Support Matrix Machines Table 1.Summary of four data sets Data sets #positive #negative dimension EEG alcoholism 77 45 256×64 EEG emotion 1286 1334 31×10 (a)B-SVM (b)R-GLM (c)SMM students face 200 200 200×200 INRIA person 607 1214 160×96 Figure 1.(a).(b)and (c)display the values of normalized regres- sion matrix of B-SVM.R-GLM and SMM respectively. The EEG alcoholism data set arises to examine EEG cor- relates of genetic predisposition to alcoholism.It contains two groups of subjects:alcoholic and control.For each subject,64 channels of electrodes are placed and the volt- age values are recorded at 256 time points. The EEG emotion data set (Zhu et al.,2014;Zheng et al., 2014)focuses on EEG emotion analysis,which is obtained by showing some positive and negative emotional movie 01 clips to persons and then recording the EEG signal via ESI (a)Synthetic data with Gaussian noise NeuroScan System from 31 pairs.Each pair contain 10 data points(two channels for one pair,and each channel contains five frequency bands).There are 2620 movie clips chosen to evoke the target emotion,such as Titanic,Kung Fu Panda and so on. The student face data set contains 400 photos of Stanford University medical students (Nazir et al..2010).which con- sists of 200 males and 200 females.Each sample is a 200 x 200 gray level image. 05 000601Q015002002☒0.0 The INRIA person data set was collected to detect whether (b)Synthetic data with salt and pepper noise there exist people in the image.We normalize the samples into 160x 96 gray images and remove the same person with Figure 2.Classification accuracy on synthetic data with different different aspects.Combining with the negative samples,we levels of noises.We use Gaussian noise with 0 mean and standard obtain 1821 samples in total. derivation from 0.01 to 1 in(a),and salt and pepper noise with density from 0.001 to 0.035 in (b). We summarize the main information of these data sets in Table 1.For the student face and INRIA person data set- s,we directly use the pixels as input features without any We add different levels of Gaussian noise and salt and pep- advanced visual features. per noise on the test data,and repeat this procedure ten For each of the compared methods,we randomly sam- times to compute the mean and standard deviation of clas- ple 70%of the data set for training and the rest for test- sification accuracy.The results are shown in Figure 2.It ing.All the hyperparameters involved are selected vi- is clear that all methods achieve comparable performance a cross validation.More specifically,we select C from on clean data,but SMM is more robust with respect to high {1×10-3,2×10-3,5×10-3,1×10-2,2×10-2,5× level of noises. 10-2..,1×103,2×10}.For each C,we tune r man- ually to make the rank of classifier matrix varied from I to 5.3.Classification Accuracy on Real-World Data the size of the matrix.We repeat this procedure ten times to We apply SMM to EEG and image classification problem- compute the mean and standard deviation of the classifica- s,and compare its performance with B-SVM(Pirsiavash tion accuracy.Table 2 shows the classification accuracy of et al.,2009).R-GLM (Zhou Li,2014).and the standard the four methods.We can see that SMM achieves the best linear SVM (L-SVM)(Cortes Vapnik,1995).We use performance on all the four data sets. four real-world matrix classification data sets:the EEG al- http://kdd.ics.uci.edu/databases/eeg/ coholism,the EEG emotion,the students face and INRIA eeg.html person. http://pascal.inrialpes.fr/data/human/Support Matrix Machines 20 40 60 80 100 10 20 30 40 50 60 70 80 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 20 40 60 80 100 10 20 30 40 50 60 70 80 −6 −4 −2 0 2 4 6 8 20 40 60 80 100 10 20 30 40 50 60 70 80 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 (a) B-SVM (b) R-GLM (c) SMM Figure 1. (a), (b) and (c) display the values of normalized regres￾sion matrix of B-SVM, R-GLM and SMM respectively. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 standard deviation of Gaussian noise accuracy B−SVM R−GLM SMM (a) Synthetic data with Gaussian noise 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 density of salt and pepper noise accuracy B−SVM R−GLM SMM (b) Synthetic data with salt and pepper noise Figure 2. Classification accuracy on synthetic data with different levels of noises. We use Gaussian noise with 0 mean and standard derivation from 0.01 to 1 in (a), and salt and pepper noise with density from 0.001 to 0.035 in (b). We add different levels of Gaussian noise and salt and pep￾per noise on the test data, and repeat this procedure ten times to compute the mean and standard deviation of clas￾sification accuracy. The results are shown in Figure 2. It is clear that all methods achieve comparable performance on clean data, but SMM is more robust with respect to high level of noises. 5.3. Classification Accuracy on Real-World Data We apply SMM to EEG and image classification problem￾s, and compare its performance with B-SVM (Pirsiavash et al., 2009), R-GLM (Zhou & Li, 2014), and the standard linear SVM (L-SVM) (Cortes & Vapnik, 1995). We use four real-world matrix classification data sets: the EEG al￾coholism, the EEG emotion, the students face and INRIA person. Table 1. Summary of four data sets Data sets #positive #negative dimension EEG alcoholism 77 45 256×64 EEG emotion 1286 1334 31×10 students face 200 200 200×200 INRIA person 607 1214 160×96 The EEG alcoholism data set2 arises to examine EEG cor￾relates of genetic predisposition to alcoholism. It contains two groups of subjects: alcoholic and control. For each subject, 64 channels of electrodes are placed and the volt￾age values are recorded at 256 time points. The EEG emotion data set (Zhu et al., 2014; Zheng et al., 2014) focuses on EEG emotion analysis, which is obtained by showing some positive and negative emotional movie clips to persons and then recording the EEG signal via ESI NeuroScan System from 31 pairs. Each pair contain 10 data points (two channels for one pair, and each channel contains five frequency bands). There are 2620 movie clips chosen to evoke the target emotion, such as Titanic, Kung Fu Panda and so on. The student face data set contains 400 photos of Stanford University medical students (Nazir et al., 2010), which con￾sists of 200 males and 200 females. Each sample is a 200 × 200 gray level image. The INRIA person data set3 was collected to detect whether there exist people in the image. We normalize the samples into 160×96 gray images and remove the same person with different aspects. Combining with the negative samples, we obtain 1821 samples in total. We summarize the main information of these data sets in Table 1. For the student face and INRIA person data set￾s, we directly use the pixels as input features without any advanced visual features. For each of the compared methods, we randomly sam￾ple 70% of the data set for training and the rest for test￾ing. All the hyperparameters involved are selected vi￾a cross validation. More specifically, we select C from {1 × 10−3 , 2 × 10−3 , 5 × 10−3 , 1 × 10−2 , 2 × 10−2 , 5 × 10−2 . . . , 1 × 103 , 2 × 103}. For each C, we tune τ man￾ually to make the rank of classifier matrix varied from 1 to the size of the matrix. We repeat this procedure ten times to compute the mean and standard deviation of the classifica￾tion accuracy. Table 2 shows the classification accuracy of the four methods. We can see that SMM achieves the best performance on all the four data sets. 2http://kdd.ics.uci.edu/databases/eeg/ eeg.html 3http://pascal.inrialpes.fr/data/human/