Classification of Structural MRI Images in Alzheimer's Disease from the Perspective of Ill-Posed Problems

Background: Machine learning neuroimaging researchers have often relied on regularization techniques when classifying MRI images. Although these were originally introduced to deal with "ill-posed'' problems it is rare to find studies that evaluate the ill-posedness of MRI image classification problems. In addition, to avoid the effects of the "curse of dimensionality'' very often dimension reduction is applied to the data. Methodology: Baseline structural MRI data from cognitively normal and Alzheimer's disease (AD) patients from the AD Neuroimaging Initiative database were used in this study. We evaluated here the ill-posedness of this classification problem across different dimensions and sample sizes and its relationship to the performance of regularized logistic regression (RLR), linear support vector machine (SVM) and linear regression classifier (LRC). In addition, these methods were compared with their principal components space counterparts. Principal Findings: In voxel space the prediction performance of all methods increased as sample sizes increased. They were not only relatively robust to the increase of dimension, but they often showed improvements in accuracy. We linked this behavior to improvements in conditioning of the linear kernels matrices. In general the RLR and SVM performed similarly. Surprisingly, the LRC was often very competitive when the linear kernel matrices were best conditioned. Finally, when comparing these methods in voxel and principal component spaces, we did not find large differences in prediction performance. Conclusions and Significance: We analyzed the problem of classifying AD MRI images from the perspective of linear ill-posed problems. We demonstrate empirically the impact of the linear kernel matrix conditioning on different classifiers' performance. This dependence is characterized across sample sizes and dimensions. In this context we also show that increased dimensionality does not necessarily degrade performance of machine learning methods. In general, this depends on the nature of the problem and the type of machine learning method.