Recursive partitioning to reduce distortion

Adaptive partitioning of a multidimensional feature space plays a fundamental role in the design of data-compression schemes. Most partition-based design methods operate in an iterative fashion, seeking to reduce distortion at each stage of their operation by implementing a linear split of a selected cell. The operation and eventual outcome of such methods is easily described in terms of binary tree-structured vector quantizers. This paper considers a class of simple growing procedures for tree-structured vector quantizers. Of primary interest is the asymptotic distortion of quantizers produced by the unsupervised implementation of the procedures. It is shown that application of the procedures to a convergent sequence of distributions with a suitable limit yields quantizers whose distortion tends to zero. Analogous results are established for treestructured vector quantizers produced from stationary ergodic training data. The analysis is applicable to procedures employing both axis-parallel and oblique splitting, and a variety of distortion measures. The results of the paper apply directly to unsupervised procedures that may be efficiently implemented on a digital computer. Appears in IEEE Transactions on Information Theory, vol. 43, no. 4, pp. 1122-1133, 1997 ∗Andrew Nobel is with the Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260. Email: [email protected]. This work was completed while he was on leave as a Beckman Institute Fellow at the Beckman Institute for Advanced Science and Technology, University of Illinois U-C, and was supported in part by National Science Foundation Grant DMS-9501926.