmission in purple).

5

 5

  .

Fig. 4

 4 Fig. 4 Statistical analysis on 𝑓 (solid line) and 𝑓 , (dashed line): HFSS™ (cyan) vs. PCE (red).

Fig. 5

 5 Fig. 5 Statistical analysis on the bandwidth at -12 dB: HFSS™ (cyan) vs. PCE (red). Solid line for 𝐵 , dashed line for 𝐵 .

Fig. 6

 6 Fig. 6 First order Sobol' indices for on 𝑓 (red circles) and 𝑓 , (blue crosses).

Fig. 7

 7 Fig. 7 First order Sobol' indices for 𝐵 (red circles) and 𝐵 (blue crosses).

Table 1 .

 1 Uncertainties of design parameters and material properties.