Deep sparse multi-task learning for feature selection in Alzheimer’s disease diagnosis

Abstract

Recently, neuroimaging-based Alzheimer’s disease (AD) or mild cognitive impairment (MCI) diagnosis has attracted researchers in the field, due to the increasing prevalence of the diseases. Unfortunately, the unfavorable high-dimensional nature of neuroimaging data, but a limited small number of samples available, makes it challenging to build a robust computer-aided diagnosis system. Machine learning techniques have been considered as a useful tool in this respect and, among various methods, sparse regression has shown its validity in the literature. However, to our best knowledge, the existing sparse regression methods mostly try to select features based on the optimal regression coefficients in one step. We argue that since the training feature vectors are composed of both informative and uninformative or less informative features, the resulting optimal regression coefficients are inevidently affected by the uninformative or less informative features. To this end, we first propose a novel deep architecture to recursively discard uninformative features by performing sparse multi-task learning in a hierarchical fashion. We further hypothesize that the optimal regression coefficients reflect the relative importance of features in representing the target response variables. In this regard, we use the optimal regression coefficients learned in one hierarchy as feature weighting factors in the following hierarchy, and formulate a weighted sparse multi-task learning method. Lastly, we also take into account the distributional characteristics of samples per class and use clustering-induced subclass label vectors as target response values in our sparse regression model. In our experiments on the ADNI cohort, we performed both binary and multi-class classification tasks in AD/MCI diagnosis and showed the superiority of the proposed method by comparing with the state-of-the-art methods.

Appendix: Affinity propagation

Here, we briefly review the affinity propagation (Frey and Dueck 2007), by which we find subclasses in each original class. Let $S_{ij}^{(h)}$ $(i,j=1,2,\dots ,N)$ denote the pairwise similarities¹³ between each pair of N samples in $\tilde{\mathbf {X}}^{(h)}$. The affinity propagation algorithm works on the similarity matrix ${\mathbf {S}}^{(h)}=[S_{ij}^{(h)}]\in \mathbb {R}^{N\times N}$ and attempts to find ‘exemplars’ that maximize the overall sum of similarities between all exemplars and their member samples. Methodologically, the algorithm defines two types of messages, namely, responsibility and availability, exchanged among samples: Responsibility $R_{ij}^{(h)}$ represents the accumulated evidence for how well-suited sample j is to serve as the exemplar for sample i; Availability $A_{ij}^{(h)}$ reflects the accumulated evidence for how appropriate it would be for sample i to choose sample j as its exemplar. Using these messages, the exemplar of sample i is determined by the one that maximizes the following objective function:

$$\begin{aligned} \mathop {{{\mathrm{argmax}}}}\limits _{j} \{R_{ij}^{(h)}+A_{ij}^{(h)}: j=1,2,\dots , N\}. \end{aligned}$$

(8)

In Algorithm 1, both ${\mathbf {R}}^{(h)}=[R_{ij}^{(h)}]$ and ${\mathbf {A}}^{(h)}=[A_{ij}^{(h)}]$ are initially set to zero matrices, and then their values are iteratively updated as below until converged:

$$\begin{aligned} {R}_{ij}^{(h)}&= \left[ \begin{array}{ll} {S}_{ij}^{(h)}-\max _{k\ne j}\{{A}_{ik}^{(h)}+{S}_{ik}^{(h)}\} &{} (i\ne j) \\ {S}_{ij}^{(h)}-\max _{k\ne j}\{{S}_{ik}^{(h)}\} &{} (i = j) \end{array} \right. \\ {A}_{ij}^{(h)}&= \left[ \begin{array}{ll} \min \{0, {R}_{jj}^{(h)}+\sum _{k\ne i,j}\max \{0, {R}_{kj}^{(h)}\}\} &{} (i\ne j)\\ \sum _{k\ne i}\max \{0, {R}_{kj}^{(h)}\} &{} (i=j) \end{array} \right. . \end{aligned}$$