## 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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## Data Availibility

All implementations and simulations underlying this article are publicly available at https://github.com/robertbeinert/curve-based-sleeve-functions.

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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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Communicated by: Holger Rauhut

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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