Variance-Preserving Orthogonal Selection (VPOS): Greedy Feature Selection via Orthogonal Deflation in PCA Loading Space

Accurate optimization of a supervised spectral objective need not produce an accurate population subspace or a better predictive representation. We investigate these distinctions for Online Kernel Supervised Principal Component Analysis (OKSPCA), which combines a centered cross-moment in finite random-feature coordinates with an Adam-style orthonormal basis update for an established objective. Fixed-map consistency, concentration and perturbation results describe the estimator and its exact subspace; same-target comparisons then assess the practical iterate separately. Across six predictive benchmarks, performance depends on the declared pipeline: replacing the tracker with the exact empirical target leaves the two regression deficits largely unchanged. Direct classification-rank models capture nearly all terminal objective energy on average, but a saved intermediate state exhibits substantial geometric deviation; a controlled sample-size study further separates empirical accuracy from population recovery. In distinct numerical-service workloads, exact on-request computation is faster in the tested classification settings, whereas Adam saves time relative to the tested full thin-SVD service for some dense wider-regression requests, alongside persistent geometric error. These diagnostics limit explanations based solely on terminal optimization accuracy and distinguish numerical cost from quality, rank coverage and freshness; they establish neither practical-tracker convergence nor predictive or deployment benefits from basis availability.

We present msPCA: an open-source R package for sparse principal component analysis with multiple components. It implements an alternating maximization algorithm to generate a set of sparse loading vectors that collectively explain a large fraction of the variance in a dataset, while remaining non-redundant. The algorithm supports two definitions of non-redundancy: either orthogonality of the loading vectors or zero pairwise correlation between principal components (PCs). In the reported benchmarks, msPCA solves sparse PCA problems with thousands of features, achieving competitive runtimes while producing sparse components with controlled feasibility violations and a high fraction of variance explained.

Unsupervised feature selection is commonly formulated as a multiobjective optimisation problem that jointly optimises subset quality and subset size. Yet the behaviour of this formulation depends critically on the choice of evaluation objective, the direction of subset-size regularisation, and the initialisation strategy. We study these factors in a controlled setting using a synthetic dataset with known informative, redundant, and irrelevant feature types. Six formulations are compared by combining three evaluation objectives: accuracy, silhouette score, and PCA reconstruction loss with subset-size minimisation or maximisation. The results show that formulation strongly affects both search dynamics and the quality of the resulting Pareto front. Silhouette-based formulations exhibit a strong bias toward trivial low-cardinality solutions and remain weak proxies for predictive performance. In contrast, the proposed PCA loss objective produces compact subsets with test accuracy comparable to subsets obtained by directly optimising supervised accuracy. Overall, the study shows that objective design is central to effective multiobjective unsupervised feature selection.