Efficient algorithms and minimax bounds for zero-delay lossy source coding

Zero-delay sequential lossy source coding schemes are considered for both individual sequences and random sources. Performance is measured by the distortion redundancy, defined as the difference between the normalized cumulative distortion of the scheme and that of the best scalar quantizer matched to the entire sequence to be encoded. Weiss-man and Merhav [2001] constructed a randomized scheme which, for any bounded individual sequence of length n, achieves a distortion redundancy O(n^-13/ logn). However, this scheme has prohibitive complexity. Here we present an algorithm with encoding complexity O(n⁴3/ logn) and distortion redundancy O(n^-13/ logn). The complexity can be made linear in the sequence length n at the price of increasing the distortion redundancy to O(n^-14/logn¹2/). We also show that for the class of bounded memoryless sources, the minimax expected distortion redundancy in zero-delay lossy coding is upper and lower bounded by (constant multiples of) n^-12/.