# 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,

what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
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LITNING
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Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
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why we need to study biomolecules, molecular biology in nanotechnology?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
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