A DISCRIMINATIVE GAUSSIAN MIXTURE MODEL WITH SPARSITY
Year: 2021

Abstract:
In probabilistic classification, a discriminative model based on the softmax function has a potential limitation in that it assumes unimodality for each class in the feature space. The mixture model can address this issue, although it leads to an increase in the number of parameters. We propose a sparse classifier based on a discriminative GMM, referred to as a sparse discriminative Gaussian mixture (SDGM). In the SDGM, a GMM-based discriminative model is trained via sparse Bayesian learning. Using this sparse learning framework, we can simultaneously remove redundant Gaussian components and reduce the number of parameters used in the remaining components during learning; this learning method reduces the model complexity, thereby improving the generalization capability. Furthermore, the SDGM can be embedded into neural networks (NNs), such as convolutional NNs, and can be trained in an end-to-end manner. Experimental results demonstrated that the proposed method outperformed the existing softmax-based discriminative models.

heading: 1 INTRODUCTION
text: In probabilistic classification, a discriminative model is an approach that assigns a class label c to an input sample x by estimating the posterior probability P (c | x). The posterior probability P (c | x) should correctly be modeled because it is not only related to classification accuracy, but also to the confidence of decision making in real-world applications such as medical diagnosis support. In general, the model calculates the class posterior probability using the softmax function after nonlinear feature extraction. Classically, a combination of the kernel method and the softmax function has been used. The recent mainstream method is to use a deep neural network for representation learning and softmax for the calculation of the posterior probability.
Such a general procedure for developing a discriminative model potentially contains a limitation due to unimodality. The softmax-based model, such as a fully connected (FC) layer with a softmax function that is often used in deep neural networks (NNs), assumes a unimodal Gaussian distribution for each class (details are shown in Appendix A). Therefore, even if the feature space is transformed into discriminative space via the feature extraction part, P (c | x) cannot correctly be modeled if the multimodality remains, which leads to a decrease in accuracy.
Mixture models can address this issue. Mixture models are widely used for generative models, with a Gaussian mixture model (GMM) as a typical example. Mixture models are also effective in discriminative models; for example, discriminative GMMs have been applied successfully in various fields, e.g., speech recognition (Tüske et al. 2015; Wang 2007). However, the number of parameters increases if the number of mixture components increases, which may lead to over-fitting and an increase in memory usage; this is useful if we can reduce the number of redundant parameters while maintaining multimodality.
In this paper, we propose a discriminative model with two important properties; multimodality and sparsity. The proposed model is referred to as the sparse discriminative Gaussian mixture (SDGM). In the SDGM, a GMM-based discriminative model is formulated and trained via sparse Bayesian
learning. This learning algorithm reduces memory usage without losing generalization capability by obtaining sparse weights while maintaining the multimodality of the mixture model.
The technical highlight of this study is twofold: One is that the SDGM finds the multimodal structure in the feature space and the other is that redundant Gaussian components are removed owing to sparse learning. Figure 1 shows a comparison of the decision boundaries with other discriminative models. The two-class data are from Ripley’s synthetic data (Ripley 2006), where two Gaussian components are used to generate data for each class. The FC layer with the softmax function, which is often used in the last layer of deep NNs, assumes a unimodal Gaussian for each class, resulting in an inappropriate decision boundary. Kernel Bayesian methods, such as the Gaussian process (GP) classifier (Wenzel et al. 2019) and relevance vector machine (RVM) (Tipping 2001), estimate nonlinear decision boundaries using nonlinear kernels, whereas these methods cannot find multimodal structures. Although the discriminative GMM finds multimodal structure, this model retains redundant Gaussian components. However, the proposed SDGM finds a multimodal structure of data while removing redundant components, which leads to an accurate decision boundary.
Furthermore, the SDGM can be embedded into NNs, such as convolutional NNs (CNNs), and trained in an end-to-end manner with an NN. The proposed SDGM is also considered as a mixture, nonlinear, and sparse expansion of the logistic regression, and thus the SDGM can be used as the last layer of an NN for classification by replacing it with the fully connected (FC) layer with a softmax activation function.
The contributions of this study are as follows:
• We propose a novel sparse classifier based on a discriminative GMM. The proposed SDGM has both multimodality and sparsity, thereby flexibly estimating the posterior distribution of classes while removing redundant parameters. Moreover, the SDGM automatically determines the number of components by simultaneously removing the redundant components during learning. • From the perspective of the Bayesian kernel methods, the SDGM is considered as the
expansion of the GP and RVM. The SDGM can estimate the posterior probabilities more flexibly than the GP and RVM owing to multimodality. The experimental comparison using benchmark data demonstrated superior performance to the existing Bayesian kernel methods. • This study connects both fields of probabilistic models and NNs. From the equivalence of
a discriminative model based on a Gaussian distribution to an FC layer, we demonstrate that the SDGM can be used as a module of a deep NN. We also demonstrate that the SDGM exhibits superior performance to the FC layer with a softmax function via end-toend learning with an NN on the image recognition task.

