In Multi-Dimensional Classification (MDC), an instance is associated with multiple class variables, each class variable belonging to different semantic spaces. Since the semantic spaces are heterogeneous, directly learning label correlations is challenging because labels from different semantic spaces cannot be directly compared. Furthermore, some existing MDC methods either focus on the global dependencies among class variables or the local label correlations. This paper addresses the MDC problem by considering global and local label correlations simultaneously. We propose a novel algorithm called MGLC (Multi-Dimensional Classification via Global and Local label Correlation). MGLC addresses the heterogeneity of semantic spaces by using one-hot label encoding to transform the original MDC output space into an encoded label space. Subsequently, a multi-dimensional classifier is induced based on sparse learning with Frobenius norm regularization. Finally, the global and local label correlations in the encoded label space are exploited to enhance the MDC. Comparative experiments on eight datasets demonstrate that MGLC outperforms state-of-the-art multi-dimensional classification algorithms.