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    1. 【95周年校慶系列講座】A Diffusion Perspective of Manifold Clustering

      時間:2020-07-29         閱讀:

      光華講壇——社會名流與企業家論壇第 5836 期

      (線上講座)

      主題A Diffusion Perspective of Manifold Clustering

      主講人伊利諾伊大學香檳分校 陳曉輝副教授

      主持人統計學院 常晉源教授

      時間2020年7月31日(周五)10:00-11:20

      直播平臺及會議IDZoom,會議ID:367 123 8320

      主辦單位:統計研究中心 數據科學與商業智能聯合實驗室 統計學院 科研處

      主講人簡介:

      Xiaohui Chen received a Ph. D. in Electrical and Computer Engineering in 2013 from the University of British Columba (UBC), Vancouver, Canada. He was a post-doctoral fellow at the Toyota Technological Institute at Chicago (TTIC), a philanthropically endowed academic computer science institute located on the University of Chicago campus. In 2013 he joined the University of Illinois at Urbana-Champaign (UIUC) as an Assistant Professor of Statistics. He is an Associate Professor of Statistics at UIUC since 2019 and a member of Discovery Partners Institute (DPI) since 2020. He held Visiting Faculty position in the Institute for Data, Systems, and Society (IDSS) at Massachusetts Institute of Technology (MIT) in 2019-2020. He received numerous notable awards, including an NSF CAREER Award in 2018, an Arnold O. Beckman Award at UIUC in 2018, an Outstanding Young Researcher Award from the International Chinese Statistical Association (ICSA) in 2019, an Associate appointment in the Center for Advanced Study at UIUC in 2020-2021, and a Simons Fellowship in Mathematics from the Simons Foundation in 2020-2021. His teaching was recognized three times by the University of Illinois List of Teachers Ranked as Excellent by Their Students.

      陳曉輝,2013年畢業于加拿大溫哥華哥倫比亞大學(UBC)獲得電子與計算機工程博士學位。他曾是芝加哥豐田技術研究所(TTIC)的博士后研究員,TTIC是位于芝加哥大學校園內的一所受慈善資助的學術計算機科學研究所。2013年,他加入了伊利諾伊大學香檳分校(UIUC),擔任統計學助理教授。自2019年起,他是UIUC的統計學副教授,自2020年起,他是Discovery Partners Institute (DPI) (DPI)的成員。2019-2020年,他在麻省理工學院(MIT)數據、系統和社會研究所(IDSS)擔任客座教授。他獲得有眾多著名的獎項,包括2018年NSF事業獎,2018UIUC的Arnold O. Beckman Award,2019年ICSA杰出青年研究學者獎,2020-2021年UIUC的高級研究中心的助理任命,2020-2021年西蒙斯基金會數學獎學金。他三次在UIUC被學生評為優秀教師。

      內容提要:

      We introduce the diffusion K-means clustering method on Riemannian submanifolds, which maximizes the within-cluster connectedness based on the diffusion distance. The diffusion K-means constructs a random walk on the similarity graph with vertices as data points randomly sampled on the manifolds and edges as similarities given by a kernel that captures the local geometry of manifolds. The diffusion K-means is a multi-scale clustering tool that is suitable for data with non-linear and non-Euclidean geometric features in mixed dimensions. Given the number of clusters, we propose a polynomial-time convex relaxation algorithm via the semidefinite programming (SDP) to solve the diffusion K-means. In addition, we also propose a nuclear norm regularized SDP that is adaptive to the number of clusters. In both cases, we show that exact recovery of the SDPs for diffusion K-means can be achieved under suitable between-cluster separability and within-cluster connectedness of the submanifolds, which together quantify the hardness of the manifold clustering problem. We further propose the localized diffusion K-means by using the local adaptive bandwidth estimated from the nearest neighbors. We show that exact recovery of the localized diffusion K-means is fully adaptive to the local probability density and geometric structures of the underlying submanifolds. Joint work with Yun Yang (UIUC).

      本文引入黎曼子流形上的擴散K均值聚類方法,以使得擴散距離的聚類內連通性最大化。擴散K均值在相似圖上會構造一個隨機游走,頂點能作為流形上隨機采樣的數據點也能作為一個核的相似性,這個核能捕獲流形的局部幾何。擴散K均值是一種多尺度聚類工具,適用于混合維度中具有非線性和非歐幾里得幾何特征的數據。在給定簇數的情況下,本文提出了一個利用半定規劃(SDP)求解擴散K均值的多項式-時間凸松弛算法。此外,本文還提出了一個能自適應簇數的核范數正則化的SDP。在這兩種情況下,本文證明了子流形在適當的簇間可分性和簇內連通性下,能夠準確恢復擴散K均值的SDPs,從而使得流形聚類問題得以量化。本文利用最近鄰估計的局部自適應帶寬,提出了局部擴散K均值算法。本文證明了局部擴散K均值能完全適應底層子流形的局部概率密度和幾何結構。本文是和Yun Yang(UIUC)一起合作完成的。

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