Understanding Sparsity Properties of Frames Using Decomposition Spaces

Understanding sparsity properties of frames using decomposition spaces

Felix Voigtlaender

http://voigtlaender.xyz

Technische Universität Berlin Applied Functional Analysis Group

Seminar “Advanced Topics in PDE and Harmonic Analysis” Bonn, 24. November 2017

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 1 / 20 Outline

1 Diﬀerent representation systems and their applications (5 slides)

2 Structured systems & Decomposition spaces (2 slides)

3 Sparsity properties characterized by decomposition spaces (7 slides)

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 2 / 20 Outline

1 Diﬀerent representation systems and their applications (5 slides)

2 Structured systems & Decomposition spaces (2 slides)

3 Sparsity properties characterized by decomposition spaces (7 slides)

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 3 / 20 The Fourier basis: The mother of all representation systems

2πihn,•i d d d Consider Fourier basis (en) d := e d on the torus = / . n∈Z n∈Z T R Z Nice properties: 2 d Fourier basis is an orthonormal basis for L (T ). Smooth functions are sparse in the Fourier basis. k d p d 2d For example: If g ∈ C , then (g (n)) d ∈ ` for p > . T b n∈Z Z d+2k Fourier basis diagonalizes translation invariant operators.

But: The coeﬃcients gb(n) = hg, eni are non-local. 1 d If g is discontinuous at a single point, then (g (n)) d ∈/ ` . b n∈Z Z d For the Fourier transform on R : 2πihξ,•i I The index set of (e ) d is uncountable. ξ∈R 2πihξ,•i p d d I We have e ∈/ L R for ξ ∈ R and p < ∞. 2πihξ,•i I The coeﬃcients gb(ξ) = hg, e i depend non-locally on g.

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 4 / 20 A wish list for representation systems

Φ = (ϕi )i∈I should be a frame for Hilbert space H, i.e.:

1 f ∈ H can be stably recovered from the analysis coeﬃcients (hf , ϕi i)i∈I . 2 The synthesis map 2 SΦ : ` (I ) → H, (ci )i∈I 7→ ∑ ci · ϕi i∈I is bounded and surjective. 2 2 Note: Each of these prop. equivalent to kf kH ∑i∈I |hf ,ϕi i| for all f ∈ H. Φ should yield sparse representations for a function class C of interest.

1 p Analysis sparsity means (hf , ϕi i)i∈I ∈ ` (I ). 2 p Synthesis sparsity means f = ∑i∈I ci ϕi with (ci )i∈I ∈ ` (I ). Φ should be “compatible” with a class of operators of interest (e.g.: operators in the class preserve sparsity w.r.t. Φ).

Characterization of function spaces using the coeﬃcients (hf ,ϕi i)i∈I . Φ should be eﬃciently implementable. Φ should be useful feature extractor (e.g. for edge detection).

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 5 / 20 Example 1: Gabor frames

2 d Given a prototype γ ∈ L R and sampling densities a,b > 0, the associated Gabor system is

n [n,k;a,b] 2πihan,•−bki d o G(γ;a,b) := γ := Lbk [Man γ] = e · γ (• − bk): n,k ∈ Z .

Illustration:

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 6 / 20 Example 1: Gabor frames

2 d Given a prototype γ ∈ L R and sampling densities a,b > 0, the associated Gabor system is

n [n,k;a,b] 2πihan,•−bki d o G(γ;a,b) := γ := Lbk [Man γ] = e · γ (• − bk): n,k ∈ Z .

Applications: Feature extraction: Gabor transform ≈ score sheet Gabor frames characterize modulation spaces, used to study ΨDOs.

[n,k;a,b] Frequency concentration: If suppγb⊂ Q, then supp F γ ⊂ Q + an.

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 6 / 20 Example 2: Wavelet frames

2 d 2 d Given low-pass ﬁlter ϕ ∈ L R , mother wavelet γ ∈ L R , and sampling density δ > 0, the associated wavelet system is

n [−1,k;δ ] d o W(ϕ,γ;δ) := γ := Lδk ϕ : k ∈ Z

n [j,k;δ] jd/2 j d o ∪ γ := 2 · γ (2 • −δk): j ∈ N0,k ∈ Z . Illustration:

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 7 / 20 Example 2: Wavelet frames

