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Thresholding and Greedy Bases

Module by: Jason Laska, Ronald DeVore. E-mail the authors

We shall next discuss some notions related to best nn-term approximation.

Thresholding

  1. Let XX be a Hilbert space. Given ff, let Λ ϵ (f)={j:|c j (f)|>ϵ}Λ ϵ (f)={j:|c j (f)|>ϵ}. The thresholding operator T ϵ T ϵ is defined by
    T ϵ f= jΛ ϵ (f) C j (f)φ j .T ϵ f= jΛ ϵ (f) C j (f)φ j .
    (1)
    It is easy to see that for each ϵϵ, T ϵ fT ϵ f is the best approximation to ff using NN terms where NN is the cardinality of Λ ϵ Λ ϵ :
    f-T ϵ X =σ N (f) X .f-T ϵ X =σ N (f) X .
    (2)
    Thresholding is easily implemented on a computer.
  2. The thresholding scheme above can be generalized if XX is not a Hilbert space provided the dictionary has some specific structure. For example, when
    1. The dictionary is the wavelet basis and X=L p X=L p , 1<p<1<p<.
    2. X=l p X=l p and the dictionary is the canonical basis δ j =φ j δ j =φ j . ex: (0,0,...,1,0)(0,0,...,1,0)
    3. For a general Banach space XX and the dictionary (φ j )(φ j ) is a greedy basis.

Greedy Bases

We briefly describe the notion of greedy basis.

definition 1

Given XX, we say (φ j )(φ j ) is a greedy basis for XX if for each ϵ>0ϵ>0,

f-T ϵ f X C(X)σ N (f) X f-T ϵ f X C(X)σ N (f) X
(3)

where NN is the cardinality of Λ ϵ Λ ϵ .

definition 2

A basis φ j φ j is said to be unconditional if

±c j φ j X Cc j φ j X ±c j φ j X Cc j φ j X
(4)

or equivalently

c j φ j X Cd j φ j X where |c j ||d j |.c j φ j X Cd j φ j X where |c j ||d j |.
(5)

This is an older concept from functional analysis. In words, this definition says that if the terms c j c j are rearranged, the series c j φ j c j φ j will still converge. This is not generally true for all bases.

definition 3

A basis φ j φ j is said to be democratic if

jΛ φ j C jΛ ' φ j , jΛ φ j C jΛ ' φ j ,
(6)

where the cardinality of Λ ' Λ ' equals the cardinality of ΛΛ.

remark 1

(φ j )(φ j ) greedy (φ j )(φ j ) is both unconditional and democratic.

Some examples involving the last two definitions:

  • The fourier basis in L p L p is not democratic, but is unconditional for l<p<l<p<.
  • The wavelet basis contains both of these properties, and is therefore greedy.

Theorem 1

If X=L p X=L p has (φ j )(φ j ) greedy, B={φ j }B={φ j }, f= j=1 c j (f)φ j f= j=1 c j (f)φ j , c j (f)=f,ψ j c j (f)=f,ψ j where ψ j ψ j is a dual basis,

fA r (B)(c j (f))w l τ ,1 τ=r+1 pfA r (B)(c j (f))w l τ ,1 τ=r+1 p
(7)

and

f A r c j (f) l τ f A r c j (f) l τ
(8)

Let us now consider a specific setting that we shall be concerned with a lot in this course. We shall examine some of the concepts we have introduced in the finite dimensional space of of all sequence (points) in R N R N . Recall that we can put many different norms on this space including the p p norms and the weak p p norms.

remark 2

Given a vector =(x 1 ,x 2 ,...,x N )R N =(x 1 ,x 2 ,...,x N )R N . The best approximation to xx from Σ n Σ n in the p p norm is to take the vector in Σ n Σ n which shares the nn largest values of xx. Its error of approximation satisfies

σ n (x) p Cn -r x w l τ ,1 τ=r+1 pσ n (x) p Cn -r x w l τ ,1 τ=r+1 p
(9)

remark 3

x w l τ x l τ ,1 τ=r+1 px w l τ x l τ ,1 τ=r+1 p.

example 1

For p=1p=1 and r=3r=3, σ n (x) 1 Cn -3 x w l τ σ n (x) 1 Cn -3 x w l τ and 1 τ=41 τ=4. In words, this equation shows what kind of ττ is needed for a given decay rate (or given some ττ, what kind of decay rate will be achieved) to approximate with certain ability.

example 2

Show σ n (x) l p Cn -r x l τ σ n (x) l p Cn -r x l τ holds with C=1C=1.

Proof: Let Λ n :={i:|x i | largest }Λ n :={i:|x i | largest },

σ n (x) l p p = iΛ n |x i | p σ n (x) l p p = iΛ n |x i | p
(10)
iΛ n |x i | p-τ |x i | τ iΛ n |x i | p-τ |x i | τ
(11)
(x w l τ n -1 τ ) p-τ (|x i | τ )(x w l τ n -1 τ ) p-τ (|x i | τ )
(12)
x l τ p-τ x l τ τ n=n -rp x l τ p x l τ p-τ x l τ τ n=n -rp x l τ p
(13)

and so

σ n (x) l p p n -rp x l τ p σ n (x) l p p n -rp x l τ p
(14)

σ n (x) l p n -r x l τ σ n (x) l p n -r x l τ

remark 4

For X=L p X=L p , {φ j }{φ j } a wavelet basis, we can say wavelet coefficients of ff are in l p l p is equivalent to ff is in a certain Besov class (roughly speaking ff has rr derivatives and f (r) L τ f (r) L τ ). We refer the reader to (Reference) for precise formulations of results of this type.

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