A unified approach to convergence rates for ℓ1-regularization and lacking sparsity
-
Jens Flemming
,
Bernd Hofmann
Abstract
In ℓ1-regularization, which is an important tool in signal and image processing, one usually is concerned with signals and images having a sparse representation in some suitable basis, e.g., in a wavelet basis. Many results on convergence and convergence rates of sparse approximate solutions to linear ill-posed problems are known, but rate results for the ℓ1-regularization in case of lacking sparsity had not been published until 2013. In the last two years, however, two articles appeared providing sufficient conditions for convergence rates in case of non-sparse but almost sparse solutions. In the present paper, we suggest a third sufficient condition, which unifies the existing two and, by the way, also incorporates the well-known restricted isometry property.
Funding source: German Research Foundation (DFG)
Award Identifier / Grant number: FL 832/1-1, HO 1454/8-2, VE 253/6-1
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Preface
- In celebration of the 60th birthday of Professor Alemdar Hasanoğlu (Hasanov)
- An inverse source problem for a damped wave equation with memory
- On the Cauchy problem for semilinear elliptic equations
- A unified approach to convergence rates for ℓ1-regularization and lacking sparsity
- Regularization of ill-posed problems by using stabilizers in the form of the total variation of a function and its derivatives
- Numerical testing in determination of sound speed from a part of boundary by the BC-method
- Inverse determination of spatially varying material coefficients in solid objects
- Determination of the initial condition in parabolic equations from boundary observations
Articles in the same Issue
- Frontmatter
- Preface
- In celebration of the 60th birthday of Professor Alemdar Hasanoğlu (Hasanov)
- An inverse source problem for a damped wave equation with memory
- On the Cauchy problem for semilinear elliptic equations
- A unified approach to convergence rates for ℓ1-regularization and lacking sparsity
- Regularization of ill-posed problems by using stabilizers in the form of the total variation of a function and its derivatives
- Numerical testing in determination of sound speed from a part of boundary by the BC-method
- Inverse determination of spatially varying material coefficients in solid objects
- Determination of the initial condition in parabolic equations from boundary observations
