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Universal lossy compression

Consider x α n . The goal of lossy compression  [link] is to describe x ˆ , also of length n but possibly defined over another reconstruction alphabet α ˆ α , such that the description requires few bits and the distortion

d ¯ ( x , x ˆ ) = 1 n i = 1 n d ( x i , x ˆ i )

is small, where d ( · , · ) is some distortion metric. It is well known that for every d ( · , · ) and distortion level D there is a minimum rate R ( D ) , such that x ˆ can be described at rate R ( D ) . The rate R ( D ) is known as the rate distortion (RD) function, it is the fundamental information theoretic limit of lossycompression  [link] , [link] .

The invention of lossy compression algorithms has been a challenging problem for decades. Despite numerous applicationssuch as image compression  [link] , [link] , video compression  [link] , and speech coding  [link] , [link] , [link] , there is a significant gap between theory and practice, and these practical lossy compressorsdo not achieve the RD function. On the other hand, theoretical constructions that achieve the RD function are impractical.

A promising recent algorithm by Jalali and Weissman  [link] is universal in the limit of infinite runtime. Its RD performance is reasonable even with modest runtime.The main idea is that the distortion version x ˆ of the input x can be computed as follows,

x ˆ = arg min w n α n { H k ( w n ) - β d ¯ ( x n , w n ) } ,

where β < 0 is the slope at the particular point of interest in the RD function, and H k ( w n ) is the empirical conditional entropy of order k ,

H k ( w n ) - 1 n a , u k n w ( u k , a ) log n w ( u k , a ) a ' α n w ( u k , a ' ) ,

where u k is a context of order k , and as before n w ( u k , a ) is the number of times that the symbol a appears following a context u k in w n . Jalali and Weissman proved  [link] that when k = o ( log ( n ) ) , the RD pair ( H k ( x ˆ n ) , d ¯ ( x n , x ˆ n ) ) converges to the RD function asymptotically in n . Therefore, an excellent lossy compression technique is to compute x ˆ and then compress it. Moreover, this compression can be universal. In particular, the choice of context order k = o ( log ( n ) ) ensures that universal compressors for context tress sources can emulate the coding length of the empirical conditional entropy H ˆ k ( x ˆ n ) .

Despite this excellent potential performance, there is still a tremendous challenge. Brute force computation of the globally minimum energysolution x n ˆ involves an exhaustive search over exponentially many sequences and is thus infeasible.Therefore, Jalali and Weissman rely on Markov chain Monte Carlo (MCMC)  [link] , which is a stochastic relaxation approach to optimization. The crux of the matter is to definean energy function,

ϵ ( w n ) = H k ( w n ) - β d ( x n , w n ) .

The Boltzmann probability mass function (pmf) is

f s ( w n ) 1 Z t exp { - 1 t ε ( w n ) } ,

where t > 0 is related to temperature in simulated annealing, and Z t is the normalization constant, which does not need to be computed.

Because it is difficult to sample from the Boltzmann pmf [link] directly, we instead use a Gibbs sampler , which computes the marginal distributions at all n locations conditioned on the rest of w n being kept fixed. For each location, the Gibbs sampler resamples from the distribution of w i conditioned on w n i { w n : n i } as induced by the joint pmf in [link] , which is computed as follows,

Questions & Answers

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Source:  OpenStax, Universal algorithms in signal processing and communications. OpenStax CNX. May 16, 2013 Download for free at http://cnx.org/content/col11524/1.1
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