heading: 2 RELATED WORK AND POSITION OF THIS STUDY
text: The position of the proposed SDGM among the related methods is summarized in Figure 2. Interestingly, by summarizing the relationships, we can confirm that the three separately developed fields, generative models, discriminative models, and kernel Bayesian methods, are related to each other. Starting from the Gaussian distribution, all the models shown in Figure 2 are connected via
four types of arrows. There is an undeveloped area in the upper right part, and the development of the area is the contribution of this study.
A (unimodal) Gaussian distribution is used as the most naive generative model in machine learning and is the foundation of this relationship diagram. A GMM is the mixture expansion of the Gaussian distributions. Since the GMM can express (almost) arbitrary continuous distributions using multiple Gaussian components, it has been utilized for a long time. Since Gaussian fitting requires numerous parameters, the sparsified versions of Gaussian (Hsieh et al. 2011) and GMM (Gaiffas & Michel 2014) have been proposed.
The discriminative models and the generative models are mutually related (Lasserre et al. 2006; Minka
2005). According to Lasserre et al. (2006), the only difference between these models is their statistical parameter constraints. Therefore, given a generative model, we can derive a corresponding discriminative model. For example, discriminative models corresponding to the Gaussian mixture model have been proposed (Axelrod et al. 2006; Bahl et al. 1996; Klautau et al. 2003; Tsai & Chang 2002; Tsuji et al. 1999; Tüske et al. 2015; Wang 2007). They indicate more flexible fitting capability for classification problems than the generative GMM because the discriminative models have a lower statistical bias than the generative models. Furthermore, as shown by Tüske et al. (2015); Variani et al. (2015), these models can be used as the last layer of the NN because these models output the class posterior probability.
From the perspective of the kernel Bayesian methods, the GP classifier (Wenzel et al. 2019) and the mixture of GPs (MGP) (Luo & Sun 2017) are the Bayesian kernelized version of the logistic regression and the discriminative GMM, respectively. The SDGM with kernelization is also regarded as a kernel Bayesian method because the posterior distribution of weights is estimated during learning instead of directly estimating the weights as points, as with the GP and MGP. The RVM (Tipping 2001) is the sparse version of the GP classifier and is the most important related study. The learning algorithm of the SDGM is based on that of the RVM; however, it is extended for the mixture model.
If we use kernelization, the SDGM becomes one of the kernel Bayesian methods and is considered as the mixture expansion of the RVM or sparse expansion of the MGP. Therefore, the classification capability and sparsity are compared with kernel Bayesian methods in Section 4.1. Otherwise, the SDGM is considered as one of the discriminative models and can be embedded in an NN. The comparison with other discriminative models is conducted in Section 4.2 via image classification by combining a CNN.

heading: 3 SPARSE DISCRIMINATIVE GAUSSIAN MIXTURE (SDGM)
text: The SDGM takes a continuous variable as its input and outputs the posterior probability of each class, acquiring a sparse structure by removing redundant components via sparse Bayesian learning. Figure 3 shows how the SDGM is trained by removing unnecessary components while maintaining discriminability. In this training, we set the initial number of components to three for each class. As the training progresses, one of the components for each class gradually becomes small and is removed.

heading: 3.1 NOTATION
text: Let x ∈ RD be a continuous input variable and tc (c ∈ {1, . . . , C}, C is the number of classes) be a discrete target variable that is coded in a one-of-C form, where tc = 1 if x belongs to class c, tc = 0
otherwise. Also, let zcm be a discrete latent variable, and zcm = 1 when x from class c belongs to the m-th component (m ∈ {1, . . . ,Mc}, Mc is the number of components for class c), zcm = 0 otherwise. For simplicity, in this paper, the probabilities for classes and components are described using only c and m; e.g., we use P (c,m | x) instead of P (tc = 1, zcm = 1 | x).

