The assessment of Alzheimer's Disease (AD) and Mild Cognitive Impairment (MCI) associated with brain changes remains a challenging task. Recent studies have demonstrated that combination of multi-modality imaging techniques can better reflect pathological characteristics and contribute to more accurate diagnosis of AD and MCI. In this paper, we propose a novel tensor-based multi-modality feature selection and regression method for diagnosis and biomarker identification of AD and MCI from normal controls. Specifically, we leverage the tensor structure to exploit high-level correlation information inherent in the multi-modality data, and investigate tensor-level sparsity in the multilinear regression model. We present the practical advantages of our method for the analysis of ADNI data using three imaging modalities (VBM-MRI, FDG-PET and AV45-PET) with clinical parameters of disease severity and cognitive scores. The experimental results demonstrate the superior performance of our proposed method against the state-of-the-art for the disease diagnosis and the identification of disease-specific regions and modality-related differences. The code for this work is publicly available at https://github.com/junfish/BIOS22.