Sparse recovery for spherical harmonic expansions

Rachel Ward 1 1 Courant Institute, New York University Workshop Sparsity and Cosmology, Nice May 31, 2011

Cosmic Microwave Background Radiation (CMB) map Temperature is measured as T (θ, ϕ) = k k=0 l= k a (l,k)yl k (θ, ϕ), where Y k l s are spherical harmonics Red band: measurements are corrupted by galactic signal

CMB map is compressible in spherical harmonics Consider the coefficient vector a = a (l,k) and T (θ, φ) n k k=0 l= k a (l,k) Yl k (θ, ϕ). This vector is predicted and observed to be compressible.

Spherical harmonics: Fourier analysis on the sphere Yl k s are products of complex exponentials and orthogonal Jacobi polynomials Yl k s are orthonormal with respect to spherical surface measure sin(ϕ)dϕdθ

CMB map inpainting via l 1 -minimization (Abrial, Moudden, Starck, Fadili, Delabrouille, Nguyen 08): Propose full-sky CMB map inpainting from partial measurements T (θ j, ϕ j ). Obtain coefficients a = a (l,k) by solving the l 1 -minimization problem: a = arg min c 1 N s.t. k k=0 l= k D = N is a prescribed maximal degree Theoretical justification? c (l,k) Y k l (θ j, ϕ j ) = T (θ j, ϕ j )

The spherical sampling matrix In matrix form, the constraints in l 1 -minimization problem are Φc = T, where Φ C m N is the spherical sampling matrix 1 Y1 1 (θ 1, ϕ 1 )... Yl k(θ 1, ϕ 1 )... 1 Y1 1 Φ = (θ 2, ϕ 2 )... Yl k(θ 2, ϕ 2 )...... 1 Y1 1(θ m, ϕ m )... Yl k(θ m, ϕ m )... We assume these measurements are underdetermined: m < N.

Restricted Isometry Property (RIP) Definition [Candès, Romberg, Tao 06] The restricted isometry constant δ s of a matrix Φ C m N is the smallest number such that for all s-sparse x C N, (1 δ s ) x 2 2 Φx 2 2 (1 + δ s ) x 2 2 Open to construct deterministic matrices satisfying the RIP in the regime m s log p (N). If Φ R m N has i.i.d. Gaussian or Bernoulli entries and m Cδ 2 (s log(n/s)) then δ s δ with high probability. [CRT 06, RV 08, R 09 ] If m = O(s log 4 (N)) the RIP holds w.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 C m N we have δ s δ 0, (δ 0 =.46 is valid), y = Φx is observed, and then x = arg min z z 1 subject to Φz = y, x x 2 x x s 1 s, where x s 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 = ψ 1 (x 1 ) ψ 2 (x 1 )...... ψ N (x 1 )... ψ 1 (x m ) ψ 2 (x m )...... ψ N (x m ) Suppose (ψ j ) N j=1 on compact domain D are orthonormal with respect to measure dν Suppose x 1,..., x m D are chosen i.i.d. from dν. Suppose max j 1...N ψ j K. Theorem (Rudelson, Vershynin 08, Rauhut 09) If m CK 2 δ 2 s log 3 (s) log(n) then the matrix 1 m Ψ satisfies δ s δ with probability at least 1 N γ log3 (s).

Examples of bounded orthonormal systems Fourier ψ j (x) = e 2πijx : D = [0, 1], dν = dx, K = 1 (also discrete analog) Chebyshev polynomials T j (x): D = [ 1, 1], dν = (1 x 2 ) 1/2 dx, K = 2 RIP for Ψ means that functions which admit s-sparse expansions with respect to the ψ j s can be recovered from their values at m sample points provided m CK 2 s log 3 (s) log(n), and functions with compressible expansions can be recovered approximately

Examples of bounded orthonormal systems [Rauhut, W 10] : preconditioned Legendre system Q(x)L j (x) L j s are normalized Legendre polynomials Q(x) = C(1 x 2 ) 1/4, dν(x) = π 1 (1 x 2 ) 1/2 dx, and K = 2 Q(x) is preconditioner; implies sparse recovery in Legendre system

Examples of bounded orthonormal systems [Rauhut,W 10] : More generally, preconditioned Jacobi system Q α (x)p α j (x) p α j s are polynomials orthonormal w.r.t. dν(x) = (1 x 2 ) α dx [Krasikov 07:] Q α p α j (x) Cα1/4 Q α (x) = (1 x 2 ) α/2+1/4, dν(x) = (1 x 2 ) 1/2 dx, and K = Cα 1/4 That is, Chebyshev sampling is universal for recovering sparse polynomial expansions

The spherical harmonics The spherical harmonics can be written as Yl k (θ, ϕ) = eikθ p k l k (cos ϕ)(sin ϕ) k, k l k, k 0 (θ, ϕ) [0, π] [0, 2π), Growth rates for complex exponentials and Jacobi polynomials give: sup 0 k N 1, k l k sin(ϕ) 1/2 Yl k (θ, ϕ) CN1/8 This implies the strategy of uniform sampling from the product measure dϕdθ.

Location of sampling points matters Figure: Phase transitions for sparse recovery on the sphere s/m 1 0.8 0.6 0.4 0.2 s/m 1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 m/n (a) 0.2 0.4 0.6 0.8 1 m/n (b) We form random s-sparse coefficient vectors c = (c l,k ) of degree D = N 1/2 = 16 and choose m sampling points from (a) product measure dϕdθ and (b) uniform spherical measure sin ϕdϕdθ. Black indicates recovery.

Sparse recovery in spherical harmonic systems Theorem (Rauhut, W 11) Suppose that (θ 1, ϕ 1 ),..., (θ m, ϕ m ), with m Cs log 3 (s)n 1/4 log(n) are drawn independently from the uniform measure on B = [0, π] [0, 2π). Let Φ be the m N spherical sampling matrix and let QΦ be its preconditioned version. With high probability the following holds for all harmonic polynomials g(θ, ϕ) = N 1/2 1 l l=0 k= l c l,kyl k (θ, ϕ). Suppose that noisy sample values y j = g(θ j, ϕ j ) + η j are observed, and that η ε. Let ĉ = arg min z 1 subject to QΦz Qy 2 mε. Then c ĉ 2 C 1σ s (c) 1 s + C 2 ε.

Conclusions Our results provide a measure of justification for good numerical results for CMB map inpainting via l 1 -minimization Our results may be of interest to other problems in geophysics, astronomy, and medical imaging.

Open problems For practical implementation, we would rather sample from a discrete grid. In experiments, the sparse recovery results for discrete vs. continuous are indistinguishable. Proof?

Open problems In our proof, we require m sn 1/4 log 4 (N) sampling points (or rows in Φ) for l 1 -minimization to be able to recover s-sparse spherical polynomials of degree N 1/2.. We should be able to improve this to m s log p (N)... In practice, different models of sparsity are more suited for the sphere, such as rotationally invariant sparsity sets, or sparsity in certain linear combinations of spherical harmonic coefficients