Termination and continuity of greedy growing for tree-structured vector quantizers

Tree-structured vector quantizers (TSVQ) provide a computationally efficient, variable-rate method of compressing vector-valued data. In applications, the problem of designing a TSVQ from empirical training data is critical. Greedy growing algorithms are a common and effective approach to the design problem. They are recursive procedures that produce a TSVQ one node at a time by optimizing a simple splitting criterion at each step. While unsupervised greedy growing algorithms are well-understood from an experimental point of view, there has been little theory to support their use, or to examine their behavior on large training sets. The authors present a rigorous analysis of a greedy growing algorithm proposed by Riskin (1990), Riskin and Gray (1991), and Balakrishnan (1991). The first part of the paper is a description of the algorithm and an examination of its asymptotic behavior as it applies to a fixed, absolutely continuous distribution. The second part of the paper establishes the structural consistency of the algorithm with respect to a convergent sequence of distributions. As an application, the authors obtain results concerning the large-sample empirical behavior of the algorithm when it is applied to stationary ergodic training vectors