Table 1 lists the average absolute error between the estimated class-prior and the true value. Overall, our proposed dimension-reduction method tends to outperform other methods, meaning that our method provides useful low-dimensional representation. Except for the ijcnn1 data set, the error of FDA tends to be larger than the other methods, implying that regarding U data as N data does not help in class-prior estimation. For the mushrooms and a9a data sets, applying the unsupervised dimension-reduction method, PCA, does not improve the estimation accuracy, while our method reduces the error of class-prior estimation. In particular, for the 20 Newsgroups data set, the existing approaches (PCA, FDA, and PNRL) perform poorly. In contrast, applying our method significantly reduces the error of class-prior estimation.

Table 1:
Average Absolute Error (with Standard Error) between the Estimated Class-Prior and the True Value on Benchmark Data Sets over 20 Trials.
PCA
Data Set$θP$None$⌊d/4⌋$$⌊d/2⌋$$⌊3d/4⌋$FDAPNRLPURL
ijcnn1 0.3 0.23 (0.02) 0.26 (0.11) 0.26 (0.11) 0.28 (0.04) 0.03 (0.010.26 (0.08) 0.21 (0.07)
0.5 0.18 (0.05) 0.14 (0.09) 0.14 (0.09) 0.17 (0.06) 0.04 (0.010.21 (0.08) 0.19 (0.07)
0.7 0.08 (0.010.11 (0.05) 0.11 (0.05) 0.10 (0.04) 0.07 (0.010.11 (0.05) 0.10 (0.01
phishing 0.3 0.02 (0.000.02 (0.000.02 (0.000.02 (0.000.04 (0.02) 0.03 (0.01) 0.02 (0.00
0.5 0.01 (0.000.01 (0.000.01 (0.000.01 (0.000.07 (0.03) 0.04 (0.02) 0.03 (0.02)
0.7 0.02 (0.000.02 (0.000.02 (0.000.02 (0.000.11 (0.04) 0.05 (0.03) 0.02 (0.00
mushrooms 0.3 0.05 (0.01) 0.05 (0.01) 0.05 (0.01) 0.05 (0.01) 0.09 (0.03) 0.03 (0.000.03 (0.00
0.5 0.05 (0.010.05 (0.010.05 (0.010.05 (0.010.16 (0.03) 0.04 (0.010.04 (0.00
0.7 0.03 (0.010.03 (0.000.03 (0.000.03 (0.010.20 (0.06) 0.03 (0.000.04 (0.03)
a9a 0.3 0.11 (0.02) 0.11 (0.02) 0.11 (0.02) 0.11 (0.02) 0.05 (0.010.08 (0.03) 0.04 (0.00
0.5 0.10 (0.02) 0.10 (0.02) 0.10 (0.02) 0.10 (0.02) 0.09 (0.04) 0.09 (0.03) 0.04 (0.01
0.7 0.08 (0.03) 0.08 (0.03) 0.08 (0.03) 0.08 (0.03) 0.18 (0.06) 0.08 (0.03) 0.04 (0.01
MNIST 0.3 0.09 (0.02) 0.09 (0.02) 0.09 (0.02) 0.09 (0.02) 0.27 (0.01) 0.01 (0.000.05 (0.02)
0.5 0.15 (0.11) 0.15 (0.11) 0.15 (0.11) 0.15 (0.11) 0.46 (0.01) 0.03 (0.000.06 (0.03)
0.7 0.60 (0.21) 0.60 (0.21) 0.60 (0.21) 0.60 (0.21) 0.65 (0.02) 0.06 (0.010.07 (0.01
F-MNIST 0.3 0.02 (0.000.02 (0.000.02 (0.000.02 (0.000.25 (0.01) 0.03 (0.000.03 (0.00
0.5 0.03 (0.000.03 (0.000.03 (0.000.03 (0.000.45 (0.01) 0.02 (0.000.04 (0.03)
0.7 0.03 (0.000.03 (0.000.03 (0.000.03 (0.000.66 (0.02) 0.02 (0.000.07 (0.03)