heading: 3.2 MODEL FORMULATION
text: The posterior probabilities of each class c given x are calculated as follows:
P (c | x) = Mc∑ m=1 P (c,m | x), P (c,m | x) = πcm exp[w T cmφ]∑C c′=1 ∑Mc′ m′=1 πc′m′ exp[w T c′m′φ] , (1)
φ = [ 1,xT, x21, x1x2, . . . , x1xD, x 2 2, x2x3, . . . , x 2 D ]T , (2)
where πcm is the mixture weight that is equivalent to the prior of each component P (c,m). It should be noted that we usewcm ∈ RH , which is the weight vector representing the m-th Gaussian component of class c. The dimension of wcm, i.e., H , is the same as that of φ; namely, H = 1 +D(D + 3)/2.
Derivation. Utilizing a Gaussian distribution as a conditional distribution of x given c and m, P (x | c,m), the posterior probability of c given x, P (c | x), is calculated as follows:
P (c | x) = Mc∑ m=1 P (c,m)P (x | c,m)∑C c=1 ∑Mc m=1P (c,m)P (x | c,m) , (3)
P (x | c,m) = 1 (2π) D 2 |Σcm| 1 2 exp
[ −1
2 (x− µcm)TΣ−1cm(x− µcm)
] , (4)
where µcm ∈ RD and Σcm ∈ RD×D are the mean vector and the covariance matrix for component m in class c. Since the calculation inside an exponential function in (4) is quadratic form, the conditional distributions can be transformed as follows:
P (x | c,m) = exp[wTcmφ], (5) where
wcm = [ − D
2 ln 2π − 1 2 ln |Σcm| − 1 2 D∑ i=1 D∑ j=1 scmijµcmiµcmj , D∑ i=1 scmi1µcmi,
. . . , D∑ i=1 scmiDµcmi,− 1 2 scm11,−scm12, . . . , −scm1D,− 1 2 scm22, . . . ,− 1 2 scmDD ]T . (6)
Here, scmij is the (i, j)-th element of Σ−1cm.

