# 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.

where we get a research paper on Nano chemistry....?
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
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
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
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
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