Weighted Eigenfunction Estimates with Applications to Compressed Sensing

  title={Weighted Eigenfunction Estimates with Applications to Compressed Sensing},
  author={Nicolas Burq and Semyon Dyatlov and Rachel A. Ward and Maciej Zworski},
  journal={SIAM J. Math. Anal.},
Using tools from semiclassical analysis, we give weighted L^\infty estimates for eigenfunctions of strictly convex surfaces of revolution. These estimates give rise to new sampling techniques and provide improved bounds on the number of samples necessary for recovering sparse eigenfunction expansions on surfaces of revolution. On the sphere, our estimates imply that any function having an s-sparse expansion in the first N spherical harmonics can be efficiently recovered from its values at m > s… 

Figures from this paper

Sparse recovery on sphere via probabilistic compressed sensing
This paper incorporates a preconditioning technique into the probabilistic approach to derive a slightly improved bound on the order of measurements for accurate recovery of spherical harmonic expansions.
Random Sampling in Bounded Orthonormal Systems
This chapter considers the recovery of signals with a sparse expansion in a bounded orthonormal system including the Fourier systems, and nonuniform sparse recovery is proved to be possible via l1-minimization.
Compressive sensing with redundant dictionaries and structured measurements
In this work, it is demonstrated that one can subsample certain bases in such a way that the D-RIP will hold without the need for random column signs.
Sparse recovery of spherical harmonic expansions from uniform distribution on sphere
It is shown that one can, with high probability, recover s-sparse spherical harmonic expansions from M ≥ s log<sup>3</sup> N measurements randomly sampled from the uniform sin θ dθ d<sub>φ</sub> measure on sphere.
Sparse recovery in Wigner-D basis expansion
A new orthonormal and bounded system is obtained for which Restricted Isometry Property (RIP) property can be established and the application of the results in the spherical near-field antenna measurement is discussed.
Near-Optimal Compressed Sensing Guarantees for Total Variation Minimization
This paper shows that a multidimensional signal x ∈ ℂNd can be reconstructed from O(s dlog(Nd)) linear measurements y = Ax using total variation minimization to a factor of the best s-term approximation of its gradient.
Coherence Bounds for Sensing Matrices in Spherical Harmonics Expansion
It will be shown that for a class of sampling patterns, the mutual coherence would be at its maximum, yielding the worst performance, and the sampling strategy is proposed to achieve the derived lower bound.
Super-Resolution on the Sphere Using Convex Optimization
It is shown that under a separation condition, one can recover an ensemble of Diracs on a sphere with high precision by a three-stage algorithm, which consists of solving a semi-definite program, root finding and least-square fitting.
Stable and Robust Sampling Strategies for Compressive Imaging
The local coherence framework developed in this paper implies that for optimal sparse recovery results, it suffices to have bounded average coherence from sensing basis to sparsity basis-as opposed to bounded maximal coherence-as long as the sampling strategy is adapted accordingly.
On Grid Compressive Sensing for Spherical Field Measurements in Acoustics
We derive a theoretically guaranteed compressive sensing method for acoustic field reconstructions using spherical field measurements on a predefined grid. This method can be used to reconstruct sparse


Weighted L∞ estimates for eigenfunctions of strictly convex surfaces of revolution give rise to new sampling techniques and provide improved bounds on the number of samples necessary for recovering sparse eigenfunction expansions on surfaces ofrevolution.
On sparse reconstruction from Fourier and Gaussian measurements
This paper improves upon best‐known guarantees for exact reconstruction of a sparse signal f from a small universal sample of Fourier measurements by showing that there exists a set of frequencies Ω such that one can exactly reconstruct every r‐sparse signal f of length n from its frequencies in Ω, using the convex relaxation.
Sparse recovery for spherical harmonic expansions
We show that sparse spherical harmonic expansions can be efficiently recovered from a small number of randomly chosen samples on the sphere. To establish the main result, we verify the restricted
Microlocal limits of plane waves and Eisenstein functions
We study microlocal limits of plane waves on noncompact Riemannian manifolds (M,g) which are either Euclidean or asymptotically hyperbolic with curvature -1 near infinity. The plane waves E(z,\xi)
Scattering Phase Asymptotics with Fractal Remainders
For a Riemannian manifold (M, g) which is isometric to the Euclidean space outside of a compact set, and whose trapped set has Liouville measure zero, we prove Weyl type asymptotics for the
Microlocal limits of Eisenstein functions away from the unitarity axis
We consider a surface M with constant curvature cusp ends and its Eisenstein functions E_j(\lambda). These are the plane waves associated to the j-th cusp and the spectral parameter \lambda, (\Delta
Microlocal Analysis for Differential Operators: An Introduction
Introduction 1. Symbols and oscillatory integrals 2. The method of stationary phase 3. Pseudodifferential operators 4. Application to elliptic operators and L2 continuity 5. Local symplectic geometry
The spectral function of an elliptic operator
In this paper we shall obtain the best possible estimates for the remainder term in the asymptotic formula for the spectral function of an arbitrary elliptic (pseudo-)differential operator. This is
Quantum ergodicity for restrictions to hypersurfaces
Quantum ergodicity theorem states that for quantum systems with ergodic classical flows, eigenstates are, on average, uniformly distributed on energy surfaces. We show that if N is a hypersurface in
Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information
It is shown how one can reconstruct a piecewise constant object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f.