# 0.5 Beyond lossless compression

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## Universal lossy compression

Consider $x\in {\alpha }^{n}$ . The goal of lossy compression  [link] is to describe $\stackrel{ˆ}{x}$ , also of length $n$ but possibly defined over another reconstruction alphabet $\stackrel{ˆ}{\alpha }\ne \alpha$ , such that the description requires few bits and the distortion

$\overline{d}\left(x,\stackrel{ˆ}{x}\right)=\frac{1}{n}\sum _{i=1}^{n}d\left({x}_{i},{\stackrel{ˆ}{x}}_{i}\right)$

is small, where $d\left(·,·\right)$ is some distortion metric. It is well known that for every $d\left(·,·\right)$ and distortion level $D$ there is a minimum rate $R\left(D\right)$ , such that $\stackrel{ˆ}{x}$ can be described at rate $R\left(D\right)$ . The rate $R\left(D\right)$ 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 $\stackrel{ˆ}{x}$ of the input $x$ can be computed as follows,

$\stackrel{ˆ}{x}=\underset{{w}^{n}\in {\alpha }^{n}}{argmin}\left\{{H}_{k}\left({w}^{n}\right)-\beta \overline{d}\left({x}^{n},{w}^{n}\right)\right\},$

where $\beta <0$ is the slope at the particular point of interest in the RD function, and ${H}_{k}\left({w}^{n}\right)$ is the empirical conditional entropy of order $k$ ,

${H}_{k}\left({w}^{n}\right)\triangleq -\frac{1}{n}\sum _{a,{u}^{k}}{n}_{w}\left({u}^{k},a\right)log\left(\frac{{n}_{w}\left({u}^{k},a\right)}{{\sum }_{{a}^{\text{'}}\in \alpha }{n}_{w}\left({u}^{k},{a}^{\text{'}}\right)}\right),$

where ${u}^{k}$ is a context of order $k$ , and as before ${n}_{w}\left({u}^{k},a\right)$ 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 $k=o\left(log\left(n\right)\right)$ , the RD pair $\left({H}_{k}\left({\stackrel{ˆ}{x}}^{n}\right),\overline{d}\left({x}^{n},{\stackrel{ˆ}{x}}^{n}\right)\right)$ converges to the RD function asymptotically in $n$ . Therefore, an excellent lossy compression technique is to compute $\stackrel{ˆ}{x}$ and then compress it. Moreover, this compression can be universal. In particular, the choice of context order $k=o\left(log\left(n\right)\right)$ ensures that universal compressors for context tress sources can emulate the coding length of the empirical conditional entropy ${\stackrel{ˆ}{H}}_{k}\left({\stackrel{ˆ}{x}}^{n}\right)$ .

Despite this excellent potential performance, there is still a tremendous challenge. Brute force computation of the globally minimum energysolution $\stackrel{ˆ}{{x}^{n}}$ 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,

$ϵ\left({w}^{n}\right)={H}_{k}\left({w}^{n}\right)-\beta d\left({x}^{n},{w}^{n}\right).$

The Boltzmann probability mass function (pmf) is

${f}_{s}\left({w}^{n}\right)\triangleq \frac{1}{{Z}_{t}}exp\left\{-\frac{1}{t}\epsilon \left({w}^{n}\right)\right\},$

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 ${w}^{n\setminus i}\triangleq \left\{{w}_{n}:\phantom{\rule{4pt}{0ex}}n\ne i\right\}$ as induced by the joint pmf in [link] , which is computed as follows,

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