<< Chapter < Page Chapter >> Page >
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

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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?
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.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, An introduction to compressive sensing. OpenStax CNX. Apr 02, 2011 Download for free at http://legacy.cnx.org/content/col11133/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'An introduction to compressive sensing' conversation and receive update notifications?