2 d 2 d Given low-pass ﬁlter ϕ ∈ L R , mother wavelet γ ∈ L R , and sampling density δ > 0, the associated wavelet system is n [−1,k;δ ] d o W(ϕ,γ;δ) := γ := Lδk ϕ : k ∈ Z

n [j,k;δ] jd/2 j d o ∪ γ := 2 · γ (2 • −δk): j ∈ N0,k ∈ Z . Applications: Feature extraction: Wavelet transform can detect singularities of functions. Sparsity: Piecewise smooth functions have sparse wavelet transform. Wavelet frames characterize Besov spaces and Triebel-Lizorkin spaces, used

to study Calderon-Zygmund operators. 23 Frequency concentration:

2 [j,k;δ ] j 2 If suppγ ⊂ Q, then supp F γ ⊂ 2 Q for j ∈ N0. b 21 20

20 21 22 23

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 7 / 20 Example 3: Shearlet frames Given low-pass ﬁlter ϕ, mother shearlet γ, and sampling density δ > 0, the associated cone-adapted shearlet system is

n [−1,k;δ] 2o SH(ϕ,γ;δ) := γ := Lδk ϕ : k ∈ Z

n [i,k;δ ] 1/2 T 2o ∪ γ := |detTi | · γ (Ti • −δk): i ∈ I0, k ∈ Z

j/2 with I0 := (j,`,ι) ∈ N0 × Z × {0,1} : |`| ≤ 2 and 0 1 2j 0 1 0 T = Rι · P · S with R = , P = , S = . j,`,ι j ` 1 0 j 0 2j/2 ` ` 1

Illustration (images courtesy of Philipp Petersen)

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 8 / 20 Example 3: Shearlet frames Given low-pass ﬁlter ϕ, mother shearlet γ, and sampling density δ > 0, the associated cone-adapted shearlet system is

n [−1,k;δ] 2o SH(ϕ,γ;δ) := γ := Lδk ϕ : k ∈ Z

What kind of spaces are characterised by shearlets? What kinds of operators can be understood using shearlets?

n [−1,k;δ] 2o SH(ϕ,γ;δ) := γ := Lδk ϕ : k ∈ Z

n [i,k;δ ] 1/2 T 2o ∪ γ := |detTi | · γ (Ti • −δk): i ∈ I0, k ∈ Z . Applications: Feature extraction: Decay of shearlet transform characterizes wavefront set. Sparsity: The shearlet coeﬃcients of cartoon-like functions are sparse. What kind of spaces are characterised by shearlets? What kinds of operators can be understood using shearlets? Frequency concentration:

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 8 / 20 Outline

1 Diﬀerent representation systems and their applications (5 slides)

2 Structured systems & Decomposition spaces (2 slides)

3 Sparsity properties characterized by decomposition spaces (7 slides)

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 9 / 20 Structured systems A structured system is of the form Γ(δ ) = γ[i,k;δ ] with i∈I ,k∈Zd

1 γ[i,k;δ] = |detT | 2 · L M γ ◦ T T i δ· −T ·k b i Ti i

for some prototype γ and a family of aﬃne maps ( Ti • + bi )i∈I . Note: All of the considered systems are (more of less) of this form!

Frequency concentration of structured system: γ[i] := γ[i,0;δ ] satisﬁes

[i] −1/2 −1 F γ = |detTi | · Lbi γb◦ Ti .

d [i] If γb is concentrated in Q ⊂ R , then F γ is concentrated in Qi := Ti Q + bi .

Q = (Qi )i∈I is the structured covering associated to the structured system.

p q Goal: Find family of Banach spaces D Q,L ,`w , parametrized by p,q,w, so that

p q (δ ) f ∈ D(Q,L ,`w ) ⇐⇒ f sparse with respect to Γ .

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 10 / 20 Decomposition spaces d Consider structured admissible covering Q = (Qi )i∈I of R . This means: d There is a ﬁxed open, bounded base set Q ⊂ R with Qi = Ti Q + bi ∀i ∈ I , ∗ ∗ |i | ≤ N for all i ∈ I , with i := {` ∈ I : Q` ∩ Qi 6= ∅}, additional technical conditions (almost always satisﬁed in practice).

Choose a regular partition of unity Φ = (ϕi )i∈I subordinate to Q. ∞ ϕi ∈ Cc (Qi ), ∑i∈I ϕi ≡ 1 and some other technical condition.

Choose a Q-moderate weight w = (wi )i∈I , i.e., wi ≤ C · wj if Qi ∩ Qj 6= ∅.

