Weighted and Controlled Frames - · PDF fileConclusion and Perspectives ... is called a frame...

Weighted andControlled

Frames

P. Balazs

IntroductionFrames

Motivation

ControlledFramesNumerical Aspects ofControlled Frames

WeightedFramesWeighted Frames asControlled Frames

Connection to FrameMultipliers

Inversion withWeights

ConclusionandPerspectives

Weighted and Controlled Frames

P. BalazsAcoustics Research Institute,

Austrian Academy of Sciences

joint work with J.P.-Antoinesupported by HASSIP

Strobl07Trends in Harmonic Analysis

June 18-22, 2007Strobl, Salzburg, AUSTRIA


Frames

P. Balazs

IntroductionFrames

Motivation







1 IntroductionFramesMotivation

2 Controlled FramesNumerical Aspects of Controlled Frames

3 Weighted FramesWeighted Frames as Controlled FramesConnection to Frame MultipliersInversion with Weights

4 Conclusion and Perspectives


Frames

P. Balazs

IntroductionFrames

Motivation












Frames

P. Balazs

IntroductionFrames

Motivation












Frames

P. Balazs

IntroductionFrames

Motivation












Frames

P. Balazs

IntroductionFrames

Motivation






Introduction


Frames

P. Balazs

IntroductionFrames

Motivation






Frames

Definition

A sequence (ψn, n ∈ Γ) is called a frame for the Hilbertspace H, if there exist constants m > 0 and M <∞ suchthat

m ‖f‖2 6∑n∈Γ

|〈f , ψn〉|2 6 M ‖f‖2 ,∀ f ∈ H.

m is a lower, M an upper frame bound. If the bounds can bechosen such that m = M, the frame is called tight.

Frame operator:

L(f ) =∑

n

〈f , ψn〉ψn.

Canonical dual frame:(ψ̃n

)=(L−1ψn

), with

f =∑n∈Γ

⟨f , ψ̃n

⟩ψn∀f ∈ H.


Frames

P. Balazs

IntroductionFrames

Motivation






Frames

Definition

A sequence (ψn, n ∈ Γ) is called a frame for the Hilbertspace H, if there exist constants m > 0 and M <∞ suchthat

m ‖f‖2 6∑n∈Γ

|〈f , ψn〉|2 6 M ‖f‖2 ,∀ f ∈ H.

m is a lower, M an upper frame bound. If the bounds can bechosen such that m = M, the frame is called tight.

Frame operator:

L(f ) =∑

n

〈f , ψn〉ψn.

Canonical dual frame:(ψ̃n

)=(L−1ψn

), with

f =∑n∈Γ

⟨f , ψ̃n

⟩ψn∀f ∈ H.


Frames

P. Balazs

IntroductionFrames

Motivation






Motivation

Mm = κ(L) condition number: numerical behavior


Frames

P. Balazs

IntroductionFrames

Motivation






Motivation

Mm = κ(L) condition number: numerical behavior

[5] : weighted and controlled frames numerical tool for spher-ical wavelets.

Here investigated in detail!


Frames

P. Balazs

IntroductionFrames

Motivation






Controlled Frames


Frames

P. Balazs

IntroductionFrames

Motivation






Controlled Frames

GL(H1,H2): bounded operators with bounded inverse.GL(+)(H): positive operators in GL(H).

Definition

C ∈ GL(H). A C-controlled frame is a family of vectorsΨ = (ψn ∈ H : n ∈ Γ), such that there exist two constantsmCL < 0 and MCL <∞ satisfying

mCL ‖f‖2 6∑

n

〈ψn, f 〉〈f ,Cψn〉 6 MCL ‖f‖2 , for all f ∈ H.

LCf =∑n∈Γ

〈ψn, f 〉Cψn is the controlled frame operator.

Equivalent to Lc ∈ GL(+)(H).


Frames

P. Balazs

IntroductionFrames

Motivation






Controlled Frames

Proposition

Ψ a C-controlled frame. Then Ψ is a classical frame.Furthermore C L = L C∗ and so∑n∈Γ

〈ψn, f 〉Cψn =∑n∈Γ

〈Cψn, f 〉ψn.

=⇒ Generalized criterion to check if a given sequenceconstitutes a frame.

