Optimal pruning for tree-structured vector quantization

Abstract

Tree-structured vector quantization (VQ) is a technique designed to represent a codebook that simplifies encoding as well as vector quantizer design. Most design algorithms for tree-structured VQ used in the past are based on heuristics that successively partition the input space. Recently, Chou, Lookabaugh and Gray proposed a tree-pruning heuristic in which a given initial tree is pruned backwards according to certain optimization criterion. We define the notion of an optimal pruned tree subject to a cost constraint and study the computational complexity of finding such an optimal tree for various cost functions. Under the assumption that all trees are equally probable, we show that, on the average, the number of pruned trees in a given tree is exponential in the number of leaves. Furthermore, we prove that finding an optimal pruned tree subject to constraints such as entropy or the expected-depth is NP-hard. However, we show that when the constraint is the number of leaves, the problem can be solved in polynomial time. We develop an algorithm to find the optimal pruned tree in O(nk) time, where n is the size of the initial tree and kis the constraint size.