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In this module we provide an overview of some of the most common greedy algorithms and their application to the problem of sparse recovery.


As opposed to solving a (possibly computationally expensive) convex optimization program, an alternate flavor to sparse recovery is to apply methods of sparse approximation . Recall that the goal of sparse recovery is to recover the sparsest vector x which explains the linear measurements y . In other words, we aim to solve the (nonconvex) problem:

min I | I | : y = i I φ i x i ,

where I denotes a particular subset of the indices i = 1 , ... , N , and φ i denotes the i th column of Φ . It is well known that searching over the power set formed by the columns of Φ for the optimal subset I * with smallest cardinality is NP-hard. Instead, classical sparse approximation methods tackle this problem by greedily selecting columns of Φ and forming successively better approximations to y .

Matching pursuit

Matching Pursuit (MP), named and introduced to the signal processing community by Mallat and Zhang  [link] , [link] , is an iterative greedy algorithm that decomposes a signal into a linear combination of elements from a dictionary. In sparse recovery, this dictionary is merely the sampling matrix Φ R M × N ; we seek a sparse representation ( x ) of our “signal” y .

MP is conceptually very simple. A key quantity in MP is the residual r R M ; the residual represents the as-yet “unexplained” portion of the measurements. At each iteration of the algorithm, we select a vector from the dictionary that is maximally correlated with the residual r :

λ k = arg max λ r k , φ λ φ λ φ λ 2 .

Once this column is selected, we possess a “better” representation of the signal, since a new coefficient indexed by λ k has been added to our signal approximation. Thus, we update both the residual and the approximation as follows:

r k = r k - 1 - r k - 1 , φ λ k φ λ k φ λ k 2 , x ^ λ k = x ^ λ k + r k - 1 , φ λ k .

and repeat the iteration. A suitable stopping criterion is when the norm of r becomes smaller than some quantity. MP is described in pseudocode form below.

Inputs: Measurement matrix Φ , signal measurements y Outputs: Sparse signal x ^ initialize: x ^ 0 = 0 , r = y , i = 0 . while ħalting criterion false do 1. i i + 1 2. b Φ T r {form residual signal estimate} 3. x ^ i x ^ i - 1 + T ( 1 ) {update largest magnitude coefficient} 4. r r - Φ x ^ i {update measurement residual} end while return x ^ x ^ i

Although MP is intuitive and can find an accurate approximation of the signal, it possesses two major drawbacks: (i) it offers no guarantees in terms of recovery error; indeed, it does not exploit the special structure present in the dictionary Φ ; (ii) the required number of iterations required can be quite large. The complexity of MP is O ( M N T )   [link] , where T is the number of MP iterations

Orthogonal matching pursuit (omp)

Matching Pursuit (MP) can prove to be computationally infeasible for many problems, since the complexity of MP grows linearly in the number of iterations T . By employing a simple modification of MP, the maximum number of MP iterations can be upper bounded as follows. At any iteration k , Instead of subtracting the contribution of the dictionary element with which the residual r is maximally correlated, we compute the projection of r onto the orthogonal subspace to the linear span of the currently selected dictionary elements. This quantity thus better represents the “unexplained” portion of the residual, and is subtracted from r to form a new residual, and the process is repeated. If Φ Ω is the submatrix formed by the columns of Φ selected at time step t , the following operations are performed:

Questions & Answers

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
why we need to study biomolecules, molecular biology in nanotechnology?
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
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
why its coecients must have a power-law rate of decay with q > 1/p. ?
Nader Reply

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Source:  OpenStax, Introduction to compressive sensing. OpenStax CNX. Mar 12, 2015 Download for free at http://legacy.cnx.org/content/col11355/1.4
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