For p,q ∈ (0,∞], and g ∈ R, deﬁne the decomposition space (quasi)-norm −1 kgk p q := wi · F (ϕi · g) p ∈ [0,∞]. D(Q,L ,`w ) b L i∈I `q The decomposition space determined by Q,p,q,w is p q D(Q,L ,` ) := g ∈ R : kgk p q < ∞ . w D(Q,L ,`w ) 0 d Here R is a suitable reservoir of distributions (think: R = S (R )). F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 11 / 20 Outline

1 Diﬀerent representation systems and their applications (5 slides)

2 Structured systems & Decomposition spaces (2 slides)

3 Sparsity properties characterized by decomposition spaces (7 slides)

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 12 / 20 Banach frames and atomic decompositions

0 I (ϕi )i∈I ⊂ X is a Banach frame for the B-space X , with coeﬀ. space Yd ≤ C if 1 Yd is solid. 2 The analysis operator

A : X → Yd ,x 7→ (hx,ϕi i)i∈I is well-deﬁned and bounded.

3 There is a bounded reconstruction operator R : Yd → X with R ◦ A = idX .

I (ϕi )i∈I ⊂ X is an atomic decomposition for X , with coeﬀ. space Yd ≤ C if 1 Yd is solid. 2 The synthesis operator

S : Yd → X ,(ci )i∈I 7→ ∑ ci · ϕi i∈I is well-deﬁned and bounded (convergence in suitable topology).

3 There is a bounded coeﬃcient operator C : X → Yd with S ◦ C = idX .

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 13 / 20 Structured Banach frame decompositions: General questions

(δ ) [i,k;δ ] p q Hope: If γ is nice, then Γ = γ is a BFD for D Q,L ,`w , where i∈I ,k∈Zd 1 h i [i,k;δ ] 2 T γ = |detTi | · L −T Mbi γ ◦ Ti . δ·Ti k

p,q I × d Question: What is the associated sequence space Cw ≤ C Z ?

p,q n I × d o Answer: C = c ∈ Z : kck p,q < ∞ , where w C Cw

(i) 1 − 1 (i) (c ) d = |detT | 2 p · w · (c ) d . k i∈I ,k∈Z p,q i i k k∈Z `p Cw i∈I `q

Question: In what sense does γ have to be “nice”? Note: Even for characterizing L2, the required “niceness” depends heavily on Q: For Gabor systems: Suﬃcient if γ belongs to the Wiener space. For wavelets, γ has to have vanishing moments.

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 14 / 20 Structured Banach frames — The theorem

Theorem (FV; 2016)

Let w = (wi )i∈I be Q-moderate and let p,q ∈ (0,∞]. There are N ∈ N and σ,τ > 0 (only dep. on p,q,d) with the following property: 1 d If γ ∈ Cc R , γb(ξ) 6= 0 for all ξ ∈ Q, and if sup ∑ Mj,i < ∞ and sup ∑ Mj,i < ∞, i∈I j∈I j∈I i∈I with !τ w τ Z j −1 σ max α β −1 Mj,i := · (1+kTj Ti k) · − |α|≤N [∂ ∂dγ ] Tj (ξ − bj ) dξ , wi Qi |β|≤1

then there is δ0 > 0, such that: [i] p q For 0 < δ ≤ δ0, the family L −T γf is Banach frame for D(Q,L ,`w ) δ·Ti k i,k p,q with coeﬃcient space Cw . Here: g˜ (x) = g (−x).

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 15 / 20 Structured atomic decompositions Theorem (FV; 2016)

Under similar conditions on γ as before, there is δ0 > 0 such that: (δ) [i,k;δ ] p q For 0 < δ ≤ δ0, the family Γ = (γ )i,k is an a.d. of D(Q,L ,`w ) p,q with coeﬃcient space Cw .

1 − 1 p,p p d Observation: For wi = |detTi | p 2 , we have Cw = ` I × Z . Thus: For 0 < p ≤ 2, for suﬃciently nice γ, and δ > 0 small, we have p p 2 d [i,k;δ,∗] p d D(Q,L ,`w ) = f ∈ L (R ): f , γ ∈ ` (I × Z ) i∈I ,k∈Zd ( ) (i) [i,k;δ] (i) p d = c · :(c ) d ∈ ` (I × ) , ∑ k γ k i∈I ,k∈Z Z i,k

1 [i] 2 T with Q = (Ti Q + bi )i∈I and γ = |detTi | · Mbi γ ◦ Ti , as well as

[i,k;δ ] [i] [i,k;δ,∗] [i] γ = L −T γ and γ = L −T γf . δ·Ti k δ·Ti k

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 16 / 20 New proofs of classical results

Corollary p,q d For Besov spaces Bs R : Structured system Wavelet system. “Horrible condition” smoothness + localization + vanishing moments. Precisely: Suﬃces to have n o d + ε |∂ α γ (ξ)| (1 + |ξ|)−L1 · min 1,|ξ|L2 for |α| ≤ , b . min{1,p}

with L2 ≥ 0 and I L2 > s to get Banach frames, −1 I L2 > p − 1 + · d − s to get atomic decompositions.

