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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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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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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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
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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.
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Smarajit 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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