  Principal component analysis (PCA) has been widely applied to dimensionality
reduction and data pre-processing for different applications in engineering,
biology and social science. Classical PCA and its variants seek for linear
projections of the original variables to obtain a low dimensional feature
representation with maximal variance. One limitation is that it is very
difficult to interpret the results of PCA. In addition, the classical PCA is
vulnerable to certain noisy data. In this paper, we propose a convex sparse
principal component analysis (CSPCA) algorithm and apply it to feature
analysis. First we show that PCA can be formulated as a low-rank regression
optimization problem. Based on the discussion, the l 2 , 1 -norm minimization
is incorporated into the objective function to make the regression coefficients
sparse, thereby robust to the outliers. In addition, based on the sparse model
used in CSPCA, an optimal weight is assigned to each of the original feature,
which in turn provides the output with good interpretability. With the output
of our CSPCA, we can effectively analyze the importance of each feature under
the PCA criteria. The objective function is convex, and we propose an iterative
algorithm to optimize it. We apply the CSPCA algorithm to feature selection and
conduct extensive experiments on six different benchmark datasets. Experimental
results demonstrate that the proposed algorithm outperforms state-of-the-art
unsupervised feature selection algorithms.
