This paper proposes a new notion of typical sequences on a wide class of abstract alphabets (so-called standard Borel spaces), which is based on approximations of memoryless sources by empirical distributions uniformly over a class of measurable “test functions.” In the finite-alphabet case, we can take all uniformly bounded functions and recover the usual notion of strong typicality (or typicality under the total variation distance). For a general alphabet, however, this function class turns out to be too large, and must be restricted. With this in mind, we define typicality with respect to any Glivenko-Cantelli function class (i.e., a function class that admits a Uniform Law of Large Numbers) and demonstrate its power by giving simple derivations of the fundamental limits on the achievable rates in several source coding scenarios, in which the relevant operational criteria pertain to reproducing empirical averages of a general-alphabet stationary memoryless source with respect to a suitable function class.
We also note that a restricted notion of typicality based on weak convergence was used by Kontoyiannis and Zamir @cite_9 in the context of universal vector quantization using entropy codes. The idea there is to consider sequences of increasing length, whose empirical distributions converge in the weak topology to the output distribution of an optimal test channel in a Shannon rate-distortion problem.
@cite_9
2138605525
We introduce a universal quantization scheme based on random coding, and we analyze its performance. This scheme consists of a source-independent random codebook (typically mismatched to the source distribution), followed by optimal entropy coding that is matched to the quantized codeword distribution. A single-letter formula is derived for the rate achieved by this scheme at a given distortion, in the limit of large codebook dimension. The rate reduction due to entropy coding is quantified, and it is shown that it can be arbitrarily large. In the special case of "almost uniform" codebooks (e.g., an independent and identically distributed (i.i.d.) Gaussian codebook with large variance) and difference distortion measures, a novel connection is drawn between the compression achieved by the present scheme and the performance of "universal" entropy-coded dithered lattice quantizers. This connection generalizes the "half-a-bit" bound on the redundancy of dithered lattice quantizers. Moreover, it demonstrates a strong notion of universality where a single "almost uniform" codebook is near optimal for any source and any difference distortion measure. The proofs are based on the fact that the limiting empirical distribution of the first matching codeword in a random codebook can be precisely identified. This is done using elaborate large deviations techniques, that allow the derivation of a new "almost sure" version of the conditional limit theorem.