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Here we use a more general definition of norm in addition to L_2. In particular, we consider L_p.

Approximation with other norms and error measures

Most of the discussion about the approximate solutions to Ax = b are about the result of minimizing the l 2 equation error | | A x - b | | 2 and/or the l 2 norm of the solution | | x | | 2 because in some cases that can be done by analytic formulas and also because the l 2 norm has a energy interpretation. However, both the l 1 and the l [link] have well known applications that are important [link] , [link] and the more general l p error is remarkably flexible [link] , [link] . Donoho has shown [link] that l 1 optimization gives essentially the same sparsity as the true sparsity measure in l 0 .

In some cases, one uses a different norm for the minimization of the equation error than the one for minimization of the solution norm. And inother cases, one minimizes a weighted error to emphasize some equations relative to others [link] . A modification allows minimizing according to one norm for one set of equations and another for a different set. A more generalerror measure than a norm can be used which used a polynomial error [link] which does not satisfy the scaling requirement of a norm, but is convex. One could even use theso-called l p norm for 1 > p > 0 which is not even convex but is an interesting tool for obtaining sparse solutions.

Different l p norms: p = .2, 1, 2, 10.

Note from the figure how the l 10 norm puts a large penalty on large errors. This gives a Chebyshev-like solution. The l 0 . 2 norm puts a large penalty on small errors making them tend to zero. This (and the l 1 norm) give a sparse solution.

The L p Norm approximation

The IRLS (iterative reweighted least squares) algorithm allows an iterative algorithm to be built from the analytical solutions of the weighted least squareswith an iterative reweighting to converge to the optimal l p approximation [link] .

The overdetermined system with more equations than unknowns

If one poses the l p approximation problem in solving an overdetermined set of equations (case 2 from Chapter 3), it comes from defining the equation error vector

e = A x - b

and minimizing the p-norm

| | e | | p = n | e n | p 1 / p


| | e | | p p = n | e n | p

neither of which can we minimize easily. However, we do have formulas [link] to find the minimum of the weighted squared error

| | W e | | 2 2 = n w n 2 | e n | 2

one of which is derived in [link] , equation [link] and is

x = [ A T W T W A ] - 1 A T W T W b

where W is a diagonal matrix of the error weights, w n . From this, we propose the iterative reweighted least squared (IRLS) error algorithmwhich starts with unity weighting, W = I , solves for an initial x with [link] , calculates a new error from [link] , which is then used to set a new weighting matrix W

W = d i a g ( w n ) ( p - 2 ) / 2

to be used in the next iteration of [link] . Using this, we find a new solution x and repeat until convergence (if it happens!).

This core idea has been repeatedly proposed and developed in different application areas over the past 50 years with a variety of success [link] . Used in this basic form, it reliably converges for 2 < p < 3 . In 1990, a modification was made to partially update the solution each iteration with

x ( k ) = q x ^ ( k ) + ( 1 - q ) x ( k - 1 )

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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?
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
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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