Linear Regression for Speaker Verification

02/12/2018 ∙ by Xiao-Lei Zhang, et al. ∙ 0

This paper presents a linear regression based back-end for speaker verification. Linear regression is a simple linear model that minimizes the mean squared estimation error between the target and its estimate with a closed form solution, where the target is defined as the ground-truth indicator vectors of utterances. We use the linear regression model to learn speaker models from a front-end, and verify the similarity of two speaker models by a cosine similarity scoring classifier. To evaluate the effectiveness of the linear regression model, we construct three speaker verification systems that use the Gaussian mixture model and identity-vector (GMM/i-vector) front-end, deep neural network and i-vector (DNN/i-vector) front-end, and deep vector (d-vector) front-end as their front-ends, respectively. Our empirical comparison results on the NIST speaker recognition evaluation data sets show that the proposed method outperforms within-class covariance normalization, linear discriminant analysis, and probabilistic linear discriminant analysis, given any of the three front-ends.

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I Introduction

IEEEexample:BSTcontrol

Speaker verification has long been a fundamental task in speech processing. A speaker verification system verifies an identity claim made by a test speaker, and decides to accept or reject the claim. It can be either text-dependent or text-independent based on its input speech materials: the former constrains a speaker to pronounce a prescribed text, while the latter does not constrain the speech contents. This paper studies text-independent speaker verification. A text-indepdent speaker verification system generally contains two components. The first component is a front-end which extracts a feature vector from a speaker utterance by some density estimator. The second component is a back-end which builds speaker models and measures the similarity of two speaker models by a classifier.

An early speaker verification front-end is feature averaging which learns a feature vector from a speaker utterance by averaging the frame-level acoustic features [1]. The method requires long speech utterances to reach stable speech statistics. Another class of front-ends is statistical models, which estimates the density of speech frames by statistical models. Early approaches of this kind build a model, e.g. vector quantization [2] or Gaussian mixture model (GMM) [3, 4], for each speaker. These approaches are inefficient when the number of speakers is large. To alleviate this problem, in [5], Reynolds et al. proposed a GMM-based universal background model (GMM-UBM) which builds a single GMM from the pool of all training speakers. GMM-UBM is a fundamental method of speaker verification in recent years. To deal with speaker and channel variability, many approaches were proposed along with GMM-UBM, where factor analysis [6] is among the effective ones. It first extracts high-dimensional supervectors of utterances which are their first- and second-order statistics produced from GMM-UBM, and then reduces the supervectors to low-dimensional identity vectors (i-vectors) by factor analysis. The above combination of GMM-UBM and i-vector is the GMM/i-vector front-end.

Recently, deep neural network (DNN) based front-ends have received much attention [7, 8, 9]. In [7], Sarkar et al. used DNN to extract frame-level bottleneck features that were then used as the input of GMM-UBM. In [9], Lei et al.

took a DNN trained for a different task, e.g. speech recognition, to generate posterior probability of speech frames, which is a supervised alternative to GMM-UBM, and then used the factor analysis

[6] to extract i-vectors from the DNN based UBM. The method is denoted as the DNN/i-vector front-end. To demonstrate the advantages of the DNN/i-vector front-end, its DNN acoustic model needs to be trained with additional data [10]. In [8], Variani et al. trained a DNN classifier to map frame-level features in a given context to the corresponding speaker identity target, and extracted a feature vector, referred as a deep vector or “d-vector”, from a speaker utterance by averaging the activations derived from the last DNN hidden layer. The method is known as the d-vector front-end.

After feature extraction by a front-end, a speaker verification back-end builds speaker models for classification. It generally contains two stages—a development stage and a test stage. The development stage builds a

speaker space from development data, where each speaker acts like a coordinate axis of the space. The test stage gets the enrollment and test speaker models of a trial from the speaker space and then evaluates the similarity of the two models by a classifier.

