On adding and subtracting eigenspaces with EVD and SVD

This paper provides two algorithms: one for adding eigenspaces, another for sub- tracting them, thus allowing for incremental updating and downdating of data mod- els. 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 approxima- tion 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. We illustrate the use of our algorithms in three generic applications, including the dynamic construction of Gaussian mixture models. In addition, we mention singular value decomposition as an alternative to eigenvalue decomposition. We show that updating SVD models comes at the cost of space resources, and argue that downdating SVD models is not possible in closed- form.