Compensated convex transforms and geometric singularity extraction from semiconvex functions
The upper and lower compensated convex transforms are `tight' one-sided approximations for a given function. We apply these transforms to the extraction of fine geometric singularities from general semiconvex/semiconcave functions and DC-functions in Rn (difference of convex functions). Well-kn...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Zhongguo Kexue Zazhishe (Science China Press)
2016
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| Online Access: | https://eprints.nottingham.ac.uk/44857/ |
| _version_ | 1848797014841622528 |
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| author | Zhang, Kewei Crooks, Elaine Orlando, Antonio |
| author_facet | Zhang, Kewei Crooks, Elaine Orlando, Antonio |
| author_sort | Zhang, Kewei |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | The upper and lower compensated convex transforms are `tight' one-sided approximations for a given function. We apply these transforms to the extraction of fine geometric singularities from general semiconvex/semiconcave functions and DC-functions in Rn (difference of convex functions). Well-known geometric examples of (locally) semiconcave functions include the Euclidean distance function and the Euclidean squared-distance function. For a locally semiconvex function f with general modulus, we show that `locally' a point is singular (a non-differentiable point) if and only if it is a scale 1-valley point, hence by using our method we can extract all fine singular points from a given semiconvex function. More precisely, if f is a semiconvex function with general modulus and x is a singular point, then locally the limit of the scaled valley transform exists at every point x and can be calculated as limλ→+∞λVλ(f)(x)=r2x/4, where rx is the radius of the minimal bounding sphere of the (Fréchet) subdifferential ∂−f(x) of the locally semiconvex f and Vλ(f)(x) is the valley transform at x. Thus the limit function V∞(f)(x):=limλ→+∞λVλ(f)(x)=r2x/4 provides a `scale 1-valley landscape function' of the singular set for a locally semiconvex function f. At the same time, the limit also provides an asymptotic expansion of the upper transform Cuλ(f)(x) when λ approaches +∞. For a locally semiconvex function f with linear modulus we show further that the limit of the gradient of the upper compensated convex transform limλ→+∞∇Cuλ(f)(x) exists and equals the centre of the minimal bounding sphere of ∂−f(x). We also show that for a DC-function f=g−h, the scale 1-edge transform, when λ→+∞, satisfies liminfλ→+∞λEλ(f)(x)≥(rg,x−rh,x)2/4, where rg,x and rh,x are the radii of the minimal bounding spheres of the subdifferentials ∂−g and ∂−h of the two convex functions g and h at x, respectively. |
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| format | Article |
| id | nottingham-44857 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T19:57:09Z |
| publishDate | 2016 |
| publisher | Zhongguo Kexue Zazhishe (Science China Press) |
| recordtype | eprints |
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| spelling | nottingham-448572017-10-12T23:24:38Z https://eprints.nottingham.ac.uk/44857/ Compensated convex transforms and geometric singularity extraction from semiconvex functions Zhang, Kewei Crooks, Elaine Orlando, Antonio The upper and lower compensated convex transforms are `tight' one-sided approximations for a given function. We apply these transforms to the extraction of fine geometric singularities from general semiconvex/semiconcave functions and DC-functions in Rn (difference of convex functions). Well-known geometric examples of (locally) semiconcave functions include the Euclidean distance function and the Euclidean squared-distance function. For a locally semiconvex function f with general modulus, we show that `locally' a point is singular (a non-differentiable point) if and only if it is a scale 1-valley point, hence by using our method we can extract all fine singular points from a given semiconvex function. More precisely, if f is a semiconvex function with general modulus and x is a singular point, then locally the limit of the scaled valley transform exists at every point x and can be calculated as limλ→+∞λVλ(f)(x)=r2x/4, where rx is the radius of the minimal bounding sphere of the (Fréchet) subdifferential ∂−f(x) of the locally semiconvex f and Vλ(f)(x) is the valley transform at x. Thus the limit function V∞(f)(x):=limλ→+∞λVλ(f)(x)=r2x/4 provides a `scale 1-valley landscape function' of the singular set for a locally semiconvex function f. At the same time, the limit also provides an asymptotic expansion of the upper transform Cuλ(f)(x) when λ approaches +∞. For a locally semiconvex function f with linear modulus we show further that the limit of the gradient of the upper compensated convex transform limλ→+∞∇Cuλ(f)(x) exists and equals the centre of the minimal bounding sphere of ∂−f(x). We also show that for a DC-function f=g−h, the scale 1-edge transform, when λ→+∞, satisfies liminfλ→+∞λEλ(f)(x)≥(rg,x−rh,x)2/4, where rg,x and rh,x are the radii of the minimal bounding spheres of the subdifferentials ∂−g and ∂−h of the two convex functions g and h at x, respectively. Zhongguo Kexue Zazhishe (Science China Press) 2016-03-24 Article PeerReviewed application/pdf en https://eprints.nottingham.ac.uk/44857/1/ZOC-semiconvex%20Zhang.pdf Zhang, Kewei, Crooks, Elaine and Orlando, Antonio (2016) Compensated convex transforms and geometric singularity extraction from semiconvex functions. Scientia Sinica Mathematica, 46 (5). pp. 747-768. ISSN 1674-7216 Compensated convex transforms ridge transform valley transform edge transform convex function semiconvex function semiconcave function linear modulus general modulus DC-functions singularity extraction minimal bounding sphere local approximation local regularity singularity landscape http://engine.scichina.com/publisher/scp/journal/SSM/46/5/10.1360/N012015-00339?slug=full text doi:10.1360/N012015-00339 doi:10.1360/N012015-00339 |
| spellingShingle | Compensated convex transforms ridge transform valley transform edge transform convex function semiconvex function semiconcave function linear modulus general modulus DC-functions singularity extraction minimal bounding sphere local approximation local regularity singularity landscape Zhang, Kewei Crooks, Elaine Orlando, Antonio Compensated convex transforms and geometric singularity extraction from semiconvex functions |
| title | Compensated convex transforms and geometric singularity extraction from semiconvex functions |
| title_full | Compensated convex transforms and geometric singularity extraction from semiconvex functions |
| title_fullStr | Compensated convex transforms and geometric singularity extraction from semiconvex functions |
| title_full_unstemmed | Compensated convex transforms and geometric singularity extraction from semiconvex functions |
| title_short | Compensated convex transforms and geometric singularity extraction from semiconvex functions |
| title_sort | compensated convex transforms and geometric singularity extraction from semiconvex functions |
| topic | Compensated convex transforms ridge transform valley transform edge transform convex function semiconvex function semiconcave function linear modulus general modulus DC-functions singularity extraction minimal bounding sphere local approximation local regularity singularity landscape |
| url | https://eprints.nottingham.ac.uk/44857/ https://eprints.nottingham.ac.uk/44857/ https://eprints.nottingham.ac.uk/44857/ |