Unsupervised Training for Deep Speech Source Separation with Kullback-Leibler Divergence Based Probabilistic Loss Function

11/11/2019
by   Masahito Togami, et al.
0

In this paper, we propose a multi-channel speech source separation with a deep neural network (DNN) which is trained under the condition that no clean signal is available. As an alternative to a clean signal, the proposed method adopts an estimated speech signal by an unsupervised speech source separation with a statistical model. As a statistical model of microphone input signal, we adopts a time-varying spatial covariance matrix (SCM) model which includes reverberation and background noise submodels so as to achieve robustness against reverberation and background noise. The DNN infers intermediate variables which are needed for constructing the time-varying SCM. Speech source separation is performed in a probabilistic manner so as to avoid overfitting to separation error. Since there are multiple intermediate variables, a loss function which evaluates a single intermediate variable is not applicable. Instead, the proposed method adopts a loss function which evaluates the output probabilistic signal directly based on Kullback-Leibler Divergence (KLD). Gradient of the loss function can be back-propagated into the DNN through all the intermediate variables. Experimental results under reverberant conditions show that the proposed method can train the DNN efficiently even when the number of training utterances is small, i.e., 1K.

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1 Introduction

0.84(0.08,0.93) ©2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

Microphone input signal in teleconferencing systems, speech diarization systems, and automatic speech recognition systems is typically a mixture of multiple speech sources and it is also contaminated by room reverberation. Thus, speech source separation techniques have been highly spotlighted. As speech source separation techniques, blind source separation (BSS)

[1, 2, 3, 4, 5, 6, 7, 8] has been actively studied. Parameters which are needed for speech source separation can be optimized in an unsupervised manner with a statistical model. A speech source model is highly important for estimating a separation filter and solving the well-known inter-frequency permutation problem [9]. There are two requirements for a speech source model in the BSS. At first, the speech source model should capture complicated spectral characteristics of a speech source. Secondly, there should be a computationally efficient algorithm for optimizing parameters based on the speech source model. However, it is highly difficult to define a statistical model which fulfills these two requirements.

As supervised speech source separation techniques, recently, deep neural network (DNN) based approaches with a training dataset in which there are microphone input signal and corresponding oracle clean data have been widely studied, e.g., deep clustering (DC) [10, 11], permutation invariant training (PIT) [12, 13], deep attractor network [14, 15], and hybrid approaches with BSS [16, 17, 18]. DNN based approaches can capture complicated spectral characteristics of a speech source. Parameter optimization can be done efficiently by forward calculation of the DNN. However, it is hard to obtain an oracle clean data in a target environment. Thus, it is highly required to train the DNN by utilizing only observed microphone input signals which contain multiple speech sources without an oracle clean data.

Recently, unsupervised DNN training techniques have been proposed [19, 20]

. These techniques estimate a time-frequency mask based on the DC. The DNN is trained without an oracle time-frequency mask. An estimated time-frequency mask by a BSS technique in an unsupervised manner is adopted as an alternative to the oracle time-frequency mask. In the BSS technique, a time-frequency mask is estimated under the assumption that each component of the microphone input signal is sparse enough at the time-frequency domain. However, when there are reverberation and background noise, the sparseness assumption does not hold and speech source separation performance degrades.

In this paper, we propose an unsupervised DNN training technique which utilizes an estimated speech signal by an unsupervised speech source separation with a time-varying spatial filter as an alternative to the clean speech signal. The time-varying spatial filter is constructed based on a time-varying spatial covariance matrix (SCM) model [5, 21]

which includes submodels about reverberation and background noise so as to increase speech source separation performance under reverberant and noisy environments. The proposed method also estimates a time-varying spatial filter via the DNN. The DNN infers intermediate variables which are utilized for constructing the time-varying spatial filter. Since there are several errors in a separated signal, both the separated signal by the unsupervised method and the separated signal via the DNN are modeled as a probabilistic signal so as to avoid overfitting to the separation errors in traing phase. The proposed method adopts a loss function which evaluates Kullback-Leibler Divergence (KLD) between the posterior probability density function (PDF) of the separated signal by the unsupervised method and that of the separated signal via the DNN. Although there are multiple intermediate variables which should be inferred by the DNN, gradient of the loss function can be back-propagated into the DNN through all the intermediate variables jointly, thanks to evaluating the output signal in the loss function. Experimental results under reverberant and noisy conditions show that the proposed method can train the DNN more effectively in an unsupervised manner than conventional methods even when the number of the training utterances is small, i.e., 1K. The proposed KLD loss function is also shown to achieve better performance than the

loss function that evaluates the output signal as a deterministic signal.

