# 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,

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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 .?
Renato
Berger describes sociologists as concerned with
what is hormones?
Wellington
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