Corollary p,q d For modulation spaces Ms R : Structured system: Gabor system. “Horrible condition” smoothness + localization.

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 17 / 20 Shearlet smoothness spaces

This is joint work with Anne Pein Shearlet smoothness spaces

p,q 2 p q The shearlet covering The spaces Ss R = D(S,L ,`w s ) are called S = (Ti Q + bi )i∈I : shearlet smoothness spaces (Labate, Mantovani, Negi; 2013). The weight w is chosen such that wi ∼ 1 + |ξ| for ξ ∈ Qi .

Q Observation: Structured family generated by ψ is a cone-adapted shearlet system (Guo, Labate, Kutyniok; 2006):

(δ ) [i,k;δ ] T Ψ := ψ = SH Mb0 (ψ ◦ T0 ), ψ; δ . i∈I ,k∈Z2

Theorem (Pein, FV; 2017)

N1 Let p0,q0 ∈ (0,1] and s0 ≥ 0. There are N1,N2 ∈ N such that if ψ1,ψ2 ∈ Cc (R) and ψ = ψ1 ⊗ ψ2 with ` d ψb (ξ) 6= 0 for ξ ∈ Q and ` ψc1 (ξ) = 0 for ` = 0,...,N2 dξ ξ =0 (δ ) then there is δ0 > 0, such that Ψ is a Banach frame and an atomic p,q 2 decomposition for Ss R for all p ≥ p0, q ≥ q0, |s| ≤ s0, and 0 < δ ≤ δ0. Application: Approximation of cartoon-like functions We consider C 2 cartoon-like functions f ∈ E2:

(image courtesy of Gitta Kutyniok)

Previously known (Guo, Labate, Lim, Kutyniok et al.): If ψ is a nice mother shearlet, then p 2 the analysis coeﬃcients of f w.r.t. SH(ψ;δ) are ` -summable for p > 3 . For suitable linear combination fN of N elements of the dual shearlet frame: −1 3/2 kf − fN kL2 ≤ Cδ,ψ · N · (1 + lnN) .

New result (Pein, FV; 2017)

For suitable linear comb. gN of N elements of the shearlet frame SH(ψ;δ): −(1−ε) kf − gN kL2 ≤ Cε,δ,ψ · N ∀ε ∈ (0,1) and N ∈ N.

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 19 / 20 Conclusion General framework: Frame construction ←→ Frequency covering ←→ Decomposition space p q Membership f ∈ D Q,L ,`w ⇐⇒ Sparsity of f w.r.t. given frame. Special cases: We recover known results about wavelet and Gabor frames. Novel results for cone-adapted shearlets.

Story for another day: Embedding theory for decomposition spaces. (Non)-transference of sparsity from one system to another.

Future work: Study operators on decomposition spaces. Try to ﬁnd “intrinsic” characterizations of decomposition spaces (e.g.: characterization of Besov spaces using moduli of continuity). Extend framework to the case of Triebel-Lizorkin type spaces. Study decomposition spaces on (bounded) domains.

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 20 / 20 Thank you!

Questions, comments, counterexamples?

, Atomic decompositions: “Similar conditions”?!

Theorem (FV; 2016)

Let w = (wi )i∈I be Q-moderate, and let p,q ∈ (0,∞]. There are N ∈ N and σ,τ > 0 and ϑ ≥ 0 (only dep. on p,q,d) with the following: Assume (1) (2) (1) d (2) 1 d γ = γ ∗ γ with γ ∈ Cc R and γ ∈ Cc R , γb6= 0 on Q, we have supi∈I ∑j∈I Ni,j < ∞ and supj∈I ∑i∈I Ni,j < ∞, with !τ w |T |ϑ Z h i τ N := i j · (1 + kT −1T k)σ · − max ∂ α γd(1) (T −1 (ξ − b )) dξ . i,j ϑ j i j j wj |Ti | Qi |α|≤N

Then there is δ0 > 0 such that (δ ) [i,k;δ] p q For 0 < δ ≤ δ0, the family Γ = (γ )i,k is an a.d. of D(Q,L ,`w ) p,q with coeﬃcient space Cw .

F. Voigtlaender Sparsity properties of frames via decomposition spaces PDE and Harmonic Analysis Seminar, Bonn 22 / 20