2 Microphone input signal model

In this paper, speech source separation is performed in a time-frequency domain. Multi-channel microphone input signal ( is the frame index and is the frequency index) is modeled as follows:

(1)

where is the number of the speech sources, is the th speech signal, is the late reverberation term, and is the multi-channel background noise term. The objective of speech source separation is estimation of .

3 Proposed method

3.1 Overview

The proposed method trains a DNN which infers parameters of speech source separation without no clean data. Block diagram of the proposed method is shown in Fig. 1.

Figure 1: Block diagram of proposed method

The proposed method consists of two major parts. In each part, an input signal is a dereverberated signal by the Weighted Prediction Error (WPE) [22]. Let be the output signal of the WPE, where ( is the tap-length of early reverberation and is the tap-length of the dereverberation filter ). The first part is a pseudo clean signal generator (PCSG). As an alternative clean signal, the PCSG generates a separated speech signal in an unsupervised manner based on the local Gaussian modeling (LGM) [5]. The PCSG regards the pseudo clean signal (PCS) as a probabilistic signal and estimates the posterior probability density function (PDF) of the PCS in which is the separation parameter that is estimated in an iterative manner. The second part is the DNN based estimation part of each speech source. In the DNN part, each speech source is also regarded as a probabilistic signal and the posterior PDF is estimated, where

is the separation parameter which is estimated via the DNN. As the PCS and the estimated signal by the DNN are both probabilistic signals, we evaluate the difference between the PCS and the estimated signal by a loss function which evaluates a difference between two posterior PDFs. By consideration of uncertainty of the PCS and the estimated signal, gradient of the loss function propagates into the DNN not only through the mean vector but also through the covariance matrix term of the posterior PDF inferred by the DNN, which leads to efficient DNN training.

3.2 Pseudo Clean Signal Generator: Unsupervised speech source separation with local Gaussian modeling

The LGM based speech source separation [5]

separates multiple speech sources assuming that the PDF of each speech source belongs to a time-varying Gaussian distribution. The PDF of the dereverberated signal is modeled as

. The multi-channel spatial covariance matrix (SCM) of the dereverberated signal is modeled as follows:

(2)

where the first term is the SCM of each speech source,

is the time-frequency variance of the

th speech source, is the multi-channel covariance matrix of the th speech source, the second term is the SCM of a residual late reverberation which is not removed by the WPE, and the third term is the SCM of the background noise. Reflecting that the amount of the late reverberation depends on the past speech source variance, the late reverberation term is modeled as a convolution of the past time-varying speech source variance with the time-invariant covariance matrix [21] as follows:

(3)

where is the tap-length of the residual late reverberation and is the time-invariant covariance matrix of the th speech source. The third term in Eq. 2 is the time-invariant SCM of the background noise. Thus, is . As all the PDFs are Gaussian distributions, the posterior PDF of the th speech source is estimated as the following Gaussian distribution:

(4)

where and are calculated as and is a identity matrix ( is the number of the microphones), and is the MWF. The separation parameter is iteratively updated so as to maximize the log likelihood function with an auxiliary function [6, 21]. After update, the inter-frequency permutation problem is solved by [23].

3.3 Posterior PDF estimation via deep neural network

In the DNN part, the posterior PDF of each speech source is also estimated based on the LGM with the time-varying multi-channel SCM model defined in Eq. 2. The posterior PDF is calculated with the estimated . and are calculated in the same way as and , respectively. In the DNN part, the parameter is estimated via the DNN. The DNN structure is shown in Fig. 2. The input feature is concatenation of log spectral of the dereverberated signal and phase difference between microphones

. Time-frequency masks and a time-frequency variance of each speech source are inferred via the DNN that contains four bidirectional long short term memory (BLSTM) layers with 1200 hidden units and five dense layers. All of the covariance matrices are estimated via time-frequency masks inferred by the DNN, i.e.,

, , and , as follows:

(5)
(6)
(7)

where ( is the Hermitian transpose of a matrix/vector).

