Adding and subtracting eigenspaces with eigenvalue decomposition and singular value decomposition

摘要

This paper provides algorithms for adding and subtracting eigenspaces, thus allowing for incremental updating and downdating of data models. Importantly, and unlike previous work, we keep an accurate track of the mean of the data, which allows our methods to be used in classification applications. The result of adding eigenspaces, each made from a set of data, is an approximation to that which would obtain were the sets of data taken together. Subtracting eigenspaces yields a result approximating that which would obtain were a subset of data used. Using our algorithms, it is possible to perform ‘arithmetic’ on eigenspaces without reference to the original data. Eigenspaces can be constructed using either eigenvalue decomposition (EVD) or singular value decomposition (SVD). We provide addition operators for both methods, but subtraction for EVD only, arguing there is no closed-form solution for SVD. The methods and discussion surrounding SVD provide the principle novelty in this paper. We illustrate the use of our algorithms in three generic applications, including the dynamic construction of Gaussian mixture models.