Canonical correlation analysis (CCA) is a typical statistical model used to analyze the correlation components between different view representations of the same objects. When the label information is available with the data representations, CCA can be extended to its discriminative counterparts by incorporating supervision in the analysis. Although most discriminative variants of CCA have achieved improved results, nearly all of their objective functions are nonconvex, implying that optimal solutions are difficult to obtain. More important, that cross-view representations from the same sample should be consistent, that is, the cross-view semantic consistency has however not been modeled. To overcome these drawbacks, in this article, we propose a discriminant semantic correlation analysis (DSCA) model by modeling the cross-view semantic consistency for each object in the sample space rather than in the commonly used feature space. To boost the nonlinear discriminating capability of DSCA, we extend it from the Euclidean to the geodesic space by transforming the metric and incorporating both the cross-view semantic and representation correlation information and consequently obtain our final model with convex objective, namely, convex DSCA (C-DSCA). Finally, with extensive experiments and comparisons, we validate the effectiveness and superiority of the proposed method.