Mathematical Characterization of Sophisticated Variants for Relevance Learning in Learning Matrix Quantization Based on Schatten-p-norms

In this paper we investigate possibilities of relevance learning in learning matrix quantization and discuss their mathematical properties. Learning matrix quantization can be seen as an extension of the learning vector quantization method, which is one of the most popular and intuitive prototype based vector quantization algorithms for classification learning. Whereas in the vector quantization approach vector data are processed, learning matrix quantization deals with matrix data as they occur in image processing of gray-scale images or in time-resolved spectral analysis. Here, we concentrate on the consideration of relevance learning when learning matrix quantization is based on the Schatten-p-norm as the data dissimilarity measure. For those matrix systems exist more relevance learning variants than for vector classification systems. We contemplate several approaches based on different matrix products as well as tensor operators. In particular, we discuss their mathematical properties related to the relevance learning task keeping in mind the stochastic gradient learning scheme for both, prototype as well as relevance learning.