报告题目∶A Unified Continuous Optimization Framework for
Center-based Clustering Methods

报告人：Prof. Marc Teboulle
        Tel-Aviv University, Israel

报告时间∶4月11日(星期五)上午∶10:00-11:00

报告地点∶蓝白楼311, 中国科学院数学与系统科学研究院

报告摘要：Clustering consists of grouping or classifying data into
similar objects. It is a fundamental technique in unsupervised
machine learning and arise in a wide spectrum of research fields
such as, astrophysics, statistics, biology, computer vision, data
compression in image processing, and information retrieval, to name
just a few. The interdisciplinary nature of the clustering problem
has generated a very large body of literature in cluster analysis,
with many formulations of the problem, and numerous clustering
algorithms which are based on various disparate motivations and
approaches. 

In this talk, we present a unifying framework for center
based clustering algorithms from an optimization theory perspective.
Starting with the standard nonconvex and nonsmooth formulation of
the clustering problem, which minimizes the sum of a finite
collection of ``min" functions, we present a systematic way for
designing center based clustering methods. Our approach builds on
two fundamental mathematical concepts: convex asymptotic functions,
and the so-called nonlinear means of Hardy, Littlewood and Polya.
This allows us to derive a generic smoothing iterative scheme for
clustering, which computationally, is as simple as the popular
k-means algorithm. We analyze its properties, and establish its
convergence. We then demonstrate that most well known center based
clustering algorithms, which were derived either heuristically,
or/and which have emerged from intuitive analogies in physics,
statistical techniques, and from information theoretic perspectives,
can be simply recovered, analyzed and extended through the proposed
framework.