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In using sparsity in posing a signal processing problem (e.g. compressive sensing), an l 1 norm can be used (or even an l 0 “pseudo norm”) to obtain solutions with zero components if possible [link] , [link] .

In addition to using side conditions to achieve a unique solution, side conditions are sometimes part of the original problem. One interesting caserequires that certain of the equations be satisfied with no error and the approximation be achieved with the remaining equations.

Moore-penrose pseudo-inverse

If the l 2 norm is used, a unique generalized solution to [link] always exists such that the norm squared of the equation error ε T * ε and the norm squared of the solution x T * x are both minimized. This solution is denoted by

x = A + b

where A + is called the Moore-Penrose inverse [link] of A (and is also called the generalized inverse [link] and the pseudoinverse [link] )

Roger Penrose [link] showed that for all A , there exists a unique A + satisfying the four conditions:

A A + A = A
A + A A + = A +
[ A A + ] * = A A +
[ A + A ] * = A + A

There is a large literature on this problem. Five useful books are [link] , [link] , [link] , [link] , [link] . The Moore-Penrose pseudo-inverse can be calculated in Matlab [link] by the pinv(A,tol) function which uses a singular value decomposition (SVD) to calculate it. There are a variety of other numerical methodsgiven in the above references where each has some advantages and some disadvantages.

Properties

For cases 2a and 2b in Figure 1, the following N by N system of equations called the normal equations [link] , [link] have a unique minimum squared equation error solution (minimum ϵ T ϵ ). Here we have the over specified case with more equations than unknowns.A derivation is outlined in "Derivations" , equation [link] below.

A T * A x = A T * b

The solution to this equation is often used in least squares approximation problems. For these two cases A T A is non-singular and the N by M pseudo-inverse is simply,

A + = [ A T * A ] - 1 A T * .

A more general problem can be solved by minimizing the weighted equation error, ϵ T W T W ϵ where W is a positive semi-definite diagonal matrix of the error weights. The solution to that problem [link] is

A + = [ A T * W T * W A ] - 1 A T * W T * W .

For the case 3a in Figure 1 with more unknowns than equations, A A T is non-singular and has a unique minimum norm solution, | | x | | . The N by M pseudoinverse is simply,

A + = A T * [ A A T * ] - 1 .

with the formula for the minimum weighted solution norm | | x | | is

A + = [ W T W ] - 1 A T A [ W T W ] - 1 A T - 1 .

For these three cases, either [link] or [link] can be directly calculated, but not both. However, they are equal so you simply use the one with the non-singularmatrix to be inverted. The equality can be shown from an equivalent definition [link] of the pseudo-inverse given in terms of a limit by

A + = lim δ 0 [ A T * A + δ 2 I ] - 1 A T * = lim δ 0 A T * [ A A T * + δ 2 I ] - 1 .

For the other 6 cases, SVD or other approaches must be used. Some properties [link] , [link] are:

  • [ A + ] + = A
  • [ A + ] * = [ A * ] +
  • [ A * A ] + = A + A * +
  • λ + = 1 / λ for λ 0 else λ + = 0
  • A + = [ A * A ] + A * = A * [ A A * ] +
  • A * = A * A A + = A + A A *

It is informative to consider the range and null spaces [link] of A and A +

  • R ( A ) = R ( A A + ) = R ( A A * )
  • R ( A + ) = R ( A * ) = R ( A + A ) = R ( A * A )
  • R ( I - A A + ) = N ( A A + ) = N ( A * ) = N ( A + ) = R ( A )
  • R ( I - A + A ) = N ( A + A ) = N ( A ) = R ( A * )

The cases with analytical soluctions

The four Penrose equations in [link] are remarkable in defining a unique pseudoinverse for any A with any shape, any rank, for any of the ten cases listed in Figure 1.However, only four cases of the ten have analytical solutions (actually, all do if you use SVD).

Questions & Answers

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.
Damian
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Professor
I think
Professor
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Brian Reply
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Rafiq
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Damian
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LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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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
Rafiq
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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
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
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sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
Daniel
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Source:  OpenStax, Basic vector space methods in signal and systems theory. OpenStax CNX. Dec 19, 2012 Download for free at http://cnx.org/content/col10636/1.5
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