Abstract
Sleeve functions are generalizations of the well-established ridge functions that play a major role in the theory of partial differential equation, medical imaging, statistics, and neural networks. Where ridge functions are non-linear, univariate functions of the distance to hyperplanes, sleeve functions are based on the squared distance to lower-dimensional manifolds. The present work is a first step to study general sleeve functions by starting with sleeve functions based on finite-length curves. To capture these curve-based sleeve functions, we propose and study a two-step method, where first the outer univariate function—the profile—is recovered, and second, the underlying curve is represented by a polygonal chain. Introducing a concept of well-separation, we ensure that the proposed method always terminates and approximates the true sleeve function with a certain quality. Investigating the local geometry, we study an inexact version of our method and show its success under certain conditions.
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Acknowledgements
The author is especially grateful to Sandra Keiper, the author of [26], for many fruitful discussions and for drawing my attention to the topic of generalized ridge and sleeve functions.
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Appendix: The set of ambiguity points
Appendix: The set of ambiguity points
In this appendix, we prove Theorem 5 for merely twice differentiable curves. Note that the proof of the original statement in [30, Prop 6] requires that \(\gamma \) is infinitely often differentiable. To study the set of ambiguity points, we use that the projection onto a Jordan \(C^2\)-curve is differentiable for most unambiguity points. These results can be found in [28], and we state it for our specific setting with \(C^2\)-curves.
Theorem 20
(Dudek–Holly [28, Thm 4.1]) Let \(\gamma \) be a Jordan \(C^2\)-arc, and let \(x \in \mathbb {R}^d\) be a point within an open neighbourhood where the projection is single-valued. If \({{\,\textrm{proj}\,}}_\gamma (x)\) is an inner point, then \({{\,\textrm{proj}\,}}_\gamma \) is differentiable at x.
The restriction to a point that is projected to an inner point is crucial since the projection becomes undifferentiable at the end points.
Counterexample 21
(End points) Consider the curve or line segment \(\gamma (t):= (t,0)^\text {T}\) with \(t \in [0,1]\). For \(x:= (0,1)^\text {T}\), the derivative in direction \((1,0)^\text {T}\) is \((1,0)^\text {T}\) but \((0,0)^\text {T}\) in direction \((-1,0)^\text {T}\). Thus, the projection is not differentiable at points \(x:= \gamma (0) + v\) with \(v \perp \dot{\gamma }(0+)\) and \({{\,\textrm{proj}\,}}_\gamma (x) = \gamma (0)\).
The ambiguity points with respect to a Jordan \(C^2\)-curve have a benign structure. The restriction \(A_2:= \{ x \in \mathbb {R}^d: \#[ {{\,\textrm{proj}\,}}_\gamma (x)] = 2 \}\) of the ambiguity set \(A:= \{ x \in \mathbb {R}^d: \#[ {{\,\textrm{proj}\,}}_\gamma (x)] > 1 \}\) consisting of all the points with exactly two projections has Lebesgue measure zero.
Lemma 22
(Ambiguity points) Let \(\gamma \) be a finite-length Jordan \(C^2\)-arc. Then, the subset \(A_2:= \{ x \in \mathbb {R}^d: \#[ {{\,\textrm{proj}\,}}_\gamma (x)] = 2 \}\) has Lebesgue measure zero.
Proof
Let x be an ambiguity point in \(A_2\) with projection \(P_1\) and \(P_2\). Since the distance function is continuous, we find a small open neighbourhood \(U_x\) such that \({{\,\textrm{dist}\,}}(y,\gamma )\) is attained by a curve point near \(P_1\) and/or \(P_2\), i.e. \({{\,\textrm{dist}\,}}(y,\gamma ) = \min \{ {{\,\textrm{dist}\,}}(y,\gamma _1), {{\,\textrm{dist}\,}}(y,\gamma _2) \}\), where \(\gamma _1\) and \(\gamma _2\) are small arcs around \(P_1\) and \(P_2\). Further, \(U_x\) may be chosen small enough such that the projection to a single arc \(\gamma _1\) or \(\gamma _2\) is single-valued in \(U_x\) such that \({{\,\textrm{proj}\,}}_{\gamma _1}\) and \({{\,\textrm{proj}\,}}_{\gamma _2}\) become continuously differentiable by Theorem 1. The ambiguity points in \(U_x\) are the zeros of the function
Since the gradient
is non-zero, \(P_1\) and \(P_2\) are distinct points, Dini’s implicit function theorem [39, Thm 1B.1] states that the ambiguity set \(A_2\) in an open neighbourhood \(\tilde{U}_x \subset U_x\) around x is the realization of a continuously differentiable map \(a_x :\mathbb {R}^{d-1} \rightarrow \mathbb {R}^d\) and is thus a Lebesgue zero set by Sard’s theorem [40]. Since the Euclidean \(\mathbb {R}^d\) is second-countable, already countably many set \(U_{x_n}\) cover \(A_2\), whose union is again a Lebesgue zero set. \(\square \)
Proposition 23
(Ambiguity points) Let \(\gamma \) be a Jordan \(C^2\)-arc. Then, the ambiguity set \(A:= \{ x \in \mathbb {R}^d: \#[ {{\,\textrm{proj}\,}}_\gamma (x)] > 1 \}\) is the closure of \(A_2\).
Proof
Since the distance to the curve is continuous, the points in \(\overline{A}_2\) are ambiguous. To show \(A \subset \overline{A}_2\), we take an ambiguity point x with \(\#[{{\,\textrm{proj}\,}}_\gamma (x)] > 2\). In two dimensions, the set \({{\,\textrm{proj}\,}}_\gamma (x)\) is located on a circle. Since \(\gamma \) is not closed, we can either shrink the circle and move it into a gap between to projection points or, if \({{\,\textrm{proj}\,}}_\gamma (x)\) lie on a half-sphere, we can move the circle outwards and enlarge it (see Fig. 12). In both cases, the centre y of the deformed circle is contained in \(A_2\). By controlling the radius, the centre may be arbitrarily close to x. This construction generalizes to \(\mathbb {R}^d\) by changing the radius and moving the sphere containing the projection points in several steps. \(\square \)
Figuratively, higher ambiguities with \(\#[{{\,\textrm{proj}\,}}_\gamma (x)] > 2\) occur at points, where the charts constructed by the implicit function theorem are glued together. Since \(A_2\) is locally a hypersurface, the Lebesgue measure of the closure remains zero, which establishes Theorem 5.
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Beinert, R. Approximation of curve-based sleeve functions in high dimensions. Adv Comput Math 49, 91 (2023). https://doi.org/10.1007/s10444-023-10088-2
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DOI: https://doi.org/10.1007/s10444-023-10088-2