Figure 2: Deep neural network structure

3.3.1 Loss function for deep neural network training

The loss function for the DNN training is set to a divergence between two posterior PDFs, i.e., the posterior PDF estimated by the LGM and the posterior PDF estimated via the DNN . As a loss function, the proposed method adopts a Kullback-Leibler divergence defined as where the utterance-level permutation invariant training (PIT) [12] is utilized similarly to conventional supervised speech source separation [13, 24], is a set of possible permutations, and D(p_i,l,k —— q_j,l,k)= (μ_q,j,l,k-μ_p,i,l,k)^HV_q,j,l,k^-1(μ_q,i,l,k-μ_p,i,l,k) +tr(V_q,j,l,k^-1V_p,i,l,k )+log—Vq,j,l,k——Vp,i,l,k—-N_m. It is shown that acts as a regularization term in Eq. 3.3.1, which leads to avoiding overfitting of the MAP estimate to that contains separation error, and gradient of the loss function propagates not only through but also through , which is favorable for the DNN training of time-frequency masks.

3.3.2 Output signal in inference phase

In the inference phase, the parameter is inferred via a DNN. After that, is iteratively updated so as to minimize the auxiliary function in the same way as the PCSG. Finally, the separated signal is obtained as a mean vector of the posterior PDF, , by the MWF.

4 Experiment

Loss SDR SIR CD FWSeg. PESQ
Func. (dB) (dB) (dB) SNR (dB)
Unprocessed -2.01 0.52 5.60 6.56 1.52
- 4.75 8.42 5.05 9.05 1.90
- 4.12 8.19 5.12 8.20 1.82
- 3.84 7.76 5.17 7.93 1.78
5.14 8.99 5.05 8.79 1.92
4.87 8.71 5.04 8.14 1.89
3.62 6.02 5.27 7.04 1.75
KLD 5.44 9.74 4.95 8.63 1.98
KLD 5.53 9.84 4.89 8.88 1.99
KLD 5.71 10.27 4.86 8.95 2.02
Table 1: Evaluation results of LGM based methods
Approaches Filtering Loss Func. Phase Diff. SDR (dB) SIR (dB) CD (dB) FWSeg.SNR (dB) PESQ
Unprocessed -2.01 0.52 5.60 6.56 1.52
CACGMM Mask - - 4.63 9.70 5.01 7.58 1.87
CACGMM MVDR - - 5.11 7.91 5.19 8.87 1.87
CACGMM Mask DC No 3.03 5.26 5.37 6.46 1.65
CACGMM MVDR DC No 3.35 4.36 5.42 7.57 1.68
CACGMM Mask DC Yes 4.10 8.32 5.15 7.06 1.79
CACGMM MVDR DC Yes 4.58 6.85 5.26 8.39 1.81
CACGMM Mask MSA+PIT No 3.41 5.74 5.33 6.87 1.66
CACGMM MVDR MSA+PIT No 3.63 4.92 5.41 7.61 1.70
CACGMM Mask MSA+PIT Yes 4.64 8.87 5.04 7.76 1.85
CACGMM MVDR MSA+PIT Yes 4.95 7.41 5.21 8.70 1.84
LGM MWF () KLD Yes 5.71 10.27 4.86 8.95 2.02

Table 2: Comparison between CACGMM based methods and proposed method

4.1 Experimental setup

Speech source separation performance of the proposed method was evaluated by using measured impulse responses in Multi-channel Impulse Response Database (MIRD) [25] and TIMIT speech corpus [26]. In the training phase, TIMIT train corpus was utilized. In the evaluation phase, TIMIT test corpus was utilized. The reverberation time was randomly set to 0.36 (sec) or 0.61 (sec). Sampling rate was set to 8000 Hz. The number of the microphone was set to . The number of the speech sources was set to . Two microphone indices were randomly selected for each sample both in the training phase and in the evaluation phase. A 3-3-3-8-3-3-3 spacing (cm) microphone array, a 4-4-4-8-4-4-4 spacing (cm) microphone array, and a 8-8-8-8-8-8-8 spacing (cm) microphone array were utilized. Frame size was 256 pt. Frame shift was 64 pt. The number of frequency bins was . The distance between speech sources and microphones was set to m. Azimuth of each talker is randomly selected for each utterance. The number of total training utterances was set to 1000, which is a smaller dataset than the conventional one, e.g., 30000 [19]