We summarize some back-ends as follows. In [5], Reynolds et al.

first built a speaker space by adapting the GMM-UBM to many speaker-dependent GMMs by the maximum a posteriori estimation in the development stage and then verifies the identity of a test speaker by a likelihood ratio test. Later on, in

[11], Campbell et al.

trained a support vector machine classifier to distinguish true speakers from imposter speakers with nuisance attribute projection

[12] for compensating session variability. In [6], Dehak et al. proposed to learn a speaker space by within class covariance normalization (WCCN) or linear discriminant analysis (LDA) and then applied cosine similarity scoring as the classifier. In [13], Kenny proposed to extract speaker models from an i-vector based front-end or LDA and then used probabilistic LDA (PLDA) as the classifier. Besides, in [14], Snyder et al. proposed an end-to-end training method to train a DNN based front-end and a PLDA-like back-end jointly.

In this paper, we propose a linear regression (LR) based back-end. LR is a traditional statistical regression model that minimizes the mean squared error between the target and its estimate with a closed-form solution. In the development stage of the back-end, we apply LR to learn a speaker space where the target of the LR model is the ground-truth indicator vectors of the speaker utterances. In the enrollment and test stages, we first extract the enrollment and test speaker models of a trial from the speaker space, and then evaluate the similarity of the two models by cosine similarity scoring. The overall back-end is denoted as the LR+cosine back-end. To evaluate its effectiveness, we propose three speaker verification systems which combine the LR+cosine back-end with the GMM/i-vector, DNN/i-vector, and d-vector front-ends, respectively.

We have conducted an extensive experiment on the NIST 2006 speaker recognition evaluation (SRE) and NIST 2008 SRE data sets. We have compared the LR+cosine back-end with the cosine similarity scoring (cosine), WCCN with the cosine similarity scoring (WCCN+cosine), LDA with the cosine similarity scoring (LDA+cosine), and LDA with the PLDA scoring (LDA+PLDA) back-ends. Our experimental results show that the proposed method outperforms the comparison methods, and the experimental conclusion is consistent in different lengths of enrollment speech.

This paper is organized as follows. In Section II, we introduce the LR-based back-end and three LR-based speaker verification systems. In Section III, we present the experiments. In Section IV, we summarize the paper.

Ii Linear regression for speaker verification

In this section, we first present the LR-based back-end in Section II-A, and then present three front-ends that will be combined respectively with the LR-based back-end in Section II-B.

The procedure of any of the three speaker verification systems is as follows. The front-end extracts a feature vector from an utterance where denotes the number of frames of the utterance. Then, the LR-based back-end first gets the speaker model from by the LR model and then verifies the identity of by the cosine similarity scoring.

Ii-a Linear regression based back-end

Suppose a labeled development corpus after processed by a front-end is given by where is the number of speakers, is the number of utterances of the -th speaker, is the feature vector of a speaker utterance produced from a front-end, and is the ground-truth label of the utterance representing the identification of the speaker, . Suppose , then we change to an -dimensional indicator vector which is a binary code with the -th dimension set to and the other dimensions set to . As a result, we can rewrite the labeled corpus as . We fit to a LR model:

(1)

where is the LR model and is the estimation error. Minimizing the squared estimation error derives the following closed-form solution:

(2)

where

In the enrollment and test stages, we apply the LR model to extract a new feature from by the following equation:

(3)

The speaker model is given by:

(4)

where is the number of utterances of the speaker.

Finally, we employ a classifier to identify the similarity of two speaker models and . Despite that many classifiers could be applied, we use the simple and effective cosine similarity scoring as an example based on the experimental conclusion of reference [6]. The cosine similarity of the two models is calculated by:

(5)

which is compared with a decision threshold . If the score is larger than , then the two models are judged as from the same speaker; otherwise, they are from different speakers.

Ii-B Front-ends

The three speaker verification systems based on the LR-based back-end use the GMM/i-vector [5, 6], DNN/i-vector [9], and d-vector [8] front-ends, respectively.

Ii-B1 GMM/i-vector front-end

The GMM/i-vector front-end contains a GMM-UBM [5, 15] which is a speaker- and channel-independent GMM trained from the pool of all speech frames of the development data, and a total variability matrix [6] that encompasses both speaker- and channel-variability. Suppose contains Gaussian mixture components, and suppose we have an utterance of frames where is a -dimensional acoustic feature. The zero-th order and centralized first-order Baum-Welch statistics of the utterance extracted from the -th component of is:

(6)
(7)

where is the mean of the -th component of . If we define as a -dimensional diagonal matrix whose diagonal blocks are , as a supervector, and as a -dimensional diagonal covariance matrix estimated during factor analysis training [6], then we obtain the i-vector by:

(8)

where and is invariant across utterances.