heading: 3.3 DUAL FORM VIA KERNELIZATION
text: Sinceφ is a second-order polynomial form, we can derive the dual form of the SDGM using polynomial kernels. By kernelization, we can treat the SDGM as the kernel Bayesian method as described in Section 2.
Let ψcm ∈ RN be a novel weight vector for the c,m-th component. Using ψcm and the training dataset {xn}Nn=1, the weight of the original form wcm is represented as
wcm = [φ(x1), · · · ,φ(xN )]ψcm, (7)
where φ(xn) is the transformation xn of using (2). Then, (5) is reformulated as follows:
P (x | c,m) = exp[wTcmφ]
= exp[ψTcm[φ(x1) Tφ(x), · · · ,φ(xN )Tφ(x)]T]
= exp[ψTcmK(X,x)], (8)
where K(X,x) is an N -dimensional vector that contains kernel functions defined as k(xn,x) = φ(xn) Tφ(x) = (xTnx + 1) 2 for its elements and X is a data matrix that has xTn in the n-th row. Whereas the computational complexity of the original form in Section 3.2 increases in the order of the square of the input dimension D, the dimensionality of this dual form is proportional to N . When we use this dual form, we use N and k(xn, ·) instead of H and φ(·), respectively.
3.4 LEARNING ALGORITHM
A set of training data and target value {xn, tnc} (n = 1, · · · , N) is given. We also define π and z as vectors that comprise πcm and zncm as their elements, respectively. As the prior distribution of the weight wcmh, we employ a Gaussian distribution with a mean of zero. Using a different precision parameter (inverse of the variance) αcmh for each weight wcmh, the joint probability of all the weights is represented as follows:
P (w | α)
= C∏ c=1 Mc∏ m=1 H∏ h=1 √ αcmh 2π exp [ −1 2 wcmh 2αcmh ] , (9)
where w and α are vectors with wcmh and αcmh as their elements, respectively. During learning, we update not only w but also α. If αcmh → ∞, the prior (9) is 0; hence a sparse solution is obtained by optimizing α as shown in Figure 4.
Using these variables, the expectation of the log-likelihood function over z, J , is defined as follows:
J = Ez [lnP (T,z | X,w,π,α)] = N∑
n=1 C∑ c=1 rncmtnc lnP (c,m | xn),
where T is a matrix with tnc as its element. The variable rncm in the right-hand side corresponds to P (m | c,xn) and can be calculated as rncm = P (c,m | xn)/P (c | xn). The posterior probability of the weight vector w is described as follows:
P (w | T,z,X,π,α) = P (T,z | X,w,π,α)P (w | α) P (T,z | X,α) (10)
An optimal w is obtained as the point where (10) is maximized. The denominator of the right-hand side in (10) is called the evidence term, and we maximize it with respect to α. However, this maximization problem cannot be solved analytically; therefore we introduce the Laplace approximation described as the following procedure.
With α fixed, we obtain the mode of the posterior distribution of w. The solution is given by the point where the following equation is maximized:
Ez [lnP (w | T,z,X,π,α)] = Ez [lnP (T,z | X,w,π,α)] + lnP (w | α) + const.
= J −wTAw + const., (11)
where A = diagαcmh. We obtain the mode of (11) via Newton’s method. The gradient and Hessian required for this estimation can be calculated as follows:
∇Ez [lnP (w | T,z,X,π,α)] = ∇J −Aw, (12)
Algorithm 1: Learning algorithm of the SDGM Input: Training data set X and teacher vector T. Output: Trained weightw obtained by maximizing (11). Initialize the weightsw, hyperparameters α, mixture coefficients π, and posterior probabilities r; while α have not converged do
Calculate J using (10); while r have not converged do
whilew have not converged do Calculate gradients using (12); Calculate Hessian (13); Maximize (11) w.r.t. w; Calculate P (c,m | xn) and P (c | xn); end rncm = P (c,m | xn)/P (c | xn);
end Calculate Λ using (16); Update α using (17); Update π using (18);
end
∇∇Ez [lnP (w | T,z,X,π,α)] = ∇∇J −A. (13) Each element of∇J and∇∇J is calculated as follows:
∂J
∂wcmh = (rncmtnc−P (c,m | xn))φh, (14)
∂2J
∂wcmh∂wc′m′h′ = P (c′,m′ | xn)(P (c,m | xn)− δcc′mm′)φhφh′ , (15)
where δcc′mm′ is a variable that takes 1 if both c = c′ andm = m′, 0 otherwise. Hence, the posterior distribution ofw can be approximated by a Gaussian distribution with a mean of ŵ and a covariance matrix of Λ, where
Λ = −(∇∇Ez [lnP (ŵ | T,z,X,π,α)])−1. (16)
Since the evidence term can be represented using the normalization term of this Gaussian distribution, we obtain the following updating rule by calculating its derivative with respect to αcmh.
αcmh ← 1− αcmhλcmh
ŵ2cmh , (17)
where λcmh is the diagonal component of Λ. The mixture weight πcm can be estimated using rncm as follows:
πcm = 1
Nc Nc∑ n=1 rncm, (18)
where Nc is the number of training samples belonging to class c. As described above, we obtain a sparse solution by alternately repeating the update of hyper-parameters, as in (17) and (18), and the posterior distribution estimation of w using the Laplace approximation. As a result of the optimization, some of αcmh approach to infinite values, and wcmh corresponding to αcmh have prior distributions with mean and variance both zero as shown in (4); hence such wcmh are removed because their posterior distributions are also with mean and variance both zero. During the procedure, the {c,m}-th component is eliminated if πcm becomes 0 or all the weights wcmh corresponding to the component become 0.
The learning algorithm of the SDGM is summarized in Algorithm 1. In this algorithm, the optimal weight is obtained as maximum a posterior solution. We can obtain a sparse solution by optimizing the prior distribution set to each weight simultaneously with weight optimization.

