Recent studies show that mental disorders change the functional organization of the brain, which could be investigated via various imaging techniques. Analyzing such changes is becoming critical as it could provide new biomarkers for diagnosing and monitoring the progression of the diseases. Functional connectivity analysis studies the covary activity of neuronal populations in different brain regions. The sparse inverse covariance estimation (SICE), also known as graphical LASSO, is one of the most important tools for functional connectivity analysis, which estimates the interregional partial correlations of the brain. Although being increasingly used for predicting mental disorders, SICE is basically a generative method that may not necessarily perform well on classifying neuroimaging data. In this paper, we propose a learning framework to effectively improve the discriminative power of SICEs by taking advantage of the samples in the opposite class. We formulate our objective as convex optimization problems for both one-class and two-class classifications. By analyzing these optimization problems, we not only solve them efficiently in their dual form, but also gain insights into this new learning framework. The proposed framework is applied to analyzing the brain metabolic covariant networks built upon FDG-PET images for the prediction of the Alzheimer's disease, and shows significant improvement of classification performance for both one-class and two-class scenarios. Moreover, as SICE is a general method for learning undirected Gaussian graphical models, this paper has broader meanings beyond the scope of brain research.