$K$ -Dimensional Coding Schemes in Hilbert Spaces

This paper presents a general coding method where data in a Hilbert space are represented by finite dimensional coding vectors. The method is based on empirical risk minimization within a certain class of linear operators, which map the set of coding vectors to the Hilbert space. Two results bounding the expected reconstruction error of the method are derived, which highlight the role played by the codebook and the class of linear operators. The results are specialized to some cases of practical importance, including K-means clustering, nonnegative matrix factorization and other sparse coding methods.