Compensated convex transforms and geometric singularity\ extraction from semiconvex functions

Abstract

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 R n (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 lim λ → + ∞ λ V λ ( f ) ( x ) = r 2 x / 4 , where r x is the radius of the minimal bounding sphere of the (Fréchet) subdifferential ∂ − f ( x ) of the locally semiconvex f and V λ ( f ) ( x ) is the valley transform at x . Thus the limit function V ∞ ( f ) ( x ) := lim λ → + ∞ λ V λ ( f ) ( x ) = r 2 x / 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 C u λ ( 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 lim λ → + ∞ ∇ C u λ ( 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 liminf λ → + ∞ λ E λ ( f ) ( x ) ≥ ( r g , x − r h , x ) 2 / 4 , where r g , x and r h , 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.

补偿凸上变换和下变换 [29, 30, 31] 是对给定函数作“紧贴”逼近的单参数单向变换。我 们将它们应用到 R n 中局部具有一般模的半凸／半凹函数和 DC-函数（既两个凸函数的差函数） 的奇点提取。（局部）半凹函数最常见的几何例子有欧氏距离函数与平方欧氏距离函数。对于局 部具有一般模的半凸函数 f，我们证明在局部意义下 x 是 f 的奇点（既不可微点）当且仅当它 是 f 的 1-阶“谷点”，因而用我们的方法可以从具有一般模的局部半凸函数中提取所有的这些 精细的几何奇点。更确切地讲，如果 f 是局部具有一般模的半凸函数，则“局部地”1-阶“山 谷”变换在每个点 x 的极限存在，而且有显示表示 lim λ→+∞ λVλ(f)(x) = r 2 x/4，其中 Vλ(f)(x) 是 f 在 x 点的“山谷”变换，rx 是 f 在 x 点次微分 ∂−f(x) 的最小包含球面的半径 [16]。所以极限 函数 V∞(f)(x) := lim λ→+∞ λVλ(f)(x) = r 2 x/4 为我们提供了一个半凸函数奇点 1-阶“谷点”的“景 观函数”。同时它也提供了补偿上凸变换 C u λ (f)(x) 当 λ 趋于 +∞ 时的一阶渐进展开式。对于具 有局部线性模的局部半凸函数，我们进一步证明，补偿凸上变换的梯度，当 λ → +∞ 时的极限 lim λ→+∞ ∇C u λ (f)(x) 存在，并且这个极限等于次微分 ∂−f(x) 的最小包含球面的中心。对于 DC-函数 f = g−h 我们证明它的 1-阶“边缘”变换，当 λ → +∞ 时满足 lim inf λ→+∞ λEλ(f)(x) ≥ (rg,x−rh,x) 2 /4， 其中 rg,x 与 rh,x 分别是次微分 ∂−g(x) 与 ∂−h(x) 的最小包含球面的半径。