We study permutation-invariant embeddings ofd-dimensional point sets, which are defined by sortingDindependent one-dimensional projections of the input. Such embeddings arise in graph deep learning where outputs should be invariant to permutations of graph nodes. Previous work showed that for large enoughDand projections in general position, this mapping is injective, and moreover satisfies a bi-Lipschitz condition. However, two gaps remain: firstly, the optimal sizeDrequired for injectivity is not yet known, and secondly, no estimates of the bi-Lipschitz constants of the mapping are known. In this paper, we make substantial progress in addressing both of these gaps. Regarding the first gap, we improve upon the best known upper bounds for the embedding dimensionDnecessary for injectivity, and also provide a lower bound on the minimal injectivity dimension. Regarding the second gap, we construct matrices of projection vectors, so that the bi-Lipschitz distortion of the mapping depends quadratically on the number of pointsn, and is completely independent of the dimensiond. We also show that for any choice of projection vectors, the distortion of the mapping will never be better than a bound proportional to the square root ofn. Finally, we show that similar guarantees can be provided even when linear projections are applied to the mapping to reduce its dimension.

The aim of this study is to optimize a 2-dof micro parallel robot. Parallel robots potential is only then efficient exploited when their structure is optimal dimensioned from geometric point of view. So, their performances depend very strong on their geometry. Thus, optimization of the geometric parameters or optimal dimensioning has become an important issue for improving the parallel robots performances. Here, intended to show the advantages of using the GA, we applied it to an optimization problem of 2 DOF parallel robot. The obtained results have shown that the use of GA in such kind of optimization problem enhances the quality of the optimization outcome, providing a better and more realistic support for the decision maker.

This paper is aimed at presenting a study on the optimization of the Biglide and Five-bar micro parallel robots, which comprises two-degree-of-freedom (DOF) micro parallel robots with constant and variable struts. The robot workspace is characterized and the inverse kinematics equation is obtained. In the paper, design optimization is implemented with Genetic Algorithms (GA) for optimization considering transmission quality index, design space and workspace. Here, intended to show the advantages of using the GA, we applied it to a multicriteria optimization problem of 2 DOF micro parallel robots. Genetic algorithms (GA) are so far generally the best and most robust kind of evolutionary algorithms. A GA has a number of advantages. It can quickly scan a vast solution set. Bad proposals do not affect the end solution negatively as they are simply discarded. The obtained results have shown that the use of GA in such kind of optimization problem enhances the quality of the optimization outcome, providing a better and more realistic support for the decision maker.