Theorem

C ∈ GL(H) self-adjoint. Ψ is a C-controlled frame if and onlyif it is a (classical) frame for H, C is positive and commuteswith the frame operator L.


Frames

P. Balazs

IntroductionFrames

Motivation






Controlled Frames

Proposition



〈Cψn, f 〉ψn.


Theorem



Frames

P. Balazs

IntroductionFrames

Motivation






Controlled Frames

Proposition



〈Cψn, f 〉ψn.


Theorem



Frames

P. Balazs

IntroductionFrames

Motivation






Numerical Aspects of Controlled Frames

L−1C C = (CL)−1 C = L−1: finding C for controlled frame ⇐⇒

preconditioning! (I.e.: Instead of solving Ax = b, solvePAx = Pb.)

If κ(LC) = MCLmCL

< Mm = κ(L) =⇒ iterative algorithms get more

efficient .

ε := MCL−mCLMCL+mCL

=⇒ For n-th iteration gn ‖f − gn‖ 6 εn ‖f‖Cn .


Frames

P. Balazs

IntroductionFrames

Motivation






Numerical Aspects of Controlled Frames

L−1C C = (CL)−1 C = L−1: finding C for controlled frame ⇐⇒

preconditioning! (I.e.: Instead of solving Ax = b, solvePAx = Pb.)

If κ(LC) = MCLmCL

< Mm = κ(L) =⇒ iterative algorithms get more

efficient .

ε := MCL−mCLMCL+mCL

=⇒ For n-th iteration gn ‖f − gn‖ 6 εn ‖f‖Cn .


Frames

P. Balazs

IntroductionFrames

Motivation






Preconditioning

Preconditioning for Gabor frames [4]:

Convergence with iteration : Relative difference of iteration steps (Gaussian

window, n = 1440, a = 32 and b = 30.)


Frames

P. Balazs

IntroductionFrames

Motivation






Weighted Frames


Frames

P. Balazs

IntroductionFrames

Motivation






Weighted Frames

Definition

Let Ψ = (ψn : n ∈ Γ) be a sequence of elements in H and(wn : n ∈ Γ) ⊆ R+

∗ a sequence of strictly positive weights. Thispair is called a w-frame of H if there exist constants m > 0 andM <∞ such that

mw ‖f‖2 6∑n∈Γ

wn |〈f , ψn〉|2 6 Mw ‖f‖2.

For (ωn) ⊆ C we call (ωnψn) a weighted frame if this sequenceforms a frame, i.e.,

momega ‖f‖2 6∑n∈Γ

|ωn|2 |〈f , ψn〉|2 6 Mω ‖f‖2.

‘weighted frame’ 6= ‘w-frame’!Related to signed frames [6].


Frames

P. Balazs

IntroductionFrames

Motivation






Weighted Frames as Controlled Frames

Weighted frames = controlled frames?

Cψk := wkψk

But:

C

(∑k

ckψk

):=∑

ckCψk =∑

ckwkψk

is in general not well defined!And

C

(∑k

⟨f , ψ̃k

⟩ψk

):=∑

k

⟨f , ψ̃k

⟩wkψk

does not ensure Cψk = wkψk! (See [3].)Example: wn = −1 ⇒ (wnψn) a frame for every frameΨ = (ψn). Not a controlled frame, because C ≡ −1 isself-adjoint, but not positive.


Frames

P. Balazs

IntroductionFrames

Motivation








Cψk := wkψk

But:

C

(∑k

ckψk

):=∑

ckCψk =∑

ckwkψk


C

(∑k

⟨f , ψ̃k

⟩ψk

):=∑

k

⟨f , ψ̃k

⟩wkψk



Frames

P. Balazs

IntroductionFrames

Motivation








Cψk := wkψk

But:

C

(∑k

ckψk

):=∑

ckCψk =∑

ckwkψk


C

(∑k

⟨f , ψ̃k

⟩ψk

):=∑

k

⟨f , ψ̃k

⟩wkψk



Frames

P. Balazs

IntroductionFrames

Motivation







Suppose: ∃C with Cψk = wkψk.

Corollary

Let C ∈ GL(H) be self-adjoint and diagonal on Ψ = (ψn) andassume it generates a controlled frame. Then the sequence(wn), which verifies the relations Cψn = wnψn, is positive andsemi-normalized, i.e. there are bounds b > a > 0, such thata 6 |cn| 6 b.


