Hierarchical clustering has been a popular method in various data analysis
applications. It partitions a data set into a hierarchical collection of
clusters, and can provide a global view of (cluster) structure behind data
across different granularity levels. A hierarchical clustering (HC) of a data
set can be naturally represented by a tree, called a HC-tree, where leaves
correspond to input data and subtrees rooted at internal nodes correspond to
clusters. Many hierarchical clustering algorithms used in practice are
developed in a procedure manner. Dasgupta proposed to study the hierarchical
clustering problem from an optimization point of view, and introduced an
intuitive cost function for similarity-based hierarchical clustering with nice
properties as well as natural approximation algorithms.
We observe that while Dasgupta's cost function is effective at
differentiating a good HC-tree from a bad one for a fixed graph, the value of
this cost function does not reflect how well an input similarity graph is
consistent to a hierarchical structure. In this paper, we present a new cost
function, which is developed based on Dasgupta's cost function, to address this
issue. The optimal tree under the new cost function remains the same as the one
under Dasgupta's cost function. However, the value of our cost function is more
meaningful. The new way of formulating the cost function also leads to a
polynomial time algorithm to compute the optimal cluster tree when the input
graph has a perfect HC-structure, or an approximation algorithm when the input
graph 'almost' has a perfect HC-structure. Finally, we provide further
understanding of the new cost function by studying its behavior for random
graphs sampled from an edge probability matrix.

提出了一种新的代价函数，基于 Dasgupta 的代价函数，解决了旧代价函数无法度量原始相似图与分层结构一致性的问题，并提出了一个多项式时间算法进行计算。