Spherical Principal Component Analysis

In many real-world applications such as text categorization and face recognition, the dimensions of data are usually very high. Dealing with high-dimensional data is computationally expensive while noise or outliers in the data can increase dramatically as the dimension increases. Dimension reduction is one of the most important and effective methods to handle high dimensional data [4, 17, 20]. Among the dimension reduction methods, Principal Component Analysis (PCA) is one of the most widely used methods due to its simplicity and effectiveness. PCA is a statistical procedure that uses an orthogonal transformation to convert a set of correlated variables into a set of linearly uncorrelated principal directions. Usually the number of principal directions is less than or equal to the number of original variables. This transformation is defined in such a way that the first principal direction has the largest possible variance (that is, accounts for as much of the variability in the data as possible), and each succeeding direction has the highest variance under the constraint that it is orthogonal to the preceding directions. The resulting vectors are an uncorrelated orthogonal basis set. When data points lie in a low-dimensional manifold and the manifold is linear or nearly-linear, the low-dimensional structure of data can be effectively captured by a linear subspace spanned by the principal PCA directions.