Frames

P. Balazs

IntroductionFrames

Motivation






Semi-normalized Weights

Lemma

Let (wn) be a semi-normalized sequence with bounds a,b.Then if (ψn) is a frame with bounds m and M, (wnψn) is alsoa frame with bounds a2m and b2M. The sequence

(wn

−1ψ̃n

)is a dual frame.


Frames

P. Balazs

IntroductionFrames

Motivation






Examples

(wn

−1ψ̃n

)is a dual, but not the canonical dual.

Example: Parseval frame, i.e. self-dual frame:

Ψ ={(

0.81650

),

(−0.40820.7071

),

(−0.4082−0.7071

)}weights: (w1,w2,w2) = ( 1

2 , 1, 2).


Frames

P. Balazs

IntroductionFrames

Motivation






Examples


Frames

P. Balazs

IntroductionFrames

Motivation






Examples

Weights (wn = ±1) =⇒ frame with same bounds.

Also valid for Non-semi-normalized weights? It is ingeneral not enough for the weights to be strictlypositive, wn > 0.Example: ONB (en) , ψn = 1

n en. This is not a frame!

(ψn = 1n en) with the weight (wn = n) ⇒ frame.


Frames

P. Balazs

IntroductionFrames

Motivation






Examples






Frames

P. Balazs

IntroductionFrames

Motivation






Examples






Frames

P. Balazs

IntroductionFrames

Motivation






Connection to Frame Multipliers

Frame multipliers [1]: Mm,ψn f =∑

nmn 〈f , ψn〉ψn.

Theorem

Let (ψn) be a sequence of elements in H. Let (wn) be a sequence ofpositive, semi-normalized weights. Then the following properties areequivalent:

1 (ψn) is a frame.

2 Mwn,ψn is a positive and invertible operator.

3 There are constants m′ > 0 and M′ <∞ such that

m′ ‖f‖2 6∑n∈Γ

〈f , ψn〉wn ψn 6 M′ ‖f‖2 .

4 (√

wnψn) is a frame.

5 M(w′n),(ψn) is a positive and invertible operator for any positive,

semi-normalized sequence (w′n).

6 (wnψn) is a frame.


Frames

P. Balazs

IntroductionFrames

Motivation






Connection to Frame Multipliers

Frame multipliers [1]: Mm,ψn f =∑

nmn 〈f , ψn〉ψn.

Theorem

Let (ψn) be a sequence of elements in H. Let (wn) be a sequence ofpositive, semi-normalized weights. Then the following properties areequivalent:

1 (ψn) is a frame.

2 Mwn,ψn is a positive and invertible operator.

3 There are constants m′ > 0 and M′ <∞ such that

m′ ‖f‖2 6∑n∈Γ

〈f , ψn〉wn ψn 6 M′ ‖f‖2 .

4 (√

wnψn) is a frame.

5 M(w′n),(ψn) is a positive and invertible operator for any positive,

semi-normalized sequence (w′n).

6 (wnψn) is a frame.


Frames

P. Balazs

IntroductionFrames

Motivation






Current development:finding weights


Frames

P. Balazs

IntroductionFrames

Motivation






Inversion with Weights

Weighted frames 6= controlled frames. Preconditioningapproach not possible!

So how can weights be found, such that the quotient of thebounds become smaller?


Frames

P. Balazs

IntroductionFrames

Motivation






Weights by ’Guessing’

Example: Suppose the frame element ψk0 is not orthogonal to therest, let

wk ={

0 k = k01 otherwise .

=⇒ weighted frame!Look at definition:

m ‖f‖2 6∑n∈Γ

|〈f , ψn〉|2 6 M ‖f‖2

Guess: use measure how ’important’ a frame element is:

w(2)n =

‖ψn‖2∑k|〈ψn, ψk〉|2

.

For the case of weighted ONBs this certainly finds the optimalsolution. Or

w(∞)n =

‖ψn‖sup

k|〈ψn, ψk〉|

.


Frames

P. Balazs

IntroductionFrames

Motivation








wk ={



m ‖f‖2 6∑n∈Γ

|〈f , ψn〉|2 6 M ‖f‖2


w(2)n =

‖ψn‖2∑k|〈ψn, ψk〉|2

.


w(∞)n =

‖ψn‖sup

k|〈ψn, ψk〉|

.