heading: 4 EXPERIMENTS

heading: 4.1 COMPARATIVE STUDY USING BENCHMARK DATA
text: To evaluate the capability of the SDGM quantitatively, we conducted a classification experiment using benchmark datasets. The datasets used in this experiment were Ripley’s synthetic data (Ripley
2006) (Ripley hereinafter) and four datasets cited from (Rätsch et al. 2001); Banana, Waveform, Titanic, and Breast Cancer. Ripley is a synthetic dataset that is generated from a two-dimensional (D = 2) Gaussian mixture model, and 250 and 1,000 samples are provided for training and test, respectively. The number of classes is two (C = 2), and each class comprises two components. The remaining four datasets are all two-class (C = 2) datasets, which comprise different data sizes and dimensionality. Since they contain 100 training/test splits, we repeated experiments 100 times and then calculated average statistics.
For comparison, we used three kernel Bayesian methods: a GP classifier, an MPG classifier (Tresp 2001; Luo & Sun 2017), and an RVM (Tipping 2001), which are closely related to the SDGM from the perspective of sparsity, multimodality, and Bayesian learning, as described in Section 2. In the evaluation, we compared the recognition error rates for discriminability and the number of nonzero weights for sparsity on the test data. The results of RVM were cited from (Tipping 2001). By way of summary, the statistics were normalized by those of the SDGM, and the overall mean was shown.
Table 1 shows the recognition error rates and the number of nonzero weights for each method. The results in Table 1 demonstrated that the SDGM achieved a better accuracy on average compared to the other kernel Bayesian methods. The SDGM is developed based on a Gaussian mixture model and is particularly effective for data where a Gaussian distribution can be assumed, such as the Ripley dataset. Since the SDGM explicitly models multimodality, it could more accurately represent the sharp changes in decision boundaries near the border of components compared to the RVM, as shown in Figure 1. Although the SDGM did not necessarily outperform the other methods in all datasets, it achieved the best accuracy on average. In terms of sparsity, the number of initial weights for the SDGM is the same as MGP, and the SDGM reduced 90.0–99.5% of weights from the initial state due to the sparse Bayesian learning, which leads to drastically efficient use of memory compared to non-sparse classifiers (GP and MGP). The results above indicated that the SDGM demonstrated generalization capability and a sparsity simultaneously.

heading: 4.2 IMAGE CLASSIFICATION
text: In this experiment, the SDGM is embedded into a deep neural network. Since the SDGM is differentiable with respect to the weights, SDGM can be embedded into a deep NN as a module and is trained in an end-to-end manner. In particular, the SDGM plays the same role as the softmax function since the SDGM calculates the posterior probability of each class given an input vector. We can show that a fully connected layer with the softmax is equivalent to the discriminative model based on a single Gaussian distribution for each class by applying a simple transformation (see Appendix A), whereas the SDGM is based on the Gaussian mixture model.
To verify the difference between them, we conducted image classification experiments. Using a CNN with a softmax function as a baseline, we evaluated the capability of SDGM by replacing softmax with the SDGM. We also used a CNN with a softmax function trained with L1 regularization, a CNN with a large margin softmax (Liu et al. 2016), and a CNN with the discriminative GMM as other baselines.
In this experiment, we used the original form of the SDGM. To achieve sparse optimization during end-to-end training, we employed an approximated sparse Bayesian learning based on Gaussian dropout proposed by Molchanov et al. (2017). This is because it is difficult to execute the learning algorithm in Section 3.4 with backpropagation due to large computational costs for inverse matrix calculation of the Hessian in (16), which takes O(N3).

heading: 4.2.1 DATASETS AND EXPERIMENTAL SETUPS
text: We used the following datasets and experimental settings in this experiment.
MNIST: This dataset includes 10 classes of handwritten binary digit images of size 28×28 (LeCun et al. 1998). We used 60,000 images as training data and 10,000 images as testing data. As a feature extractor, we used a simple CNN that consists of five convolutional layers with four max pooling layers between them and a fully connected layer. To visualize the learned CNN features, we first set the output dimension of the fully connected layer of the baseline CNN as two (D = 2). Furthermore, we tested by increasing the output dimension of the fully connected layer from two to ten (D = 10).
Fashion-MNIST: Fashion-MNIST (Xiao et al. 2017) includes 10 classes of binary fashion images with a size of 28 × 28. It includes 60,000 images for training data and 10,000 images for testing data. We used the same CNN as in MNIST with 10 as the output dimension.
CIFAR-10 and CIFAR-100: CIFAR-10 and CIFAR-100 (Krizhevsky & Hinton 2009) consist of 60,000 32 × 32 color images in 10 classes and 100 classes, respectively. There are 50,000 training images and 10,000 test images for both datasets. For these datasets, we trained DenseNet (Huang et al. 2017) with a depth of 40 and a growth rate of 12 as a baseline CNN.
ImageNet: ImageNet classification dataset (Russakovsky et al. 2015) includes 1,000 classes of generic object images with a size of 224 × 224. It consists of 1,281,167 training images, 50,000 validation images, and 100,000 test images. For this dataset, we used MobileNet (Howard et al. 2017) as a baseline CNN.
It should be noted that we did not employ additional techniques to increase classification accuracy such as hyperparameter tuning and pre-trained models; therefore, the accuracy of the baseline model did not reach the state-of-the-art. This is because we considered that it is not essential to confirm the effectiveness of the proposed method.

