Karhunen-Loeve Subspace

The Karhunen-Loeve expansion is a well-known technique in the pattern recognition field, however, there is no exact proof of the K-L expansion in the context of approximation to a set of patterns. This paper provides an exact proof of the following well-known theorem: an N-dimensional subspace provides the best approximation to a set of patterns if and only if it is spanned by the first N-number of eigenelements of the covariance operator of the pattern set.<>