# 5.2 Greedy algorithms  (Page 2/3)

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$\begin{array}{cc}\hfill {x}_{k}& =arg\underset{x}{min}{\parallel y-{\Phi }_{\Omega }x\parallel }_{2},\hfill \\ \hfill {\stackrel{^}{\alpha }}_{t}& ={\Phi }_{\Omega }{x}_{t},\hfill \\ \hfill {r}_{t}& =y-{\stackrel{^}{\alpha }}_{t}.\hfill \end{array}$

These steps are repeated until convergence. This is known as Orthogonal Matching Pursuit (OMP)  [link] . Tropp and Gilbert  [link] proved that OMP can be used to recover a sparse signal with high probability using compressive measurements. The algorithm converges in at most $K$ iterations, where K is the sparsity, but requires the added computational cost of orthogonalization at each iteration. Indeed, the total complexity of OMP can be shown to be $O\left(MNK\right).$

While OMP is provably fast and can be shown to lead to exact recovery, the guarantees accompanying OMP for sparse recovery are weaker than those associated with optimization techniques . In particular, the reconstruction guarantees are not uniform , i.e., it cannot be shown that a single measurement matrix with $M=CKlogN$ rows can be used to recover every possible $K-$ sparse signal with $M=CKlogN$ measurements. (Although it is possible to obtain such uniform guarantees when it is acceptable to take more measurements. For example, see [link] .) Another issue with OMP is robustness to noise; it is unknown whether the solution obtained by OMP will only be perturbed slightly by the addition of a small amount of noise in the measurements. Nevertheless, OMP is an efficient method for CS recovery, especially when the signal sparsity $K$ is low. A pseudocode representation of OMP is shown below.

Inputs: Measurement matrix $\Phi$ , signal measurements $y$ Outputs: Sparse representation $\stackrel{^}{x}$ Initialize: ${\stackrel{^}{\theta }}_{0}=0$ , $r=y$ , $\Omega =\varnothing$ , $i=0$ . while ħalting criterion false do 1. $i←i+1$ 2. $b←{\Phi }^{T}r$ {form residual signal estimate} 3. $\Omega ←\Omega \cup \mathrm{supp}\left(\mathbf{T}\left(b,1\right)\right)$ {add index of residual's largest magnitude entry to signal support} 4. ${\stackrel{^}{x}}_{i}{|}_{\Omega }←{\Phi }_{\Omega }^{†}x$ , ${\stackrel{^}{x}}_{i}{|}_{{\Omega }^{C}}←0$ {form signal estimate} 5. $r←y-\Phi {\stackrel{^}{x}}_{i}$ {update measurement residual} end while return $\stackrel{^}{x}←{\stackrel{^}{x}}_{i}$

## Stagewise orthogonal matching pursuit (stomp)

Orthogonal Matching Pursuit is ineffective when the signal is not very sparse as the computational cost increases quadratically with the number of nonzeros $K$ . In this setting, Stagewise Orthogonal Matching Pursuit (StOMP)  [link] is a better choice for approximately sparse signals in a large-scale setting.

StOMP offers considerable computational advantages over ${\ell }_{1}$ minimization and Orthogonal Matching Pursuit for large scale problems with sparse solutions. The algorithm starts with an initial residual ${r}_{0}=y$ and calculates the set of all projections ${\Phi }^{T}{r}_{k-1}$ at the ${k}^{th}$ stage (as in OMP). However, instead of picking a single dictionary element, it uses a threshold parameter $\tau$ to determine the next best set of columns of $\Phi$ whose correlations with the current residual exceed $\tau$ . The new residual is calculated using a least squares estimate of the signal using this expanded set of columns, just as before.

Unlike OMP, the number of iterations in StOMP is fixed and chosen before hand; $S=10$ is recommended in [link] . In general, the complexity of StOMP is $O\left(KNlogN\right)$ , a significant improvement over OMP. However, StOMP does not bring in its wake any reconstruction guarantees. StOMP also has moderate memory requirements compared to OMP where the orthogonalization requires the maintenance of a Cholesky factorization of the dictionary elements.

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
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
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
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
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 to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
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