Ii-B2 DNN/i-vector front-end

The difference between the DNN/i-vector front-end [9] and the GMM/i-vector front-end is that the GMM-UBM in the DNN/i-vector front-end is estimated by a DNN acoustic model trained for automatic speech recognition. Specifically, the DNN acoustic model is used to estimate the senone posteriors of acoustic features, where a senone is used to model the tied states of a set of triphones that are close in the acoustic space. If we model the posterior distribution of a senone by a Gaussian mixture component of the GMM-UBM, then we can use the senone posteriors to train the GMM-UBM in the following way.

Suppose the development corpus contains utterances, and the -th utterance has frames . The parameters of the GMM-UBM are estimated by:

(9)
(10)
(11)
(12)

where represent the alignments of which are the posteriors of produced by the DNN acoustic model, and are the prior and covariance of the -th mixture component, respectively.

The DNN acoustic model is trained in a supervised mode, where the ground-truth labels of the speech frames are the alignments produced by a hidden-Markov-model-GMM (HMM-GMM) speech recognition system. It usually adopts a contextual window with a window size of, e.g.

, to expand the input from to where is the half-window length.

Ii-B3 D-vector front-end

The d-vector front-end [8] averages the frame-level features of an utterance produced from the top hidden layer of a DNN classifier for an utterance-level d-vector . The DNN is trained to minimize the classification error of speech frames, where the ground-truth label of a speech frame is the indicator vector of the speaker that the speech frame belongs to. The DNN adopts a large contextual window with a window size of to expand its input acoustic feature from to , which is important in improving the effectiveness and robustness of the d-vector front-end.

Iii Experiments

In this section, we present the databases and evaluation metrics at first in Section

III-A, then the experimental setup in Section III-B, and finally the experimental results in Sections III-C and III-D.

Iii-a Databases and evaluation metrics

We took the 8conv condition of NIST 2006 speaker recognition evaluation (SRE) database as the development set, and the 8conv condition of NIST 2008 SRE for enrollment and test. The 8conv condition of NIST 2006 SRE contains 402 female speakers and 298 male speakers. The 8conv condition of NIST 2008 SRE contains 395 female speakers and 240 male speakers. Each speaker has 8 conversations. A speaker utterance in a conversation was about 1 to 2 minutes long after removing the silence segments by VAD, where we took its ASR transcript as its VAD label. We split all speech signals into 15 second segments.

To illustrate the global performance of the proposed method in terms of detection error tradeoff (DET) curves, we built an initial test condition as follows. We selected the first 150 second speech of the first conversation of a speaker as the enrollment data of the speaker, and split the last 30 second speech of the 6-th conversation of the speaker into two test segments with each segment as an individual test. We took each speaker as a claimant with the remaining speakers acting as imposters, and rotated through the tests of all speakers. We conducted the experiment on females and males respectively. The number of claimant and imposter trials are summarized in Table II. The closer the DET curve approaches to the origin, the better the performance is.

To investigate how the performance of the proposed method varies with the length of the enrollment speech, we conducted experiments in six test conditions described in Table I. Specifically, for each speaker in the 8conv condition of the NIST 2008 SRE, we first randomly picked 2 segments from a randomly selected conversation with each segment as an individual test; then, we randomly selected segments from the remaining 7 conversations as the enrollment data of the speaker, where we set to 1, 2, 3, 5, 10, and 15 for the six test conditions respectively. For a given test condition, we built the claimant and imposter trials in the same way as the initial test condition. Therefore, the number of the trials are the same as that in Table II. Because the enrollment and test speech of a trial was selected randomly, we ran the experiments on each test condition 100 times and reported the average results so as to prevent biased conclusions. We used equal error rate (EER), minimum detection cost function (DCF) with SRE’08 parameters (DCF08), and minimum DCF with SRE’10 parameters (DCF10) as the evaluation metrics. The smaller the EER or DCF is, the better the performance is.

Name Length of enrollment speech Length of test speech 15"-15" 15 seconds 15 seconds 30"-15" 30 seconds 15 seconds 45"-15" 45 seconds 15 seconds 75"-15" 75 seconds 15 seconds 150"-15" 150 seconds 15 seconds 225"-15" 225 seconds 15 seconds

TABLE I: Description of test conditions.