heading: 4.2.2 RESULTS
text: Figure 5 shows the two-dimensional feature embeddings on the MNIST dataset. Different feature embeddings were acquired for each method. When softmax was used, the features spread in a fan shape and some parts of the distribution overlapped around the origin. However, when the SDGM was used, the distribution for each class exhibited an ellipse shape and margins appeared between the
class distributions. This is because the SDGM is based on a Gaussian mixture model and functions to push the features into a Gaussian shape.
Table 2 shows the recognition error rates on each dataset. SDGM achieved better performance than softmax. Although sparse learning was ineffective in two out of six comparisons according to the comparison with the discriminative GMM, replacing softmax with SDGM was effective in all the comparisons. As shown in Figure 5, SDGM can create margins between classes by pushing the features into a Gaussian shape. This phenomenon positively affected classification capability. Although large-margin softmax, which has the effect of increasing the margin, and the discriminative GMM, which can represent multimodality, also achieved relatively high accuracy, the SDGM can achieve the same level of accuracy with sparse weights.

heading: 5 CONCLUSION
text: In this paper, we proposed a sparse classifier based on a Gaussian mixture model (GMM), which is named sparse discriminative Gaussian mixture (SDGM). In the SDGM, a GMM-based discriminative model was trained by sparse Bayesian learning. This learning algorithm improved the generalization capability by obtaining a sparse solution and automatically determined the number of components by removing redundant components. The SDGM can be embedded into neural networks (NNs) such as convolutional NNs and could be trained in an end-to-end manner.
In the experiments, we demonstrated that the SDGM could reduce the number of weights via sparse Bayesian learning, thereby improving its generalization capability. The comparison using benchmark datasets suggested that SDGM outperforms the conventional kernel Bayesian classifiers. We also demonstrated that SDGM outperformed the fully connected layer with the softmax function when it was used as the last layer of a deep NN.
One of the limitations of this study is that the proposed sparse learning reduces redundant Gaussian components but cannot obtain the optimal number of components, which should be improved in future work. Since the learning of the proposed method can be interpreted as the incorporation of the EM algorithm into the sparse Bayesian learning, we will tackle a theoretical analysis by utilizing the proofs for the EM algorithm (Wu 1983) and the sparse Bayesian learning (Faul & Tipping 2001). Furthermore, we would like to tackle the theoretical analysis of error bounds using the PAC-Bayesian theorem. We will also develop a sparse learning algorithm for a whole deep NN structure including the feature extraction part. This will improve the ability of the CNN for larger data classification. Further applications using the probabilistic property of the proposed model such as semi-supervised learning, uncertainty estimation, and confidence calibration will be considered.

heading: ACKNOWLEDGMENTS
text: This work was supported in part by JSPS KAKENHI Grant Number JP17K12752 and JST ACT-I Grant Number JPMJPR18UO.

heading: A RELATIONSHIP BETWEEN THE DISCRIMINATIVE GAUSSIAN AND LOGISTIC REGRESSION
text: We show that a fully connected layer with the softmax function, or logistic regression, can be regarded as a discriminative model based on a Gaussian distribution by utilizing transformation of the equations. Let us consider a case in which the class-conditional probability P (x|c) is a Gaussian distribution. In this case, we can omit m from the equations (3)–(6).
If all classes share the same covariance matrix and the mixture weight πcm, the terms πcm in (1), x21, x1x2, . . . , x1xD, x 2 2, x2x3, . . . , x2xD, . . . , x 2 D in (2), and − 12sc11, . . .,− 1 2scDD in (6) can be canceled; hence the calculation of the posterior probability P (c|x) is also simplified as
P (c|x) = exp(wc Tφ)∑C
c=1 exp(wc Tφ)
,
where
wc = [logP (c)− 1
2 D∑ i=1 D∑ j=1 scijµciµcj + D 2 log 2π + 1 2 log |Σc|, D∑ i=1 sci1µci, · · ·, D∑ i=1 sciDµci] T,
φ = [ 1,xT ]T .
This is equivalent to a fully connected layer with the softmax function, or linear logistic regression.

