joint work with Yurii Nesterov
In this work, we propose p-order methods for minimization of convex functions that are p-times differentiable with $\nu$-Hölder continuous p-th derivatives. For the schemes without acceleration, we establish iteration complexity bounds of $O(\epsilon^{-1/(p+\nu-1)})$. Assuming that $\nu$ is known, we obtain an improved complexity bound of $O(\epsilon^{-1/(p+\nu)})$ for the corresponding accelerated scheme. For the case in which $\nu$ is unknown, we present a universal accelerated tensor scheme with complexity of $O(\epsilon^{-p/(p+1)(p+\nu-1)})$. A lower complexity bound is also obtained.
joint work with Xin Liu, Yaxiang Yuan
Clustering is an important topic in data science. The popular approaches including K-means algorithm and nonnegative matrix factorization (NMF) directly use data points as input and consider problems in Euclidean space. In this paper, we propose a new approach which considers Gaussian kernel based on graph construction. Moreover, the final cluster labels can be directly obtained without post-processing. The experimental results on real data sets have shown that the proposed approach achieves better clustering results in terms of accuracy and mutual information.