#speakers #true trials #imposter trials Female 395 790 311,260 Male 240 480 114,720

TABLE II: Number of claimant and imposter trials.

Iii-B Experimental setup

Iii-B1 Acoustic features

We set the frame length to 25 ms and the frame shift to 10 ms. We extracted 19-dimensional mel-frequency cepstral coefficients (MFCC), 13-dimensional relative spectral filtered perceptual linear predictive cepstral coefficients (RASTA-PLP) and 1-dimensional log energy, as well as their delta and delta-delta coefficients from each frame, which produced a total of 99-dimensional acoustic feature per frame.

Iii-B2 Front-ends

For the GMM/i-vector front-end, we used gender-dependent UBMs containing 2048 Gaussian mixtures and 400 total factors defined by the total variability matrix . We followed the MSR identity toolbox for the implementation of the GMM/i-vector front-end.

For the DNN/i-vector front-end, we trained a DNN acoustic model from the Switchboard-1 database. The alignments of the frames for the DNN training, which contained 8730 senones, were generated by a HMM-GMM speech recognition system implemented in the Kaldi pipeline. The half-window length of the DNN was set to 3, which expanded the acoustic features to 693 dimensions. As a result, the DNN acoustic model used the 693-dimensional feature as the input and its corresponding 8730 dimensional alignment as the ground-truth label. The DNN has hidden layers, each of which consists of rectified linear units. The output units of the DNN are the softmax functions. The DNN was optimized by the minimum cross-entropy criterion. The number of epoches for backpropagation training was set to . The batch size was set to

. The learning rate of the stochastic gradient descent was set to

. The momentum was set to 0.5 for the first 10 epoches, and set to 0.9 for the other epoches. The dropout rate of the hidden units was set to 0.2. We used the posterior probability of the development data produced by the DNN acoustic model to train gender-dependent UBMs. Because many senones have small posterior probabilities, we truncated the UBMs from 8730 Gaussian mixtures to 3096 Gaussian mixtures by discarding the mixtures that have small zero-th order Baum-Welsh statistics. We used 400 total factors to generate the i-vectors.

For the d-vector front-end, we trained gender-dependent DNNs on the development data, where the two DNNs have the same parameter setting as follows. The half-window length was set to 20, which expanded the acoustic feature to 4059 dimensions. Each DNN has hidden layers, each of which consists of 400 rectified linear units. The output dimensions of the two DNNs are 395 for the females and 240 for the males, respectively. The learning rate of the stochastic gradient descent was set to . All other parameters were set to the same values as those in the DNN/i-vector front-end.

Iii-B3 Back-ends in comparison

We compared the LR+cosine back-end with the following back-ends:

  • Cosine similarity scoring (cosine): The cosine back-end evaluates the cosine similarity of two speaker models directly where the speaker model is simply an average of the utterance-level feature vectors of the speaker produced from a front-end [6].

  • WCCN+cosine: WCCN helps compensate for channel variability [16]. It was first proposed for a SVM based back-end, and then applied to the cosine similarity scoring by Dehak et al. [6]. Here we compared with the WCCN+cosine method [6].

  • LDA+cosine: LDA is a supervised dimensionality reduction method. Dehak et al. [6] applied LDA to the cosine similarity scoring. Here we set the output dimension of LDA to 200 in all evaluations, which is a common experimental setting in literature.

  • LDA+PLDA: The PLDA classifier was first introduced to speaker verification by Kenny in [13]. LDA is usually used as a feature extractor for PLDA. We set the output dimension of LDA to 200 in all evaluations.

Iii-C Results

We report the comparison results in the initial test condition in Figs. 1 to 6 respectively. From the figures, we observe that the proposed method outperforms the comparison methods significantly when the GMM/i-vector or DNN/i-vector front-end is used (Figs. 1, 2, 4 and 5), and outperforms the comparison methods slightly when the d-vector front-end is used (Figs. 3 and 6).

