The use of a tree-structured piecewise linear filter as an adaptive equalizer is proposed. In the tree equalizer, each node in a tree is associated with a linear filter restricted to a polygonal domain, and each subtree is associated with a piecewise linear filter. A training sequence is used to adaptively update the filter coefficients and domains at each node, and to select the best subtree and corresponding piecewise linear filter. The tree-structured approach offers several advantages. First, it makes use of standard linear adaptive filtering techniques at each node to find the corresponding conditional linear filter. Second, it allows for efficient selection of the subtree and corresponding piecewise linear filter of appropriate complexity. Overall, the approach is computationally efficient and conceptually simple. Numerical experiments are performed to show the advantages of tree-structured piecewise linear and piecewise decision feedback equalizers over linear, polynomial, and decision feedback equalizers for the equalization of channels with severe intersymbol interference.>

The authors propose and analyze a novel architecture for nonlinear adaptive filters. These nonlinear filters are piecewise linear filters obtained by arranging linear filters and thresholds in a tree structure. A training algorithm is used to adaptively update the filter coefficients and thresholds at the nodes of the tree, and to prune the tree. The resulting tree-structured piecewise linear adaptive filter inherits the robust estimation and fast adaptation of linear adaptive filters, along with the approximation and model-fitting properties of tree-structured regression models. A rigorous analysis of the training algorithm for the tree-structured filter is performed. Some techniques are developed for analyzing hierarchically organized stochastic gradient algorithms with fixed gains and nonstationary dependent data. Simulation results show the significant advantages of the tree-structured piecewise linear filter over linear and polynomial filters for adaptive echo cancellation.>

The idea of a tree-structured piecewise linear adaptive filter is proposed. The tree-structured filter is constructed as follows. Each node in a binary tree is associated with a linear filter restricted to a polygonal domain, and this is done in such a way that each subtree is associated with a piecewise linear filter. A training sequence is used to adaptively update the filter coefficients and domains at each node, and to select the best subtree and the corresponding piecewise linear filter. The authors summarize the derivation and analysis of the tree-structured piecewise linear adaptive filter. They then investigate the performance of the tree-structured filter as an adaptive echo canceler and compare it with that of linear and truncated Volterra series adaptive echo cancelers through computer simulations.>

A critical issue in classification tree design-obtaining right-sized trees, i.e. trees which neither underfit nor overfit the data-is addressed. Instead of stopping rules to halt partitioning, the approach of growing a large tree with pure terminal nodes and selectively pruning it back is used. A new efficient iterative method is proposed to grow and prune classification trees. This method divides the data sample into two subsets and iteratively grows a tree with one subset and prunes it with the other subset, successively interchanging the roles of the two subsets. The convergence and other properties of the algorithm are established. Theoretical and practical considerations suggest that the iterative free growing and pruning algorithm should perform better and require less computation than other widely used tree growing and pruning algorithms. Numerical results on a waveform recognition problem are presented to support this view.>

An efficient iterative method is proposed to grow and prune classification trees. This method divides the data sample into two subsets and iteratively grows a tree with one subset and prunes it with the other subset, successively interchanging the roles of the two subsets. The convergence and other properties of the algorithm are established. Theoretical and practical considerations suggest that the iterative tree growing and pruning algorithm should perform better and require less computation than other widely used tree growing and pruning algorithms. Numerical results on a waveform recognition problem are presented to support this view.>