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The unintentional electromagnetic (EM) radiation from electronic information equipment will cause the leakage of information. EM information leakage identification is an important research area of EM information security. Previous studies only focused on a particular type of information leakage, ignoring that there are multiple types of information contained in the leaked EM signal. This paper presents a multi-dimensional information leakage model for the first time, which reveals the multiple information leakage risks in an unintentional EM radiation. The EM information leakage of computer monitors is taken as an example to verify the model. Furthermore, a Convolutional Neural Network (CNN) based multi-dimensional information identification method is proposed. This novel method designed a unified neural network architecture to identify multiple information leaked by the same EM radiation from the monitor. It overcomes the limitation of available identification methods that need to definite leakage characteristics for each type of information leakage. The effectiveness of the method is validated by contrast experiments.