Fig. 1: DET curves of the back-ends with the GMM/i-vector front-end on the female part of the initial test condition.
Fig. 2: DET curves of the back-ends with the DNN/i-vector front-end on the female part of the initial test condition.
Fig. 3: DET curves of the back-ends with the d-vector front-end on the female part of the initial test condition.
Fig. 4: DET curves of the back-ends with the GMM/i-vector front-end on the male part of the initial test condition.
Fig. 5: DET curves of the back-ends with the DNN/i-vector front-end on the male part of the initial test condition.
Fig. 6: DET curves of the back-ends with the d-vector front-end on the male part of the initial test condition.

To prevent a biased conclusion that the proposed method happens to have some advantage in the initial test condition, we ran a comparison in the 6 test conditions described in Table I, where each test condition has 100 independent implementations randomly generated from the NIST 2008 SRE database. We report the average results on the male and female parts of the implementations in Tables III and IV respectively. From the tables, we observe that the proposed LR+cosine back-end outperforms the comparison methods when the enrollment speech is longer than 15 seconds, and is comparable to LDA+PLDA when the enrollment speech is 15 seconds long, given any of the three front-ends.

EER (in %) DCF08 DCF10
GMM/i-vector DNN/i-vector d-vector GMM/i-vector DNN/i-vector d-vector GMM/i-vector DNN/i-vector d-vector
15"-15" Cosine 16.52 12.14 9.87 6.2423 4.8058 4.2655 0.0940 0.0882 0.0873 [t]
WCCN+cosine 3.78 4.26 9.87 1.8382 2.2631 4.2655 0.0625 0.0764 0.0873
LDA+cosine 9.70 9.21 10.03 3.9029 3.8131 4.0192 0.0805 0.0833 0.0855
LDA+PLDA 3.08 2.84 7.70 1.3656 1.3634 3.1572 0.0550 0.0575 0.0831
LR+cosine 2.80 3.10 7.45 1.2911 1.4950 2.9859 0.0528 0.0578 0.0748 [b]
30"-15" Cosine 10.54 7.14 7.80 4.2643 3.0554 3.5023 0.0833 0.0760 0.0808 [t]
WCCN+cosine 2.44 2.71 7.80 1.1682 1.4803 3.5023 0.0485 0.0629 0.0808
LDA+cosine 5.72 5.54 7.72 2.4234 2.4642 3.2372 0.0655 0.0706 0.0789
LDA+PLDA 2.05 1.85 5.80 0.9270 0.9054 2.4158 0.0455 0.0471 0.0750
LR+cosine 1.59 1.77 5.10 0.7326 0.8503 2.1988 0.0391 0.0435 0.0645 [b]
45"-15" Cosine 7.63 5.07 7.02 3.2110 2.2481 3.2053 0.0745 0.0679 0.0778 [t]
WCCN+cosine 1.98 2.22 7.02 0.9423 1.1902 3.2053 0.0413 0.0563 0.0778
LDA+cosine 4.17 4.12 6.91 1.7938 1.9061 2.9395 0.0563 0.0636 0.0752
LDA+PLDA 1.73 1.57 5.15 0.7831 0.7733 2.1448 0.0410 0.0434 0.0715
LR+cosine 1.18 1.34 4.34 0.5670 0.6585 1.9069 0.0331 0.0380 0.0592 [b]
75"-15" Cosine 5.07 3.29 6.57 2.2101 1.5529 2.9559 0.0629 0.0586 0.0745 [t]
WCCN+cosine 1.64 1.79 6.57 0.7537 0.9592 2.9559 0.0355 0.0493 0.0745
LDA+cosine 3.01 2.94 6.36 1.3081 1.4282 2.7023 0.0480 0.0554 0.0719
LDA+PLDA 1.53 1.40 4.62 0.6779 0.6855 1.9399 0.0376 0.0396 0.0674
LR+cosine 0.94 1.07 3.70 0.4384 0.5065 1.6927 0.0274 0.0327 0.0545 [b]
150"-15" Cosine 2.92 1.89 6.06 1.2964 0.9553 2.7485 0.0493 0.0484 0.0713 [t]
