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The training process can be formulated as an optimization problem (1) where (x0,wL) is a pair of training data. Each transformation is differentiable, thus the gradient-based learning algorithm can be employed for training CNNs. The gradients with respect to each transformation inputs and parameters can be formulated as follows: (2) (3) where is the Jacobian of fn with respect to x evaluated at the point (xn-1 wn), and is the Jacobian of fn with respect to w evaluated at the same point. (3) are the main ideas of Backpropagation (BP) algorithm.

G. rectified linear unit [5], local contrast normalization [6], local response normalization [7] and dropout [8]. On the other hand, engineering studies have never been stopped. Handwritten character recognition [9, 10], natural image processing [7, 11], etc. are well-known engineering application of CNNs. The most interesting work had been done by Krizhevsky et al. who won the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) in 2012 [7]. 1% with handcrafted features. In ILSVRC 2013, an approach from Zeiler et al.

In general, 30% labeled samples leads to good results on both unlabeled training data and test data which indicates more label information makes a contribution to better recognition rate. As shown in Table 1- 3, the proposed method results in a better recognition rate in both unlabeled train data and the test data on these three databases. Fig. (4)) Recognition rate of unlabeled train data with 20% labeled data on PIE. Table 1 Recognition rates (%) on Yale. 2 (14) Table 2 Recognition rates (%) on ORL.