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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

what is variations in raman spectra for nanomaterials
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Crow Reply
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RAW Reply
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I think
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Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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scanning tunneling microscope
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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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
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Smarajit Reply
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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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