Hebbian learning approaches based on general inner products and distance measures in non-Euclidean spaces

The topic of this thesis is to define a unified and generalized scheme for Hebbian approaches in non-Euclidean spaces for unsupervised and supervised learning. This can be realized in different ways. One possibility is the replacement of the inner product by a semi-inner product (SIP). A SIP relaxes the strict properties of an inner product but preserves the linear aspect in the first argument. Thus, these SIPs are natural equivalents of inner products generating Banach spaces instead of Hilbert spaces for inner products. In this work SIPs for Banach spaces are considered for unsupervised Hebbian like learning approaches.

Further, the learning scheme of the supervised Learning Vector Quantization (LVQ) network, which is originally designed for applications in Euclidean data space, can be interpreted under specific circumstances as a Hebbian like learning, too. It is shown that, non-Euclidean metrics applied in LVQ can improve the performance of classification learning compared to Euclidean variants.

The previously addressed Hebbian learning methods are vectorial approaches. However, if the data space is a vector space of matrices equipped with a respective matrix norm, then matrix approaches for Hebbian like learning methods become of interest. The extension of these methods in non-Euclidean spaces of matrices to process matrix data is the last main point of this thesis.