However, not all of the data are representative and meaningful, so the analysis and disposal of large-scale data occupies an increasingly important position in scientific research and social life [1]. Cluster analysis is an important unsupervised learning method in machine learning. Its basic idea is grouping a set of objects into clusters, in a way that objects in the same cluster share more similarity than those from separated clusters, in terms of distances of a certain space. In the evolution of clustering, due to the differences of data types and clustering strategies, cluster analysis can be divided into two main branches, namely, traditional clustering algorithms and modern clustering algorithms. Traditional clustering algorithms include clustering algorithm based on partition, density, model, fuzzy theory and so on [2, 3].

Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in non-convex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem limited. We explore how generally we can apply Banach's fixed point theorem to establish the convergence of iterative methods when pairing it with carefully designed metrics. Our first result is a strong converse of Banach's theorem, showing that it is a universal analysis tool for establishing global convergence of iterative methods to unique fixed points, and for bounding their convergence rate. In other words, we show that, whenever an iterative map globally converges to a unique fixed point, there exists a metric under which the iterative map is contracting and which can be used to bound the number of iterations until convergence. We illustrate our approach in the widely used power method, providing a new way of bounding its convergence rate through contraction arguments. We next consider the computational complexity of Banach's fixed point theorem. Making the proof of our converse theorem constructive, we show that computing a fixed point whose existence is guaranteed by Banach's fixed point theorem is CLS-complete. We thus provide the first natural complete problem for the class CLS, which was defined in [Daskalakis, Papadimitriou 2011] to capture the complexity of problems such as P-matrix LCP, computing KKT-points, and finding mixed Nash equilibria in congestion and network coordination games.

Sequential decision problems that involve multiple objectives are prevalent. Consider for example a driver of a semi-autonomous car who may want to optimize competing objectives such as travel time and the effort associated with manual driving. We introduce a rich model called Lexicographic MDP (LMDP) and a corresponding planning algorithm called LVI that generalize previous work by allowing for conditional lexicographic preferences with slack. We analyze the convergence characteristics of LVI and establish its game theoretic properties. The performance of LVI in practice is tested within a realistic benchmark problem in the domain of semi-autonomous driving. Finally, we demonstrate how GPU-based optimization can improve the scalability of LVI and other value iteration algorithms for MDPs.