WCCN+cosine 1.35 1.46 6.06 0.5999 0.7758 2.7485 0.0308 0.0430 0.0713
LDA+cosine 2.10 2.12 5.81 0.9249 1.0437 2.4882 0.0402 0.0484 0.0683
LDA+PLDA 1.34 1.24 4.25 0.6010 0.6022 1.7482 0.0356 0.0371 0.0640
LR+cosine 0.74 0.86 3.25 0.3375 0.3967 1.4872 0.0233 0.0281 0.0500 [b]
225"-15" Cosine 2.14 1.45 5.88 0.9864 0.7626 2.6622 0.0430 0.0438 0.0700 [t]
WCCN+cosine 1.24 1.37 5.88 0.5564 0.7024 2.6622 0.0286 0.0409 0.0700
LDA+cosine 1.82 1.85 5.65 0.8148 0.9222 2.4214 0.0371 0.0449 0.0672
LDA+PLDA 1.33 1.22 4.07 0.5771 0.5913 1.6874 0.0344 0.0364 0.0619
LR+cosine 0.69 0.78 3.10 0.3155 0.3674 1.4063 0.0218 0.0265 0.0476 [b]
TABLE III: Comparison results of the back-ends on the female parts of the 6 test conditions.
EER (in %) DCF08 DCF10
GMM/i-vector DNN/i-vector d-vector GMM/i-vector DNN/i-vector d-vector GMM/i-vector DNN/i-vector d-vector
15"-15" Cosine 15.74 7.95 10.16 5.9013 3.2361 4.0586 0.0898 0.0755 0.0883 [t]
WCCN+cosine 4.14 4.13 10.16 1.8184 1.9061 4.0586 0.0661 0.0713 0.0883
LDA+cosine 9.89 6.55 10.29 3.8431 2.7328 3.8002 0.0781 0.0717 0.0785
LDA+PLDA 3.72 3.33 8.58 1.5178 1.4099 3.2521 0.0630 0.0654 0.0858
LR+cosine 3.55 3.44 8.37 1.5312 1.4960 3.1300 0.0572 0.0557 0.0793 [b]
30"-15" Cosine 10.03 4.31 8.12 3.9863 1.8684 3.3539 0.0780 0.0598 0.0837 [t]
WCCN+cosine 2.63 2.68 8.12 1.1601 1.2544 3.3539 0.0523 0.0588 0.0837
LDA+cosine 5.87 3.77 7.97 2.3780 1.6657 3.0572 0.0630 0.0571 0.0720
LDA+PLDA 2.50 2.25 6.46 1.0336 0.9715 2.5377 0.0544 0.0588 0.0822
LR+cosine 2.05 2.08 5.90 0.9079 0.9157 2.3789 0.0448 0.0446 0.0727 [b]
45"-15" Cosine 7.38 2.94 7.25 3.0176 1.3420 3.0528 0.0682 0.0516 0.0810 [t]
WCCN+cosine 2.08 2.12 7.25 0.9033 1.0171 3.0528 0.0451 0.0507 0.0810
LDA+cosine 4.29 2.81 7.00 1.7848 1.2687 2.7608 0.0545 0.0498 0.0682
LDA+PLDA 2.08 1.83 5.59 0.8581 0.8065 2.2553 0.0508 0.0550 0.0799
LR+cosine 1.54 1.57 4.99 0.6975 0.7030 2.0807 0.0385 0.0385 0.0689 [b]
75"-15" Cosine 4.80 1.93 6.68 2.0268 0.9133 2.7933 0.0564 0.0423 0.0785 [t]
WCCN+cosine 1.65 1.70 6.68 0.7231 0.8063 2.7933 0.0384 0.0431 0.0785
LDA+cosine 3.07 2.11 6.25 1.3144 0.9545 2.4982 0.0452 0.0417 0.0638
LDA+PLDA 1.74 1.56 5.06 0.7340 0.6900 1.9961 0.0477 0.0516 0.0778
LR+cosine 1.14 1.20 4.13 0.5295 0.5530 1.7971 0.0327 0.0323 0.0644 [b]
150"-15" Cosine 2.92 1.28 6.28 1.2581 0.6091 2.6047 0.0442 0.0339 0.0752 [t]
WCCN+cosine 1.38 1.42 6.28 0.6140 0.6831 2.6047 0.0333 0.0373 0.0752
LDA+cosine 2.27 1.61 5.83 0.9524 0.7351 2.3034 0.0381 0.0352 0.0606
LDA+PLDA 1.57 1.41 4.69 0.6839 0.6470 1.8220 0.0458 0.0502 0.0744
LR+cosine 0.93 1.03 3.73 0.4423 0.4682 1.6328 0.0290 0.0285 0.0602 [b]
225"-15" Cosine 2.23 1.03 6.04 0.9779 0.4954 2.5024 0.0383 0.0302 0.0746 [t]
WCCN+cosine 1.28 1.29 6.04 0.5518 0.6083 2.5024 0.0309 0.0346 0.0746
LDA+cosine 1.89 1.42 5.53 0.8172 0.6262 2.2010 0.0352 0.0320 0.0590
LDA+PLDA 1.49 1.29 4.41 0.6461 0.6154 1.7348 0.0453 0.0488 0.0737
LR+cosine 0.86 0.94 3.43 0.3913 0.4057 1.5194 0.0267 0.0262 0.0582 [b]
TABLE IV: Comparison results of the back-ends on the male parts of the 6 test conditions.

