In this paper, the connection between Menger probabilistic norms and H"{o}hle probabilistic norms is discussed. In addition, the correspondence between probabilistic norms and Wu-Fang fuzzy (semi-) norms is established. It is shown that a probabilistic norm (with triangular norm $min$) can generate a Wu-Fang fuzzy semi-norm and conversely, a Wu-Fang fuzzy norm can generate a probabilistic norm.

We consider the problem of fitting a parametric curve to a given point cloud (e.g., measurement data). Least-squares approximation, i.e., minimization of the l2 norm of residuals (the Euclidean distances to the data points), is the most common approach. This is due to its computational simplicity [1]. However, in the case of data that is affected by noise or contains outliers, this is not alway...

We consider continuous functions that are deened by programs for their evaluation. The basic arithmetic operations and univariate special functions are real analytic in the interior of their domains. However , another frequent ingredient, the absolute value function, has a kink at the origin, which occurs similarly for max and min. A slightly more serious complication arises with the introducti...

We present minimax and stochastic formulations of some linear approximation problems with uncertain data in R n equipped with the Euclidean (l2), Absolute-sum (l1) or Chebyshev (l1) norms. We then show that these problems can be solved using convex optimization. Our results parallel and extend the work of El-Ghaoui and Lebret on robust least squares 3], and the work of Ben-Tal and Nemirovski on...

We investigate techniques for similarity analysis of spatio-temporal trajectories for mobile objects. Such kind of data may contain a great amount of outliers, which degrades the performance of Euclidean and Time Warping Distance. Therefore, here we propose the use of non-metric distance functions based on the Longest Common Subsequence (LCSS), in conjunction with a sigmoidal matching function....

We prove sharp inequalities for the average number of affine diameters through the points of a convex body K in Rn. These inequalities hold if K is a polytope or of dimension two. An example shows that the proof given in the latter case does not extend to higher dimensions. The example also demonstrates that for n ≥ 3 there exist norms and convex bodies K ⊂ Rn such that the metric projection on...

To avoid the undesired effects of distance concentration in high-dimensional spaces, previous work has already advocated the use of fractional p norms instead of the ubiquitous Euclidean norm. Closely related to concentration is the emergence of hub and anti-hub objects. Hub objects have a small distance to an exceptionally large number of data points while anti-hubs lie far from all other data...

This paper adresses the problem of least-square fitting with rational pole curves. The issue is to minimize a sum of squared Euclidean norms with respect to three types of unknowns: the control points, the node values, and the weights. A new iterative algorithm is proposed to solve this problem. The method alternates between three steps to converge towards a solution. One step uses the projecti...

The kinematic sensitivity is a unit-consistent measure that has been recently proposed as a mechanism performance index to compare robot architectures. This paper presents a robust geometric approach for computing this index for the case of planar parallel mechanisms. The physical meaning of the kinematic sensitivity is investigated through different combinations of the Euclidean and infinity n...

We show the optimality of sphere-separable partitions for problems where n vectors in d-dimensional space are to be partitioned into p categories to minimize a cost function which is dependent in the sum of the vectors in each category; the sum of the squares of their Euclidean norms; and the number of elements in each category.We further show that the number of these partitions is polynomial i...