Non-Euclidean principal component analysis by Hebbian learning

Principal component analysis based on Hebbian learning is originally designed for data processing in Euclidean spaces. We present in this contribution an extension of Oja׳s Hebbian learning approach for non-Euclidean spaces. We show that for Banach spaces the Hebbian learning can be carried out using the underlying semi-inner product. Prominent examples for such Banach spaces are the lp-spaceslp-spaces for p≠2p≠2. For kernels spaces, as applied in support vector machines or kernelized vector quantization, this approach can be formulated as an online learning scheme based on the differentiable kernel. Hence, principal component analysis can be explicitly carried out in the respective data spaces but now equipped with a non-Euclidean metric. In the article we provide the theoretical framework and give illustrative examples.