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The social and financial costs associated with Alzheimer's disease (AD) result in significant burdens on our society. In order to understand the causes of this disease, public-private partnerships such as the Alzheimer's Disease Neuroimaging Initiative (ADNI) release data into the scientific community. These data are organized into various modalities (genetic, brain-imaging, cognitive scores, diagnoses, etc.) for analysis. Many statistical learning approaches used in medical image analysis do not explicitly take advantage of this multimodal data structure. In this work we propose a novel objective function and optimization algorithm that is designed to handle multimodal information for the prediction and analysis of AD. Our approach relies on robust matrix-factorization and row-wise sparsity provided by the ℓ2,1- norm in order to integrate multimodal data provided by the ADNI. These techniques are jointly optimized with a classification task to guide the feature selection in our proposed Task Balanced Multimodal Feature Selection method. Our results, when compared against some widely used machine learning algorithms, show improved balanced accuracies, precision, and Matthew's correlation coefficients for identifying cognitive decline. In addition to the improved prediction performance, our method is able to identify brain and genetic biomarkers that are of interest to the clinical research community. Our experiments validate existing brain biomarkers and single nucleotide polymorphisms located on chromosome 11 and detail novel polymorphisms on chromosome 10 that, to the best of the authors' knowledge, have not previously been reported. We anticipate that our method will be of interest to the greater research community and have released our method's code online.11Code is provided at: https://github.com/minds-mines/TBMFSjl