<< Chapter < Page Chapter >> Page >

Review: denoising in smooth function spaces i - method of sieves

Suppose we make noisy measurements of a smooth function:

Y i = f * ( x i ) + W i , i = { 1 , ... , n } ,


W i i . i . d . N 0 , σ 2


x i = i n .

The unknown function f * is a map

f * : [ 0 , 1 ] R .

In Lecture 4 , we consider this problem in the case where f * was Lipschitz on [ 0 , 1 ] . That is, f * satisfied

| f * ( t ) - f * ( s ) | L | t - s | , t , s [ 0 , 1 ]

where L > 0 is a constant. In that case, we showed that by using a piecewise constant function on a partition of n 1 3 equal-size bins [link] we were able to obtain an estimator f ^ n whose mean square error was

E f * - f ^ n 2 = O n - 2 3 .
Example of the piecewise constant approximation of f *

In this lecture we will use the Maximum Complexity-Regularized Likelihood Estimation result we derived in Lecture 14 to extend our denoising scheme in several important ways.

To begin with let's consider a broader class of functions.

Hölder spaces

For 0 < α < 1 , define the space of functions

H α ( C α ) = | f | < C α : sup x , h | f ( x + h ) - f ( x ) | | h | α C α

for some constant C α < and where f L . H α above contains functions that are bounded, but less smooth than Lipschitz functions. Indeed, the space of Lipschitzfunctions can be defined as H 1 ( α = 1 )

H 1 ( C 1 ) = | f | < C 1 : sup x , h | f ( x + h ) - f ( x ) | | h | C 1

for C 1 < . Functions in H 1 are continuous, but those in H α , α < 1 , are not in general.

Let's also consider functions that are smoother than Lipschitz. If α = 1 + β , where 0 < β < 1 , then define

H α ( C α ) = f H 1 ( C α ) : f x H β ( C α ) .

In other words, H α , 1 < α < 2 , contains Lipschitz functions that are also differentiable and their derivatives are Hölder smooth with smoothness β = α - 1 .

And finally, let

H 2 ( C 2 ) = f : f x H 1 ( C 2 )

contain functions that have continuous derivatives, but that are notnecessarily twice-differentiable.

If f H α ( C α ) , 0 < α 2 , then we say that f is Hölder - α smooth with Hölder constant C α . The notion of Hölder smoothness can also be extended to α > 2 in a straightforward way.

Note: If α 1 < α 2 then

f H α 2 f H α 1 .

Summarizing, we can describe Hölder spaces as follows. If f * H α C α for some 0 < α 2 and C α < , then

  • 0 < α 1                    | f * ( t ) - f * ( s ) | C α | t - s | α
  • 1 < α 2                    f * x ( t ) - f * x ( s ) C α | t - s | α - 1

Note that in general there is a natural relationship between the Hölder space containing the function and the approximation classused to estimate the function. Here we will consider functions which are Hölder - α smooth where 0 < α 2 and work with piecewise linear approximations. If we were to consider smootherfunctions, α > 2 we would need consider higher order approximation functions, i.e. quadratic, cubic, etc.

Denoising example for signal-plus-gaussian noise observation model

Now let's assume f * H α ( C α ) for some unknown α ( 0 < α 2 ) ; i.e. we don't know how smooth f * is. We will use our observations

Y i = f * ( x i ) + W i , i = { 1 , ... , n } ,

to construct an estimator f ^ n . Intuitively, the smoother f * is, the better we should be able to estimate it. Can we take advantage ofextra smoothness in f * if we don't know how smooth it is? The smoother f * is, the more averaging we can perform to reduce noise. In other words for smoother f * we should average over larger bins. Also, we will need to exploit the extra smoothnessin our approximation of f * . To that end, we will consider candidate functions that are piecewiselinear functions on uniform partitions of [ 0 , 1 ] . Let

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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
How can I make nanorobot?
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
how can I make nanorobot?
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
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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Statistical learning theory. OpenStax CNX. Apr 10, 2009 Download for free at http://cnx.org/content/col10532/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Statistical learning theory' conversation and receive update notifications?