Global growth of bandlimited local approximations

Reyes, Noli N.; Vallejo, Louie John D.

doi:10.1016/j.jmaa.2012.10.006

Cited by 5 publications

(4 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We illustrate several instances of (2), develop tractable reformulations of the form (8), and solve them using (12), where x 0 = 0 and the relaxation strategy is that recommended in [4, Chapter 5], namely…”

Section: Applicationsmentioning

confidence: 99%

“…which fulfills Assumption 1(i) and yields [9]. We apply (12) with ( 14) and the following control scheme. We split K into 12 blocks of 100 consecutive indices, and select I n by periodically sweeping through the blocks, hence satisfying (13) with M = 12.…”

Section: B Reconstruction From Thresholded Scalar Productsmentioning

confidence: 99%

“…alternating projection algorithm. The extension of (3) to recovery problems with several transformations modeled as linear projections r k = proj V k x is discussed in [8], [12]. More broadly, if all the operators (R k ) k∈K are linear, reliable algorithms are available to solve (2).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Fixed Point Framework for Recovering Signals from Nonlinear Transformations

Combettes¹,

Woodstock²

2020

Preprint

View full text Add to dashboard Cite

We consider the problem of recovering a signal from nonlinear transformations, under convex constraints modeling a priori information. Standard feasibility and optimization methods are ill-suited to tackle this problem due to the nonlinearities. We show that, in many common applications, the transformation model can be associated with fixed point equations involving firmly nonexpansive operators. In turn, the recovery problem is reduced to a tractable common fixed point formulation, which is solved efficiently by a provably convergent, block-iterative algorithm. Applications to signal and image recovery are demonstrated.

show abstract

Section: Applicationsmentioning

confidence: 99%

Section: B Reconstruction From Thresholded Scalar Productsmentioning

confidence: 99%

See 1 more Smart Citation

A Fixed Point Framework for Recovering Signals from Nonlinear Transformations

Combettes¹,

Woodstock²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Let H be a real Hilbert space with scalar product • | • and associated norm • , let x 0 ∈ H, let U and V be closed vector subspaces of H with projection operators proj U and proj V , respectively, and let p ∈ V . The basic best approximation problem minimize x − x 0 subject to x ∈ U and proj V x = p (1.1) covers a wide range of scenarios in areas such as harmonic analysis, signal processing, interpolation theory, and optics [3,22,32,35,38,40,43,52,59]. In this setting, a function of interest x ∈ H is known to lie in the subspace U and its projection p onto the subspace V is known.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of Functions from Prescribed Proximal Points

Combettes¹,

Woodstock²

2021

Preprint

View full text Add to dashboard Cite

Under investigation is the problem of finding the best approximation of a function in a Hilbert space subject to convex constraints and prescribed nonlinear transformations. We show that in many instances these prescriptions can be represented using firmly nonexpansive operators, even when the original observation process is discontinuous. The proposed framework thus captures a large body of classical and contemporary best approximation problems arising in areas such as harmonic analysis, statistics, interpolation theory, and signal processing. The resulting problem is recast in terms of a common fixed point problem and solved with a new block-iterative algorithm that features approximate projections onto the individual sets as well as an extrapolated relaxation scheme that exploits the possible presence of affine constraints. A numerical application to signal recovery is demonstrated.

show abstract

Reconstruction of functions from prescribed proximal points

Combettes

Woodstock

2021

Journal of Approximation Theory

View full text Add to dashboard Cite

Global growth of bandlimited local approximations

Cited by 5 publications

References 9 publications

A Fixed Point Framework for Recovering Signals from Nonlinear Transformations

A Fixed Point Framework for Recovering Signals from Nonlinear Transformations

Reconstruction of Functions from Prescribed Proximal Points

Reconstruction of functions from prescribed proximal points

Contact Info

Product

Resources

About