We drew the curves of the relative improvement scores of the proposed method over the best comparison methods in Figs. 7 and 8, where the relative improvement score is defined by:

with and denoted as the EERs of the proposed method and best comparison method respectively. From the figures, we observe the following phenomena. (i) The relative improvement is getting larger when the enrollment speech is getting longer. An exception is that, when the DNN/i-vector is used as the front-end, the relative improvement is not always increased for the females. This is caused by the fast performance improvement of the cosine similarity scoring when the enrollment speech is getting longer. (ii) The highest relative improvement happens with the GMM/i-vector front-end, which reaches 44.3% for the females in the 225"-15" test condition and 33.0% for the males in the 150"-15" test condition.

Fig. 7: Relative EER improvement of the LR+cosine back-end over the best comparison methods in the female parts of the 6 test conditions.
Fig. 8: Relative EER improvement of the LR+cosine back-end over the best comparison methods in the male parts of the 6 test conditions.

We also drew the soft decision scores produced from the LR+cosine and LDA+PLDA back-ends for the females in Fig. 9

where we have normalized the decision scores to a range where the mean values of the decision scores of the imposter and true trials are zero and one respectively. From the figure, we observe that the scores produced by LR+cosine have smaller with-in class variances and smaller overlaps than those produced by LDA+PLDA.

Fig. 9: Histograms of the soft decision scores produced by LR+cosine and LDA+PLDA in the female parts of the 6 test conditions, where the decision scores have been normalized so that the mean values of the imposter and true trials are zero and one respectively.

Iii-D Effects of back-ends in fusion systems

Fusing the decision scores produced from multiple base methods is an effective way for further improving the performance of the base methods. This subsection studies the approach of averaging the soft decision scores produced from the systems that use GMM/i-vector and DNN/i-vector as the front-ends, respectively. Figures 10 and 11 show the DET curves of the fusion systems with different back-ends on the initial test condition. Tables V and VI list the comparison results of the fusion systems on the 6 test conditions defined in Table I. From the figures and tables, we observe the same experimental phenomena as those in Section III-C, which supports the effectiveness of the LR+cosine back-end in the fusion systems.

Note that we have also evaluated the fusion systems that fuse the GMM/i-vector, DNN/i-vector, and d-vector front-ends together. The experimental conclusions are similar with the above.

