The basic input for many real shapes is a finite cloud of unordered points. The strongest equivalence between shapes in practice is Euclidean motion. The recent polynomial-time classification of point clouds required a Lipschitz continuous function that vanishes on degenerate simplices, while the usual volume is not Lipschitz. We define the strength of any geometric simplex and prove its continuity under perturbations with explicit bounds for Lipschitz constants.
Many applications deal with point configurations or clouds of points obtained as edge pixels or feature points of objects across all scales from galaxies to molecules [1]. Positions of points in a Euclidean space, such as atomic centres, are always uncertain due to measurement noise or thermal vibrations. Molecular dynamics simulates trajectories of atomic clouds, while machine learning tries to predict molecular properties that depend on atomic geometry.
The robustness of algorithmic predictions means that an output only continuously changes under perturbations of given points. The classical εδ concept of continuity is very weak in the sense that all standard functions are continuous on domains, where they are defined. For example, f (x) = 1 x is continuous for any x 0 because, for any
x 2 ≤ ε. However, for small x > 0, the chosen delta δ is much smaller than ε, so f (x) = 1 x grows too fast close to 0. The Lipschitz continuity below is much more practical by restricting the growth of a function via a constant and an amount of perturbation. Definition 1.1 (Lipschitz continuity). A function f : R m → R is Lipschitz continuous if there is a Lipschitz constant λ > 0 such that | f (x)f (y)| ≤ λ|x -y| for any x, y ∈ R n , where |x -y| denotes the Euclidean distance.
Then f (x) = 1 x is not Lipschitz continuous because for any λ > 0, we can set c = max{1, λ},
Though the Lipschitz continuity makes sense for maps between metric spaces, we consider only scalar functions on simplices in R n . A simplex can be defined as a finite set of elements whose every subset is also a simplex. We consider only geometric realizations of a simplex, still called a simplex.
Definition 1.2 (a geometric simplex A on n + 1 points in R n ). The (geometric) simplex A on any n + 1 ordered points
with ordered vertices p 0 , . . . , p n . An orientation of A is the sign of the determinant of the n × n matrix with the columns p 1p 0 , . . . , p np 0 , and is denoted by sign(A).
In dimension n = 1, the simplex on any points p 0 , p 1 ∈ R is the line segment connecting p 0 and p 1 . If points p 0 , . . . , p n ∈ R n are affinely independent, i.e. there is no (n -1)-dimensional affine subspace of R n containing all p 0 , . . . , p n and hence the simplex A, then A is n-dimensional. However, Definition 1.2 makes sense for any points.
The volume vol(A) of a simplex A or an arbitrary polyhedron is often used as a shape descriptor and detects affine independence in the sense that A is degenerate if and only if vol(A) = 0. However, the volume and all other distance-based descriptors do not distinguish mirror images, which have different signs of orientation.
When A goes through a degenerate configuration, an orientation of A can discontinuously change the sign. This discontinuity is an obstacle to recognizing simplices (or more general clouds) that are nearly mirror-symmetric.
One attempt to resolve this discontinuity is to consider the signed volume sign(A)vol(A), because vol(A) vanishes only on degenerate simplices. However, vol(A) is not Lipschitz continuous for any n ≥ 2, as illustrated below.
Example 1.3 (the area of a triangle is not Lipschitz continuous). For any large l > 0 and real ε close to 0, let T (l, ε) ⊂ R 2 be the 2D simplex (triangle) on the vertices (0, ε) and (±l, 0). The signed area lε of T (l, ε) distinguishes mirror images T (l, ±ε) but is not Lipschitz continuous under perturbations. Indeed, as ε → 0, the triangle T (l, ε) degenerates to a straight line, while the area drops to 0 too quickly so that vol(T (l, ε))vol(T (l, 0))
bounded. Hence, if given points are not restricted to a fixed bounded region, a small change in their positions may lead to a large change for the area of a triangle, and similarly for the volume of a high-dimensional simplex in R n .
Problem 1.4 (Lipschitz continuous detection of degenerate simplices). Find a Lipschitz continuous real-valued function f (A) for all simplices A on n + 1 points p 0 , . . . , p n ∈ R n such that f (A) = 0 if and only if A is degenerate.
We solve Problem 1.4 by introducing the strength function whose Lipschitz continuity is proved in Theorem 2.4.
Hence, the strength of T (l, ε) is Lipschitz continuous with respect to perturbations of ε.
of a normalized triangle over the region ∆. Recall that an isometry is any distance-preserving transformation of R n , which decomposes into translations and orthogonal maps from the group O(R n ). A rigid motion is any composition of translations and rotations from the special orthogonal group SO(R n ) The strength of a simplex was essentially used to define a Lipschitz continuous metric on invariants of n-dimensional clouds of m unordered points, which are complete under rigid motion in R n and can be computed in a polynomial time of m, for a fixed dimension n [2, Theorem 4.7], see detailed proofs in [3].
Theorem 2.4 (strength properties: invariance, computational time, and Lipschitz continuity). (a) The strength σ(A) and s
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