Sparse Principal Component Analysis (SPCA) is widely used in data processing and dimension reduction; it uses the lasso to produce modified principal components with sparse loadings for better interpretability. However, sparse PCA never considers an additional grouping structure where the loadings share similar coefficients (i.e., feature grouping), besides a special group with all coefficients being zero (i.e., feature selection). In this paper, we propose a novel method called Feature Grouping and Sparse Principal Component Analysis (FGSPCA) which allows the loadings to belong to disjoint homogeneous groups, with sparsity as a special case. The proposed FGSPCA is a subspace learning method designed to simultaneously perform grouping pursuit and feature selection, by imposing a non-convex regularization with naturally adjustable sparsity and grouping effect. To solve the resulting non-convex optimization problem, we propose an alternating algorithm that incorporates the difference-of-convex programming, augmented Lagrange and coordinate descent methods. Additionally, the experimental results on real data sets show that the proposed FGSPCA benefits from the grouping effect compared with methods without grouping effect.

The use of complex models --with many parameters-- is challenging with high-dimensional small-sample problems: indeed, they face rapid overfitting. Such situations are common when data collection is expensive, as in neuroscience, biology, or geology. Dedicated regularization can be crafted to tame overfit, typically via structured penalties. But rich penalties require mathematical expertise and entail large computational costs. Stochastic regularizers such as dropout are easier to implement: they prevent overfitting by random perturbations. Used inside a stochastic optimizer, they come with little additional cost. We propose a structured stochastic regularization that relies on feature grouping. Using a fast clustering algorithm, we define a family of groups of features that capture feature covariations. We then randomly select these groups inside a stochastic gradient descent loop. This procedure acts as a structured regularizer for high-dimensional correlated data without additional computational cost and it has a denoising effect. We demonstrate the performance of our approach for logistic regression both on a sample-limited face image dataset with varying additive noise and on a typical high-dimensional learning problem, brain image classification.