The ℓ¹-regularized empirical risk minimizer achieves minimax-optimal convergence rate n^{-r/(1+b-br)} for nonlinear statistical inverse learning under variational source conditions and polynomial effective-dimension decay.
Compressed sensing for in- verse problems II: applications to deconvolution, source recovery, and MRI
3 Pith papers cite this work. Polarity classification is still indexing.
3
Pith papers citing it
abstract
This paper extends the sample complexity theory for ill-posed inverse problems developed in a recent work by the authors [`Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform', J. Eur. Math. Soc., to appear], which was originally focused on the sparse Radon transform. We demonstrate that the underlying abstract framework, based on infinite-dimensional compressed sensing and generalized sampling techniques, can effectively handle a variety of practical applications. Specifically, we analyze three case studies: (1) The reconstruction of a sparse signal from a finite number of pointwise blurred samples; (2) The recovery of the (sparse) source term of an elliptic partial differential equation from finite samples of the solution; and (3) A moderately ill-posed variation of the classical sensing problem of recovering a wavelet-sparse signal from finite Fourier samples, motivated by magnetic resonance imaging. For each application, we establish rigorous recovery guarantees by verifying the key theoretical requirements, including quasi-diagonalization and coherence bounds. Our analysis reveals that careful consideration of balancing properties and optimized sampling strategies can lead to improved reconstruction performance. The results provide a unified theoretical foundation for compressed sensing approaches to inverse problems while yielding practical insights for specific applications.
representative citing papers
Derives explicit reconstruction error bounds for inverse problems on Riemannian manifolds from Marcinkiewicz-Zygmund point samples, with detailed results for convolutions on two-point homogeneous spaces including the sphere.
Stochastic generalized sampling uses leverage-score sampling and a new matrix Bernstein inequality to guarantee stable recovery at m ≳ n log n samples with high probability, even for redundant frames, and demonstrates near-exponential convergence on analytic function recovery from Fourier data.
citing papers explorer
-
Statistical inverse learning and $\ell^1$-regularization
The ℓ¹-regularized empirical risk minimizer achieves minimax-optimal convergence rate n^{-r/(1+b-br)} for nonlinear statistical inverse learning under variational source conditions and polynomial effective-dimension decay.
-
Sampling theorems for inverse problems on Riemannian manifolds
Derives explicit reconstruction error bounds for inverse problems on Riemannian manifolds from Marcinkiewicz-Zygmund point samples, with detailed results for convolutions on two-point homogeneous spaces including the sphere.
-
Stochastic Generalized Sampling
Stochastic generalized sampling uses leverage-score sampling and a new matrix Bernstein inequality to guarantee stable recovery at m ≳ n log n samples with high probability, even for redundant frames, and demonstrates near-exponential convergence on analytic function recovery from Fourier data.