Convexification of Permutation-Invariant Sets and an Application to Sparse Principal Component Analysis

We develop techniques to convexify a set that is invariant under permutation and/or change of sign of variables and discuss applications of these results. First, we convexify the intersection of the unit ball of a permutation and sign-invariant norm with a cardinality constraint. This gives a nonlinear formulation for the feasible set of sparse principal component analysis (PCA) and an alternative proof of the K-support norm. Second, we characterize the convex hull of sets of matrices defined by constraining their singular values. As a consequence, we generalize an earlier result that characterizes the convex hull of rank-constrained matrices whose spectral norm is below a given threshold. Third, we derive convex and concave envelopes of various permutation-invariant nonlinear functions and their level sets over hypercubes, with congruent bounds on all variables. Finally, we develop new relaxations for the exterior product of sparse vectors. Using these relaxations for sparse PCA, we show that our relaxation closes 98% of the gap left by a classical semidefinite programming relaxation for instances where the covariance matrices are of dimension up to 50 × 50.

Using ℓ1-Relaxation and Integer Programming to Obtain Dual Bounds for Sparse PCA

Dual Bounds of Sparse Principal Component Analysis Sparse principal component analysis (PCA) is a widely used dimensionality reduction tool in machine learning and statistics. Compared with PCA, sparse PCA enhances the interpretability by incorporating a sparsity constraint. However, unlike PCA, conventional heuristics for sparse PCA cannot guarantee the qualities of obtained primal feasible solutions via associated dual bounds in a tractable fashion without underlying statistical assumptions. In “Using L1-Relaxation and Integer Programming to Obtain Dual Bounds for Sparse PCA,” Santanu S. Dey, Rahul Mazumder, and Guanyi Wang present a convex integer programming (IP) framework of sparse PCA to derive dual bounds. They show the worst-case results on the quality of the dual bounds provided by the convex IP. Moreover, the authors empirically illustrate that the proposed convex IP framework outperforms existing sparse PCA methods of finding dual bounds.

Lymphocyte–monocyte–neutrophil index: a predictor of severity of coronavirus disease 2019 patients produced by sparse principal component analysis

Background It is important to recognize the coronavirus disease 2019 (COVID-19) patients in severe conditions from moderate ones, thus more effective predictors should be developed. Methods Clinical indicators of COVID-19 patients from two independent cohorts (Training data: Hefei Cohort, 82 patients; Validation data: Nanchang Cohort, 169 patients) were retrospected. Sparse principal component analysis (SPCA) using Hefei Cohort was performed and prediction models were deduced. Prediction results were evaluated by receiver operator characteristic curve and decision curve analysis (DCA) in above two cohorts. Results SPCA using Hefei Cohort revealed that the first 13 principal components (PCs) account for 80.8% of the total variance of original data. The PC1 and PC12 were significantly associated with disease severity with odds ratio of 4.049 and 3.318, respectively. They were used to construct prediction model, named Model-A. In disease severity prediction, Model-A gave the best prediction efficiency with area under curve (AUC) of 0.867 and 0.835 in Hefei and Nanchang Cohort, respectively. Model-A’s simplified version, named as LMN index, gave comparable prediction efficiency as classical clinical markers with AUC of 0.837 and 0.800 in training and validation cohort, respectively. According to DCA, Model-A gave slightly better performance than others and LMN index showed similar performance as albumin or neutrophil-to-lymphocyte ratio. Conclusions Prediction models produced by SPCA showed robust disease severity prediction efficiency for COVID-19 patients and have the potential for clinical application.