Sparse recovery for spherical harmonic expansions

Sparse recovery for spherical harmonic expansions

Rachel Ward1

1Courant Institute, New York University

Workshop Sparsity and Cosmology, NiceMay 31, 2011

Cosmic Microwave Background Radiation (CMB) map

• Temperature is measured asT (θ, ϕ) =

∑∞k=0

∑k`=−k a(`,k)Y

k` (θ, ϕ), where Y k

` ’s arespherical harmonics

• Red band: measurements are corrupted by galactic signal

Rachel Ward1 1Courant Institute, New York University Sparse recovery for spherical harmonic expansions

CMB map is compressible in spherical harmonics

• Consider the coefficient vector a = a(`,k) and

T (θ, φ) ≈n∑

k=0

k∑`=−k

a(`,k)Yk` (θ, ϕ).

• This vector is predicted and observed to be compressible.


Spherical harmonics: Fourier analysis on the sphere

• Y k` ’s are products of complex exponentials and orthogonal

Jacobi polynomials• Y k

` ’s are orthonormal with respect to spherical surfacemeasure sin(ϕ)dϕdθ


CMB map inpainting via `1-minimization

• (Abrial, Moudden, Starck, Fadili, Delabrouille, Nguyen ′08):Propose full-sky CMB map inpainting from partialmeasurements T (θj , ϕj). Obtain coefficients a = a(`,k) bysolving the `1-minimization problem:

a = arg min ‖c‖1 s.t.

√N∑

k=0

k∑`=−k

c(`,k)Yk` (θj , ϕj) = T (θj , ϕj)

• D =√

N is a prescribed maximal degree• Theoretical justification?


The spherical sampling matrix

• In matrix form, the constraints in `1-minimization problem areΦc = T, where Φ ∈ Cm×N is the spherical sampling matrix

Φ =

1 Y 1

1 (θ1, ϕ1) . . . Y k` (θ1, ϕ1) . . .

1 Y 11 (θ2, ϕ2) . . . Y k

` (θ2, ϕ2) . . ....

......

1 Y 11 (θm, ϕm) . . . Y k

` (θm, ϕm) . . .

• We assume these measurements are underdetermined: m < N.

• Compressed sensing etc: If Φ acts as approximate isometry onsparse vectors, then compressible vectors are stably recoveredvia `1-minimization


Restricted Isometry Property (RIP)

Definition

[Candes, Romberg, Tao ′06] The restricted isometry constant δs ofa matrix Φ ∈ Cm×N is the smallest number such that for alls-sparse x ∈ CN ,

(1− δs)‖x‖22 ≤ ‖Φx‖22 ≤ (1 + δs)‖x‖22

• Open to construct deterministic matrices satisfying the RIP inthe regime m � s logp(N).

• If Φ ∈ Rm×N has i.i.d. Gaussian or Bernoulli entries andm ≥ Cδ−2(s log(N/s)) then δs ≤ δ with high probability.

• [CRT ′06, RV ′08, R ′09 ] If m = O(s log4(N)) the RIP holdsw.h.p. for Φ associated to bounded orthonormal systems.


RIP matrices are good for sparse recovery

• [CRT ′06, C′08, Foucart ′10] If for Φ ∈ Cm×N we haveδs ≤ δ0, (δ0 = .46 is valid), y = Φx is observed, and

x = arg minz ‖z‖1 subject to Φz = y,

then

‖x− x‖2 �‖x− xs‖1√

s,

where xs is the best s-term approximation to x.

• If x is s-sparse, then x = x is recovered exactly.

• If x is well-approximated by an s-sparse vector, then x ≈ x.


Sparse recovery for bounded orthonormal systems

Ψ =

ψ1(x1) ψ2(x1) . . . . . . ψN(x1)...

......

ψ1(xm) ψ2(xm) . . . . . . ψN(xm)

• Suppose (ψj)Nj=1 on compact domain D are orthonormal with

respect to measure dν

• Suppose x1, . . . , xm ∈ D are chosen i.i.d. from dν.

• Suppose maxj∈1...N ‖ψj‖∞ ≤ K .

Theorem (Rudelson, Vershynin ’08, Rauhut ’09)

If m ≥ CK 2δ−2s log3(s) log(N) then the matrix 1√m

Ψ satisfies

δs ≤ δ with probability at least 1− N−γ log3(s).


