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.

💡 Research Summary

The paper introduces a unified theoretical framework for representing data that live in an (possibly infinite‑dimensional) Hilbert space (\mathcal H) by low‑dimensional coding vectors in (\mathbb R^{K}). The authors formalize the coding process as a pair ((T,c)) where (c\in\mathcal C\subset\mathbb R^{K}) is a code drawn from a finite or suitably bounded codebook and (T\in\mathcal T\subset L(\mathbb R^{K},\mathcal H)) is a linear operator that maps the code back into the original space. The learning objective is empirical risk minimization (ERM) of the reconstruction loss
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