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-   -   Related to Legendre polynomials (http://book.caltech.edu/bookforum/showthread.php?t=4347)

skwong 06-11-2013 07:29 AM

Related to Legendre polynomials
 
In lect 12, we learn the regularization for a nonlinear transformation which is
composed of Legendre polynomials. But then in hw6, Q2 to Q6,
The nonlinear transform doesn't talk anything about Legendre polynomials.

Does it mean, Legendre polynomials is used in deriving the theory,
and any function, including the \phi in hw6, can actually be
expressed in terms of Legendre polynomials ? Further, does it only hold true
for the range -1 to 1 ?

yaser 06-11-2013 12:20 PM

Re: Related to Legendre polynomials
 
Quote:

Originally Posted by skwong (Post 11112)
In lect 12, we learn the regularization for a nonlinear transformation which is
composed of Legendre polynomials. But then in hw6, Q2 to Q6,
The nonlinear transform doesn't talk anything about Legendre polynomials.

Does it mean, Legendre polynomials is used in deriving the theory,
and any function, including the \phi in hw6, can actually be
expressed in terms of Legendre polynomials ? Further, does it only hold true
for the range -1 to 1 ?

Any polynomial, including the one in HW 6, can be expressed in terms Legendre polynomials. The polynomial can be defined outside the range [-1,+1] but the orthognality of Legendre polynomials is within [-1,+1].

skwong 06-11-2013 06:58 PM

Re: Related to Legendre polynomials
 
Thank you very much for the prompt reply. Does it mean that if we want to use
this regulariser, then we need to normalise the input data to range -1 to 1 ?

yaser 06-11-2013 07:36 PM

Re: Related to Legendre polynomials
 
Quote:

Originally Posted by skwong (Post 11130)
Thank you very much for the prompt reply. Does it mean that if we want to use
this regulariser, then we need to normalise the input data to range -1 to 1 ?

The regularizer can be used for any domain. It just has a nice interpretation in the [-1,+1] domain because of the properties of Legendre polynomials in that domain.


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