, because small number of required utterances is preferable in practice. The number of total test utterances was 200. As a background noise signal, white Gaussian noise is added. Signal to noise Ratio (S/N) was randomly set from 20 dB to 30 dB. S/N between two speakers was randomly set from -5 dB to 5 dB. Mini-batch size was set to 128. Each utterance was split in every 100-frames segment. Neural network parameters were updated by

times. Adam optimizer [27] (learning rate was

) with gradient clipping was utilized. The proposed method calculates complex-valued gradient by Tensorflow

[28]. In each method, WPE was utilized (tap length was and was set to ).

4.2 Evaluation measures and comparative methods

We utilized Cepstrum distance (CD), Frequency-weighted segmental SNR (FWSegSNR), and PESQ as dereverberation performance measures. For speech source separation performance evaluation, we utilized SDR and SIR from BSS_EVAL [29]

. Four methods were evaluated, i.e., 1) Conventional unsupervised training method with complex angular central Gaussian mixture model (CACGMM)

[19]: Time-frequency mask of each source is inferred with the sparseness assumption. This model does not have any reverberation model. A loss function which evaluates an intermediate variable is adopted. The DNN has four BLSTM layers. Only the output dense layer of the DNN is different from that of the proposed method. 2) Unsupervised speech source separation based on LGM without DNN: The separation parameter is updated iteratively based on [6]. This method is also identical to PCGS in the proposed method. 3) Unsupervised training with LGM based PCGS and loss function: The loss function evaluates difference between the MAP estimate of the PCS posterior PDF and that of the estimated posterior PDF via the DNN, i.e., . 4) Proposed unsupervised training with LGM based PCGS and KLD loss function.

4.3 Experimental results

At first, we evaluated three types of LGM based methods. The number of the covariance matrices of residual late reverberation was set to 1, 4, or 8. In Table 1, experimental results for LGM based methods are shown. The proposed unsupervised training methods with KLD loss function is shown to be more effective than the unsupervised training methods with loss function. The proposed method also outperformed the LGM without the DNN (PCGS). This result confirmed that the proposed method is robust against separation error of the PCGS. In the loss function cases, when is 4 or 8, performance was degraded. It can be interpreted that the DNN parameters were not correctly learned by back-propagation only through the mean vector of the posterior PDF. In the proposed KLD loss function cases, performance monotonically increased in accordance with the number of . It is shown that back-propagation via the covariance matrix term of the posterior PDF is effective.

In Table 2, we compared the proposed KLD loss function based method with and the conventional unsupervised DNN training method with CACGMM. Unlike the proposed method, CACGMM does not have a reverberation model. Originally, a CACGMM based method without phase difference feature was proposed in [19]. However, the proposed method utilizes phase difference between microphones as an input feature. To evaluate each method fairly, we also evaluated CACGMM based methods with the phase difference feature. In addition to deep clustering (DC) based methods in which the dimension of the embedding vector was set to , PIT based methods which evaluate time-frequency masks were also evaluated. The time-frequency mask is evaluated by the magnitude spectrum approximation (MSA), because the pseudo oracle time-frequency mask is real-valued and the phase-sensitive spectrum approximation (PSA) [30] cannot be utilized. We also evaluated the original CACGMM [8] without no DNN training. As an output signal, we evaluated time-frequency masking results and minimum variance distortionless response (MVDR) results. It is shown that the proposed method outperformed all variants of CACGMM based methods. This result confirmed that effectiveness of the proposed reverberation and background noise models and DNN training with the proposed probabilistic loss function based on KLD.

5 Conclusions

We proposed an unsupervised multi-channel speech source separation method in which the deep neural network (DNN) is trained with no oracle clean signal. As a pseudo clean signal, the proposed method adopts the separated signal by the conventional unsupervised local Gaussian modeling. So as to reduce reverberation and background noise effectively, the proposed method estimates a time-varying covariance matrix of microphone input signal which contains reverberation and background noise components. Since both the pseudo clean signal and the estimated signal via the DNN are probabilistic signals, we proposed a loss function which evaluates the Kullback-Leibler divergence (KLD) between two posterior probability density functions. Experimental results showed that the proposed method can separate speech sources more accurately than the conventional methods under a reverberant and noisy environment.

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