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UID:20250709T225036EDT-4296oL92pc@132.216.98.100
DTSTAMP:20250710T025036Z
DESCRIPTION:An embedding theorem: differential analysis behind massive data
  analysis\n\nHigh-dimensional data can be difficult to analyze. Assume dat
 a are distributed on a low-dimensional manifold. The Vector Diffusion Mapp
 ing (VDM)\, introduced by Singer-Wu\, is a non-linear dimension reduction 
 technique and is shown robust to noise. It has applications in cryo-electr
 on microscopy and image denoising and has potential application in time-fr
 equency analysis. In this talk\, I will present a theoretical analysis of 
 the effectiveness of the VDM. Specifically\, I will discuss parametrisatio
 n of the manifold and an embedding which is equivalent to the truncated VD
 M. In the differential geometry language\, I use eigen-vector fields of th
 e connection Laplacian operator to construct local coordinate charts that 
 depend only on geometric properties of the manifold. Next\, I use the coor
 dinate charts to embed the entire manifold into a finite-dimensional Eucli
 dean space. The proof of the results relies on solving the elliptic system
  and provide estimates for eigenvector fields and the heat kernel and thei
 r gradients.High-dimensional data can be difficult to analyze. Assume data
  are distributed on a low-dimensional manifold. The Vector Diffusion Mappi
 ng (VDM)\, introduced by Singer-Wu\, is a non-linear dimension reduction t
 echnique and is shown robust to noise. It has applications in cryo-electro
 n microscopy and image denoising and has potential application in time-fre
 quency analysis. In this talk\, I will present a theoretical analysis of t
 he effectiveness of the VDM. Specifically\, I will discuss parametrisation
  of the manifold and an embedding which is equivalent to the truncated VDM
 . In the differential geometry language\, I use eigen-vector fields of the
  connection Laplacian operator to construct local coordinate charts that d
 epend only on geometric properties of the manifold. Next\, I use the coord
 inate charts to embed the entire manifold into a finite-dimensional Euclid
 ean space. The proof of the results relies on solving the elliptic system 
 and provide estimates for eigenvector fields and the heat kernel and their
  gradients.\n
DTSTART:20170426T173000Z
DTEND:20170426T183000Z
LOCATION:Room 920\, Burnside Hall\, CA\, QC\, Montreal\, H3A 0B9\, 805 rue 
 Sherbrooke Ouest
SUMMARY:Chen-Yun Lin\, University of Toronto
URL:/mathstat/channels/event/chen-yun-lin-university-t
 oronto-267816
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