Frames

P. Balazs

IntroductionFrames

Motivation








wk ={



m ‖f‖2 6∑n∈Γ

|〈f , ψn〉|2 6 M ‖f‖2


w(2)n =

‖ψn‖2∑k|〈ψn, ψk〉|2

.


w(∞)n =

‖ψn‖sup

k|〈ψn, ψk〉|

.


Frames

P. Balazs

IntroductionFrames

Motivation






Weights by Best Approximation with FrameMultiplier

This question can be “translated” to the frame multipliercontext: Approximate the identity as a frame multiplier?

The symbol of the best approximation of the identity by aframe multiplier using Hilbert-Schmidt topology will be usedas the weights w(mult).

The best approximation of an operator T ∈ HS(H) by aframe multiplier Mw(m),ψk

fixing the frame is given [2] by

P(T) =∑

k

[(|Gψk |

2)†〈Tψk, ψk〉H

]ψk ⊗ ψk


Frames

P. Balazs

IntroductionFrames

Motivation






Weights by Best Approximation with FrameMultiplier

This question can be “translated” to the frame multipliercontext: Approximate the identity as a frame multiplier?

The symbol of the best approximation of the identity by aframe multiplier using Hilbert-Schmidt topology will be usedas the weights w(mult).

The best approximation of an operator T ∈ HS(H) by aframe multiplier Mw(m),ψk

fixing the frame is given [2] by

P(T) =∑

k

[(|Gψk |

2)†〈Tψk, ψk〉H

]ψk ⊗ ψk


Frames

P. Balazs

IntroductionFrames

Motivation






Numerical tests

Sample data for frames with 4 elements in 2 dimensions:

Out of 100000 runs the guessed weight improved the condition number 100000 times.Weights by approximation with multiplers improved the condition number 59686 times.Weights by approximation with multiplers were the best solution 31244 times.


Frames

P. Balazs

IntroductionFrames

Motivation






Numerical tests

M = 4, dim = 3


Frames

P. Balazs

IntroductionFrames

Motivation






Numerical tests

M = 6, dim = 3


Frames

P. Balazs

IntroductionFrames

Motivation






Numerical tests

M = 11, dim = 10


Frames

P. Balazs

IntroductionFrames

Motivation






Numerical tests

M = 20, dim = 10


Frames

P. Balazs

IntroductionFrames

Motivation






Perspectives


Frames

P. Balazs

IntroductionFrames

Motivation






Perspectives

Best approximation by multipliers in operator norm, e.g.using LMIs.

Further numerical tests to find better guesses.Investigate partial algebra structure of multipliers


Frames

P. Balazs

IntroductionFrames

Motivation






Perspectives




Frames

P. Balazs

IntroductionFrames

Motivation






Perspectives




Frames

P. Balazs

IntroductionFrames

Motivation






Conclusions

Introduced controlled framesThey give a generalized way to check frame condition.Are equivalent to preconditioning.

Introduced weighted framesIn general not controlled frames.Gave first data how to find good weights to find a (notthe) dual.


Frames

P. Balazs

IntroductionFrames

Motivation






Conclusions

Introduced controlled framesThey give a generalized way to check frame condition.Are equivalent to preconditioning.

Introduced weighted framesIn general not controlled frames.Gave first data how to find good weights to find a (notthe) dual.


Frames

P. Balazs

IntroductionFrames

Motivation






Thank you.

P. Balazs.Basic definition and properties of Bessel multipliers.Journal of Mathematical Analysis and Applications, 325(1):571–585, January 2007.

P. Balazs.Hilbert-Schmidt operators and frames - classification and approximation.submitted, 2007.

P. Balazs.Matrix-representation of operators using frames.accepted for Sampling Theory in Signal and Image Processing (STSIP), 2007.

P. Balazs, H. G. Feichtinger, M. Hampejs, and G. Kracher.Double preconditioning for Gabor frames.IEEE Transactions on Signal Processing, 54(12):4597–4610, December 2006.

I. Bogdanova, P. Vandergheynst, J.-P. Antoine, L. Jacques, and M. Morvidone.Stereographic wavelet frames on the sphere.Applied and Computational Harmonic Analysis, 19:223–252, 2005.

I. Peng and S. Waldon.Signed frames and Hadamard products of Gram matrices.Linear Algebra Appl., 347:131–157, 2002.

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