20 News 0.3 0.04 (0.000.04 (0.010.04 (0.000.04 (0.000.29 (0.00) 0.29 (0.09) 0.03 (0.01
0.5 0.08 (0.03) 0.06 (0.010.07 (0.010.08 (0.03) 0.49 (0.00) 0.25 (0.07) 0.05 (0.01
0.7 0.69 (0.00) 0.69 (0.00) 0.69 (0.00) 0.69 (0.00) 0.69 (0.00) 0.13 (0.030.07 (0.01
PCA
Data Set$θP$None$⌊d/4⌋$$⌊d/2⌋$$⌊3d/4⌋$FDAPNRLPURL
ijcnn1 0.3 0.23 (0.02) 0.26 (0.11) 0.26 (0.11) 0.28 (0.04) 0.03 (0.010.26 (0.08) 0.21 (0.07)
0.5 0.18 (0.05) 0.14 (0.09) 0.14 (0.09) 0.17 (0.06) 0.04 (0.010.21 (0.08) 0.19 (0.07)
0.7 0.08 (0.010.11 (0.05) 0.11 (0.05) 0.10 (0.04) 0.07 (0.010.11 (0.05) 0.10 (0.01
phishing 0.3 0.02 (0.000.02 (0.000.02 (0.000.02 (0.000.04 (0.02) 0.03 (0.01) 0.02 (0.00
0.5 0.01 (0.000.01 (0.000.01 (0.000.01 (0.000.07 (0.03) 0.04 (0.02) 0.03 (0.02)
0.7 0.02 (0.000.02 (0.000.02 (0.000.02 (0.000.11 (0.04) 0.05 (0.03) 0.02 (0.00
mushrooms 0.3 0.05 (0.01) 0.05 (0.01) 0.05 (0.01) 0.05 (0.01) 0.09 (0.03) 0.03 (0.000.03 (0.00
0.5 0.05 (0.010.05 (0.010.05 (0.010.05 (0.010.16 (0.03) 0.04 (0.010.04 (0.00
0.7 0.03 (0.010.03 (0.000.03 (0.000.03 (0.010.20 (0.06) 0.03 (0.000.04 (0.03)
a9a 0.3 0.11 (0.02) 0.11 (0.02) 0.11 (0.02) 0.11 (0.02) 0.05 (0.010.08 (0.03) 0.04 (0.00
0.5 0.10 (0.02) 0.10 (0.02) 0.10 (0.02) 0.10 (0.02) 0.09 (0.04) 0.09 (0.03) 0.04 (0.01
0.7 0.08 (0.03) 0.08 (0.03) 0.08 (0.03) 0.08 (0.03) 0.18 (0.06) 0.08 (0.03) 0.04 (0.01
MNIST 0.3 0.09 (0.02) 0.09 (0.02) 0.09 (0.02) 0.09 (0.02) 0.27 (0.01) 0.01 (0.000.05 (0.02)
0.5 0.15 (0.11) 0.15 (0.11) 0.15 (0.11) 0.15 (0.11) 0.46 (0.01) 0.03 (0.000.06 (0.03)
0.7 0.60 (0.21) 0.60 (0.21) 0.60 (0.21) 0.60 (0.21) 0.65 (0.02) 0.06 (0.010.07 (0.01
F-MNIST 0.3 0.02 (0.000.02 (0.000.02 (0.000.02 (0.000.25 (0.01) 0.03 (0.000.03 (0.00
0.5 0.03 (0.000.03 (0.000.03 (0.000.03 (0.000.45 (0.01) 0.02 (0.000.04 (0.03)
0.7 0.03 (0.000.03 (0.000.03 (0.000.03 (0.000.66 (0.02) 0.02 (0.000.07 (0.03)
20 News 0.3 0.04 (0.000.04 (0.010.04 (0.000.04 (0.000.29 (0.00) 0.29 (0.09) 0.03 (0.01
0.5 0.08 (0.03) 0.06 (0.010.07 (0.010.08 (0.03) 0.49 (0.00) 0.25 (0.07) 0.05 (0.01
0.7 0.69 (0.00) 0.69 (0.00) 0.69 (0.00) 0.69 (0.00) 0.69 (0.00) 0.13 (0.030.07 (0.01

Notes: “None” means that the class-prior is estimated without dimension-reduction methods, PCA is the principal component analysis, FDA is Fisher's discriminant analysis, and PNRL is the supervised counterpart of the proposed method. The class-prior is estimated by the method based on kernel mean embedding. The boldface denotes the best and comparable approaches in terms of the average absolute error according to the t-test at the significance level 5%.

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