In this paper, we specify the performance of our pioneering billiard robot in pulling game training. We discuss the important role of practical demonstration in sports education, and introduce the robot technology to teaching demonstration of sports item. And put forward our algorithm for the fast image calculation of the ball obstructed by other balls, the problem that must be resolved while t...

We study the problem of estimating the leading eigenvectors of a high-dimensional population covariance matrix based on independent Gaussian observations. We establish a lower bound on the minimax risk of estimators under the l2 loss, in the joint limit as dimension and sample size increase to infinity, under various models of sparsity for the population eigenvectors. The lower bound on the ris...

We state below two results that we use frequently in our proofs. The first is well-known consequence of the CS decomposition. It relates the canonical angles between subspaces to the singular values of products and differences of their corresponding projection matrices.

The assessment of a cluster’s significance given the null (configuration) model relies on the estimation of the probability described in Eq. 1 of the main text. This function has to be evaluated many times along the execution of OSLOM in order to clean up each cluster and to evaluate the clusters at the different hierarchical levels. We explain here how the values of the distribution function c...