Hierarchical Filters in a Collaborative Filtering Framework for System Identification and Knowledge Retrieval
This chapter provides a critical review of hierarchical filters and the associated adaptive learning algorithms. Hierarchical filters are collaborative adaptive filtering architectures where short-length adaptive transversal filters are combined into layers, which are then combined into a multilayered structures. These structures offer potentially faster speed of convergence compared to the standard finite impulse response (FIR) filters, which is due to the small order of their constituting sub-filters. Several approaches can be used to adapt the coefficients of hierarchical filters. These include the use of the standard least mean square (LMS) algorithm for every sub-filter, via a variant of linear backpropagation, through to using a different algorithm for every layer and every sub-filter within the layer. Unless the input signal is white or the unknown channel is sparse, hierarchical filters converge to biased solutions. We make use of this property to propose a collaborative approach to the identification of sparse channels. The performances of these algorithms are evaluated for a variety of applications, including system identification and sparsity detection. The benefits and limitations of hierarchical adaptive filtering in this context are highlighted.
KeywordsInput Layer Finite Impulse Response Less Mean Square Recursive Little Square Gradient Descent Algorithm
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