A General Framework for Unsupervised Processing of Structured Data

Self-organization constitutes an,important paradigm in machine learning with successful applications e.g. in data- and web-mining. Most approaches, however, have been proposed for processing data contained in a fixed and finite dimensional vector space. In this article, we will focus on extensions to more general data structures like sequences and tree structures. Various modifications of the standard self-organizing map (SOM) to sequences or tree structures have been proposed in the literature some of which are the temporal Kohonen map, the recursive SOM, and SOM for structured data. These methods enhance the standard SOM by utilizing recursive connections. We define a general recursive dynamic in this article which provides recursive processing of complex data structures by recursive computation of internal representations for the given context. The above mentioned mechanisms of SOMs for structures are special cases of the proposed general dynamic. Furthermore, the dynamic covers the supervised case of recurrent and recursive networks. The general framework offers an uniform notation for training mechanisms such as Hebbian learning. Moreover, the transfer of computational alternatives such as vector quantization or the neural gas algorithm to structure processing networks can be easily achieved. One can formulate general cost functions corresponding to vector quantization, neural gas, and a modification of SOM. The cost functions can be compared to Hebbian learning which can be interpreted as an approximation of a stochastic gradient descent. For comparison, we derive the exact gradients for general cost functions