The Canonical Distortion Measure for Vector Quantization and Function Approximation
This work addresses the issue of inappropriate metrics like Hamming or Euclidean for natural signals such as speech or images, offering a domain-specific solution for vector quantization.
The paper tackles the problem of measuring distortion for vector quantization and function approximation by introducing a canonical distortion measure (CDM) derived from an environment of functions, showing that optimizing reconstruction error with CDM leads to optimal piecewise constant approximations. Experimental results with neural networks implementing CDM are described as encouraging.
To measure the quality of a set of vector quantization points a means of measuring the distance between a random point and its quantization is required. Common metrics such as the {\em Hamming} and {\em Euclidean} metrics, while mathematically simple, are inappropriate for comparing natural signals such as speech or images. In this paper it is shown how an {\em environment} of functions on an input space $X$ induces a {\em canonical distortion measure} (CDM) on X. The depiction 'canonical" is justified because it is shown that optimizing the reconstruction error of X with respect to the CDM gives rise to optimal piecewise constant approximations of the functions in the environment. The CDM is calculated in closed form for several different function classes. An algorithm for training neural networks to implement the CDM is presented along with some encouraging experimental results.