High-resolution functional quantization

Suppose a function of N real source variables X₁^N = (X₁, X₂, ..., X_N) is desired at a destination constrained to receive a limited number of bits. If the result of evaluating the function, Y = G(X₁^N), can be itself encoded, this is the optimal strategy - the origin of Y becomes irrelevant to the communication problem. We consider two alternative scenarios: distributed quantization, in which each X_i must be separately encoded: and linear transform coding of X₁^N. Optimal fixed- and variable-rate scalar quantizers are derived under the conventional assumptions of high-resolution quantization theory, and we find optimal transforms for transform coding. For certain classes of functions, examples demonstrate large improvements over using quantizers designed to minimize distortion of the X_iS.