Examples of bounded orthonormal systems

• Fourier ψj(x) = e2πijx : D = [0, 1], dν = dx , K = 1(also discrete analog)

• Chebyshev polynomials Tj(x): D = [−1, 1],dν = (1− x2)−1/2dx , K =

√2

• RIP for Ψ means that functions which admit s-sparseexpansions with respect to the ψj ’s can be recovered fromtheir values at m sample points provided

m ≥ CK 2s log3(s) log(N),

and functions with compressible expansions can be recoveredapproximately



[Rauhut, W ′10] : preconditioned Legendre system Q(x)Lj(x)

• Ljs are normalized Legendre polynomials

• Q(x) = C (1− x2)1/4, dν(x) = π−1(1− x2)−1/2dx , andK = 2

• Q(x) is preconditioner; implies sparse recovery in Legendresystem



[Rauhut,W ′10] : More generally, preconditioned Jacobi systemQα(x)pαj (x)

• pαj s are polynomials orthonormal w.r.t. dν(x) = (1− x2)αdx

• [Krasikov 07:] |Qαpαj (x)| ≤ Cα1/4

• Qα(x) = (1− x2)α/2+1/4, dν(x) = (1− x2)−1/2dx , andK = Cα1/4

That is, Chebyshev sampling is universal for recovering sparsepolynomial expansions


The spherical harmonics

• The spherical harmonics can be written as

Y k` (θ, ϕ) = e ikθp

|k|`−|k|(cosϕ)(sinϕ)|k|, (θ, ϕ) ∈ [0, π]× [0, 2π),

−k ≤ ` ≤ k , k ≥ 0

• Growth rates for complex exponentials and Jacobi polynomialsgive:

sup0≤k≤

√N−1,−k≤`≤k

| sin(ϕ)1/2Y k` (θ, ϕ)| ≤ CN1/8

• This implies the strategy of uniform sampling from theproduct measure dϕdθ.


Location of sampling points matters

Figure: Phase transitions for sparse recovery on the sphere

m/N

s/m

0.2 0.4 0.6 0.8 1

0.20.40.60.81

(a)

m/N

s/m

0.2 0.4 0.6 0.8 1

0.20.40.60.81

(b)

• We form “random” s-sparse coefficient vectors c = (c`,k) ofdegree D = N1/2 = 16 and choose m sampling points from(a) product measure dϕdθ and (b) uniform spherical measuresinϕdϕdθ. Black indicates recovery.


Sparse recovery in spherical harmonic systems

Theorem (Rauhut, W ’11)

Suppose that (θ1, ϕ1), . . . , (θm, ϕm), withm ≥ Cs log3(s)N1/4 log(N) are drawn independently from theuniform measure on B = [0, π]× [0, 2π).Let Φ be the m × N spherical sampling matrix and let QΦ be itspreconditioned version. With high probability the following holdsfor all harmonic polynomials

g(θ, ϕ) =∑N1/2−1

`=0

∑`k=−` c`,kY k

` (θ, ϕ). Suppose that noisysample values yj = g(θj , ϕj) + ηj are observed, and that ‖η‖∞ ≤ ε.Let

c = arg min ‖z‖1 subject to ‖QΦz −Qy‖2 ≤√

mε.

Then

‖c − c‖2 ≤C1σs(c)1√

s+ C2ε.


Conclusions

• Our results provide a measure of justification for goodnumerical results for CMB map inpainting via `1-minimization

• Our results may be of interest to other problems ingeophysics, astronomy, and medical imaging.


Open problems

For practical implementation, we would rather sample from adiscrete grid. In experiments, the sparse recovery results fordiscrete vs. continuous are indistinguishable.Proof?


Open problems

In our proof, we require m � sN1/4 log4(N) sampling points (orrows in Φ) for `1-minimization to be able to recover s-sparsespherical polynomials of degree N1/2. .

We should be able to improve this to m � s logp(N)...

In practice, different models of sparsity are more suited for thesphere, such as rotationally invariant sparsity sets, or sparsity incertain linear combinations of spherical harmonic coefficients


Sparse recovery for spherical harmonic expansions

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