On Finite Memory Universal Data Compression and Classification of Individual Sequences

Abstract

Consider the case where consecutive blocks of N letters of a semi-infinite individual sequence X over a finite-alphabet are being compressed into binary sequences by some one-to-one mapping. No a-priori information about X is available at the encoder, which must therefore adopt a universal data-compression algorithm. It is known that if the universal LZ77 data compression algorithm is successively applied to N-blocks then the best error-free compression for the particular individual sequence X is achieved, as $N$ tends to infinity. The best possible compression that may be achieved by any universal data compression algorithm for finite N-blocks is discussed. It is demonstrated that context tree coding essentially achieves it. Next, consider a device called classifier (or discriminator) that observes an individual training sequence X. The classifier's task is to examine individual test sequences of length N and decide whether the test N-sequence has the same features as those that are captured by the training sequence X, or is sufficiently different, according to some appropriatecriterion. Here again, it is demonstrated that a particular universal context classifier with a storage-space complexity that is linear in N, is essentially optimal. This may contribute a theoretical "individual sequence" justification for the Probabilistic Suffix Tree (PST) approach in learning theory and in computational biology.