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Distance-Preserving Property of Random Projection for Low Dimensional Subspaces

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In the big data era we confront with large-scale problems dealing with data points or features in high dimensional vector spaces. In the enduring effort of trying to decrease the solving complexity of such large problems, dimension reduction has played an essential role. The well-known Johnson-Lindenstrauss (JL) Lemma and the Restricted Isometry Property (RIP) allow the use of random projection to reduce the space dimension while keeping the Euclidean distance between any two data points, which leads to the boom of Compressed Sensing and the field of sparsity related signal processing. Recently, successful applications of sparse models in computer vision and machine learning have increasingly hinted that the underlying structure of high dimensional data looks more like a union of subspaces (UoS) than a single subspace.

In this talk I will introduce our recent work on dimension reduction motivated by subspace clustering and JL Lemma. We study for the first time the distance-preserving property of Gaussian random projection matrices for two low-dimensional subspaces. We theoretically prove that with high probability the affinity and the distance, between two compressed subspaces is concentrated on its estimate. Numerical experiments on both synthetic and real-world data verify our theoretical work.

Short Bio: Yuantao Gu received the B.E. degree from Xi’an Jiaotong University in 1998, and the Ph.D. degree with honor from Tsinghua University in 2003, both in Electronic Engineering. He joined the faculty of Tsinghua in 2003 and is now a Tenured Associate Professor with Department of Electronic Engineering. He was a visiting scientist at Microsoft Research Asia during 2005 to 2006, Research Laboratory of Electronics at Massachusetts Institute of Technology during 2012 to 2013, and Department of Electrical Engineering and Computer Science at the University of Michigan in Ann Arbor during 2015. His research interests include high-dimensional statistics, sparse signal recovery, temporal-space and graph signal processing, related topics in wireless communications and information networks. He has been an Associate Editor of the IEEE Transactions on Signal Processing since 2015 and an Elected Member of the IEEE Signal Processing Theory and Methods Technical Committee since 2016. He received the Best Paper Award of IEEE Global Conference on Signal and Information Processing (GlobalSIP) in 2015, the Award for Best Presentation of Journal Paper of IEEE International Conference on Signal and Information Processing (ChinaSIP) in 2015, and Zhang Si-Ying (CCDC) Outstanding Youth Paper Award (with his student) in 2017.

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