Fig. 10: DET curves of the fusion systems on the female part of the initial test condition.
Fig. 11: DET curves of the fusion systems on the male part of the initial test condition.
EER (in %) DCF08 DCF10
15"-15" Cosine 9.85 3.9613 0.0816 [t]
WCCN+cosine 3.34 1.6565 0.0625
LDA+cosine 6.72 2.7861 0.0720
LDA+PLDA 2.64 1.1850 0.0491
LR+cosine 2.46 1.1496 0.0501 [b]
30"-15" Cosine 5.32 2.3171 0.0669 [t]
WCCN+cosine 2.12 1.0516 0.0486
LDA+cosine 3.78 1.6648 0.0575
LDA+PLDA 1.73 0.7877 0.0399
LR+cosine 1.43 0.6557 0.0369 [b]
45"-15" Cosine 3.59 1.6360 0.0582 [t]
WCCN+cosine 1.76 0.8402 0.0416
LDA+cosine 2.69 1.2355 0.0498
LDA+PLDA 1.48 0.6643 0.0359
LR+cosine 1.07 0.5102 0.0317 [b]
75"-15" Cosine 2.22 1.0728 0.0487 [t]
WCCN+cosine 1.44 0.6726 0.0356
LDA+cosine 1.88 0.8991 0.0424
LDA+PLDA 1.31 0.5844 0.0328
LR+cosine 0.89 0.3925 0.0269 [b]
150"-15" Cosine 1.27 0.6418 0.0393 [t]
WCCN+cosine 1.18 0.5437 0.0308
LDA+cosine 1.35 0.6384 0.0358
LDA+PLDA 1.16 0.5038 0.0308
LR+cosine 0.72 0.3108 0.0230 [b]
225"-15" Cosine 0.97 0.5041 0.0349 [t]
WCCN+cosine 1.11 0.4976 0.0289
LDA+cosine 1.17 0.5593 0.0332
LDA+PLDA 1.13 0.4923 0.0300
LR+cosine 0.66 0.2911 0.0217 [b]
TABLE V: Comparison results of the back-ends in the fusion systems on the female parts of the 6 test conditions.
EER (in %) DCF08 DCF10
15"-15" Cosine 7.75 3.0791 0.0713 [t]
WCCN+cosine 3.60 1.6093 0.0636
LDA+cosine 5.92 2.3919 0.0646
LDA+PLDA 3.21 1.3305 0.0592
LR+cosine 3.04 1.3140 0.0531 [b]
30"-15" Cosine 4.16 1.7299 0.0557 [t]
WCCN+cosine 2.36 1.0279 0.0502
LDA+cosine 3.23 1.3784 0.0496
LDA+PLDA 2.17 0.9102 0.0528
LR+cosine 1.78 0.8054 0.0422 [b]
45"-15" Cosine 2.77 1.2232 0.0479 [t]
WCCN+cosine 1.85 0.8134 0.0428
LDA+cosine 2.38 1.0190 0.0421
LDA+PLDA 1.78 0.7465 0.0486
LR+cosine 1.37 0.6244 0.0358 [b]
75"-15" Cosine 1.71 0.7997 0.0388 [t]
WCCN+cosine 1.46 0.6495 0.0359
LDA+cosine 1.68 0.7529 0.0346
LDA+PLDA 1.51 0.6403 0.0456
LR+cosine 1.04 0.4770 0.0304 [b]
150"-15" Cosine 1.08 0.5223 0.0309 [t]
WCCN+cosine 1.27 0.5602 0.0313
LDA+cosine 1.26 0.5737 0.0295
LDA+PLDA 1.34 0.5974 0.0442
LR+cosine 0.89 0.4074 0.0269 [b]
225"-15" Cosine 0.85 0.4194 0.0274 [t]
WCCN+cosine 1.12 0.4976 0.0286
LDA+cosine 1.06 0.4904 0.0267
LDA+PLDA 1.27 0.5681 0.0429
LR+cosine 0.80 0.3586 0.0247 [b]
TABLE VI: Comparison results of the back-ends in the fusion systems on the male parts of the 6 test conditions.

Iv Conclusions

In this paper, we have presented a speaker verification back-end based on linear regression. Linear regression is a simple linear model that minimizes the mean squared estimation error between the target and its estimate with a closed form solution, where the target for our speaker verification problem is defined as the ground-truth indicator vectors of utterances. The proposed LR+cosine back-end first learns speaker models by the LR model, and then applies the cosine similarity scoring to evaluate the similarity of a pair of speaker models. We have further proposed three LR-based speaker verification systems by combining the LR+cosine back-end with the GMM/i-vector, DNN/i-vector, and d-vector front-ends respectively. We have conducted an extensive experiment on the NIST 2006 SRE and NIST 2008 SRE data sets, where we used the 8conv condition of the NIST 2006 SRE for development and the 8conv condition of the NIST 2008 SRE for enrollment and test. To prevent a biased experimental conclusion on a particular evaluation environment, the experiment was carried out with different lengths of enrollment speech covering a range from 15 seconds to 225 seconds and repeated 100 times. The experimental results show that the proposed LR+cosine back-end outperforms several common back-ends including the cosine, WCCN+cosine, LDA+cosine, and LDA+PLDA back-ends in most cases in terms of DET curves, EER, DCF08, and DCF10.

V Acknowledgements

This work was supported in part by the Natural Science Foundation of China under Grant No. 61671381.

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