With brain disorders being highly heterogeneous and affecting both the structure and function of multiple subsystems of the brain, it is crucial to develop learning frameworks to study, characterize, and utilize the interplay of associated information from different modalities and brain subsystems. However, most learning frameworks for neuroimaging data either do not consider the target assessment information during the unsupervised extraction of lower-dimensional brain subspaces, or can extract only high-dimensional importance associations as an ordered list of involved features when making diagnostic predictions, making manual interpretation at the level of subspaces difficult. Starting with the use of learning models to study how structural and functional feature modalities differ in terms of their saliency towards diagnostic classification, this work presents novel subspace learning frameworks to understand various active and independent subspaces within the brain. This is achieved by performing a decomposition in the saliency space to extract robust multimodal subspaces that define the most significant change in a given cognitive or biological trait. Through rigorous cross-validation procedures on Alzheimer's disease (AD) data, the framework not only uncovers AD-related brain regions in the associated brain subspaces, but also enables automated identification of multiple collectively varying structural and functional sub-systems of the brain. This framework is further extended to find independently salient subspaces using deep learning models that can handle high-dimensional voxel-level neuroimaging features to automatically uncover intrinsic networks in the brain associated with disease-specific clinical assessments. Additionally, the utilization and study of sub-domain structures in deep learning models for neuroimaging are also explored by introducing a flexible deep learning framework to effectively incorporate multimodal features while accounting for and exploiting the heterogeneity in the sub-domains of the brain. Experiments with this framework demonstrate that the discriminatory information from structural and functional sub-domains can be better recovered and analyzed if the complexity of the subspace structure in the model can be tuned to reflect the extent of non-linearity with which each sub-domain encodes the information.