heading: B EVALUATION OF CHARACTERISTICS USING SYNTHETIC DATA
text: To evaluate the characteristics of the SDGM, we conducted classification experiments using synthetic data. The dataset comprises two classes. The data were sampled from a Gaussian mixture model with eight components for each class. The numbers of training data and test data were 320 and 1,600, respectively. The scatter plot of this dataset is shown in Figure 6.
In the evaluation, we calculated the error rates for the training data and the test data, the number of components after training, the number of nonzero weights after training, and the weight reduction ratio (the ratio of the number of the nonzero weights to the number of initial weights), by varying the number of initial components as 2, 4, 8, . . . , 20. We repeated evaluation five times while regenerating the training and test data and calculated the average value for each evaluation criterion. We used the dual form of the SDGM in this experiment.
Figure 6 displays the changes in the learned class boundaries according to the number of initial components. When the number of components is small, such as that shown in Figure 6(a), the decision boundary is simple; therefore, the classification performance is insufficient. However, according to the increase in the number of components, the decision boundary fits the actual class boundaries. It is noteworthy that the SDGM learns the GMM as a discriminative model instead of a generative model; an appropriate decision boundary was obtained even if the number of components for the model is less than the actual number (e.g., 6(c)).
Figure 7 shows the evaluation results of the characteristics. Figures 7(a), (b), (c), and (d) show the recognition error rate, number of components after training, number of nonzero weights after training, and weight reduction ratio, respectively. The horizontal axis shows the number of initial components in all the graphs.
In Figure 7(a), the recognition error rates for the training data and test data are almost the same with the few numbers of components and decrease according to the increase in the number of initial components while it is 2 to 6. This implied that the representation capability was insufficient when the number of components was small, and that the network could not accurately separate the classes. Meanwhile, changes in the training and test error rates were both flat when the number of initial components exceeded eight, even though the test error rates were slightly higher than the training error rate. In general, the training error decreases and the test error increases when the complexity of
the classifier is increased. However, the SDGM suppresses the increase in complexity using sparse Bayesian learning, thereby preventing overfitting.
In Figure 7(b), the number of components after training corresponds to the number of initial components until the number of initial components is eight. When the number of initial components exceeds ten, the number of components after training tends to be reduced. In particular, eight components are reduced when the number of initial components is 20. The results above indicate the SDGM can reduce unnecessary components.
From the results in Figure 7(c), we confirm that the number of nonzero weights after training increases according to the increase in the number of initial components. This implies that the complexity of the trained model depends on the number of initial components, and that the minimum number of components is not always obtained.
Meanwhile, in Figure 7(d), the weight reduction ratio increases according to the increase in the number of initial components. This result suggests that the larger the number of initial weights, the more weights were reduced. Moreover, the weight reduction ratio is greater than 99 % in any case. The results above indicate that the SDGM can prevent overfitting by obtaining high sparsity and can reduce unnecessary components.

heading: C DETAILS OF INITIALIZATION
text: In the experiments during this study, each trainable parameters for the m-th component of the c-th class were initialized as follows (H = 1 + D(D + 3)/2, where D is the input dimension, for the original form and H = N , where N is the number of the training data, for the kernelized form):
• wcm (for the original form): A zero vector 0 ∈ RH .
• ψcm (for the kernelized form): A zero vector 0 ∈ RH .
• αcm: An all-ones vector 1 ∈ RH . • πcm: A scalar 1∑C c=1 Mc
, where C is the number of classes and Mc is the number of components for the c-th class.
• rncm: Initialized based on the results of k-means clustering applied to the training data; rncm = 1 if the n-th sample belongs to class c and is assigned to the m-th component by k-means clustering, rncm = 0 otherwise.

Review summaries:
The paper proposes a sparse classifier via  discriminative GMM. This model is trained based on sparse Bayesian learning. The sparsity constraint removes redundant Gaussian components which results in  reducing the number of parameters and improving the generalization. This framework can potentially be embedded into the deep models and trained in an end-to-end fashion. The main motivation is that the proposed model (i.e., SDGM,)  can consider multimodal data while conventional softmax classifiers only assume unimodality for each class. Experimental results show the superiority of the SDGM over existing softmax-based discriminative models.