An Analysis of the Superiorization Method via the Principle of Concentration of Measure

Censor, Yair; Levy, Eliahu

doi:10.1007/s00245-019-09628-4

Cited by 8 publications

(12 citation statements)

References 31 publications

(56 reference statements)

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“…For big i, thus small β i , the contribution might again be negligible. This shows that the main contribution in (8) of [3] seems to come from intermediate terms.…”

Section: Conclusion 5 For a Productmentioning

confidence: 84%

Accumulated Random Distances in High Dimensions -- Ways of Calculation

Levy

2020

Preprint

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In this note we refine and improve some of the calculations in our 2019 article with Yair Censor (Applied Mathematics and Optimization, accepted for publication) where an analysis of the superiorization method is made via the principle of concentration of measure. Some paragraphs there are repeated here for the sake of completeness. Yet, for the case of accumulating 'steps' on the sphere, reference to distances as done there is replaced by reference to the angles, which makes simpler expressions. The treatment here of the action of a random transformation is also rather 'cleaner'. For some standard deviations, precise inequalities are here obtained rather then just Oexpressions. Some further settings are mentioned, of no direct interest as per the latter article, showing results that similar calculations yield.

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“…For big i, thus small β i , the contribution might again be negligible. This shows that the main contribution in (8) of [3] seems to come from intermediate terms.…”

Section: Conclusion 5 For a Productmentioning

confidence: 84%

Accumulated Random Distances in High Dimensions -- Ways of Calculation

Levy

2020

Preprint

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“…A mathematical guarantee has not been found to date that the overall process of the superiorized version of the basic algorithm will not only retain its feasibility-seeking nature but also preserve globally the target function reductions. This fundamental question of the SM, called "the guarantee problem of the SM", received recently partial answers in [22,29]. However, numerous works cited in [19] show that this global function reduction of the SM occurs in practice in many real-world applications.…”

Section: Definition 23 (Bounded Perturbation Resilience)mentioning

confidence: 99%

“…All these culminated in Ran Davidi's 2010 PhD dissertation [31] and the many papers since then cited in [19]. Introductory and advanced materials about the SM, accompanied by relevant references, can be found in the recent papers [22,20], in particular, [20,Section 2] contains the basics of the superiorization methodology in condensed form. A comprehensive overview of the state-of-the-art and current research on superiorization appears in our continuously updated online bibliography that currently contains 110 items [19].…”

mentioning

confidence: 99%

“…Later papers continue research on perturbation resilience, which lies at the heart of the SM, see, e.g., [10]. Another recent work attempts at analysing the behavior of the SM via the concept of concentration of measure [22]. Efforts to understand the very good performance of the SM in practice recently motivated Byrne [13] to notice and study similarities between some SM algorithms and optimization methods.…”

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confidence: 99%

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Superiorization vs. accelerated convex optimization: The superiorized / regularized least-squares case

2020

J. Appl. Numer. Optim.

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We conduct a study and comparison of superiorization and optimization approaches for the reconstruction problem of superiorized / regularized least-squares solutions of underdetermined linear equations with nonnegativity variable bounds. Regarding superiorization, the state of the art is examined for this problem class, and a novel approach is proposed that employs proximal mappings and is structurally similar to the established forward-backward optimization approach. Regarding convex optimization, accelerated forward-backward splitting with inexact proximal maps is worked out and applied to both the natural splitting least-squares term / regularizer and to the reverse splitting regularizer / least-squares term. Our numerical findings suggest that superiorization can approach the solution of the optimization problem and leads to comparable results at significantly lower costs, after appropriate parameter tuning. On the other hand, applying accelerated forward-backward optimization to the reverse splitting slightly outperforms superiorization, which suggests that convex optimization can approach superiorization too, using a suitable problem splitting.

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“…Acknowledgment. This work is an offshoot of a collaboration with Prof. Yair Censor of the University of Haifa (see [CenLe], [CenLe19]). I am very much indebted to Professor Censor for raising the questions and many helpful discussions and advice.…”

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confidence: 99%

`Ultrafilters Relaxed' -- Species of Families of Subsets

Levy¹

2021

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An assortments of 'species' of families of subsets of a set and some of their properties are investigated, with an eye on the logic and limit role they may play as 'relaxed parallels' to ultrafilters. Introduction, Points vs. FamiliesLet X be a set. For points x ∈ X and subsets S ⊂ X, there is the truth-value whether X ∈ S or not.But with families of subsets F ⊂ P(X), where P(x) is the set of subsets of X, there is also a truth-value -whether S ∈ F or not.Indeed, for every x ∈ X, the family'does the same as x' for that matter: by definition, the truth-value whether x ∈ S is the same as the truth-value whether S ∈ U x .Key words and phrases. families of subsets, ultrafilters and 'relaxing them', logical connectives, filters, eventual families, limits with respect to them, inner and outer (finitely-additive) 'measures', multi-sets and multi-families, common fixed-points, Hausdorff topological spaces.

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An Analysis of the Superiorization Method via the Principle of Concentration of Measure

Cited by 8 publications

References 31 publications

Accumulated Random Distances in High Dimensions -- Ways of Calculation

Accumulated Random Distances in High Dimensions -- Ways of Calculation

Superiorization vs. accelerated convex optimization: The superiorized / regularized least-squares case

`Ultrafilters Relaxed' -- Species of Families of Subsets

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