Efficient realization of classification using modified Haar DWT
The Haar discrete wavelet transform is used as inspiration for a new simple and fast algorithm to train multiclass dyadic decision trees. A localized ordering of the training data changes a multidimensional Haar transform into the one dimensional case, and avoids cache misses. The resulting tree has very simple structure convenient for direct implementation as a fixed-depth threshold network and very fast evaluation of the classification function. The simple structure of the tree is also conducive to good compression of the function. We implement and test the learning algorithm and threshold network but without incorporating sampled interpolated-value points for improved generalization. Compression appears good in low dimensions.
A weightedL2 norm is introduced in which Markov operators, e.g., associated with noisy maps, are contracting provided the kernel (i.e., the transitional distribution) is smooth enough. This results in strong relaxational properties of noisy maps. Similar to this norm, integral functionals appear useful when studying spatiotemporal chaos and random fields.