Dimensionality Reduction Based on Geodesic Distance Hao - PowerPoint PPT Presentation

Apr 06, 2023 •155 likes •341 views

Dimensionality Reduction Based on Geodesic Distance Hao Li,515030910494 Yifan Shen,515030910491 2018.5.28 Background Dimensionality reduction method: Principle Component Analysis PCA Isometric Mapping ISOMAP Locally Linear Embedding LLE

Dimensionality Reduction Based on Geodesic Distance Hao Li,515030910494 Yifan Shen,515030910491 2018.5.28
Background Dimensionality reduction method: Principle Component Analysis PCA Isometric Mapping ISOMAP Locally Linear Embedding LLE
Background Distance on the sample space: Euclidian distance Path distance Geodesic distance
Our goal IN MDS or ISOMAP Path distance <- Geodesic distance Heat Method on point cloud
The heat method
Time discretization --We’ll talk about the solution of this later, after we acquire all the necessary operators…
Apply to point cloud
The discrete LBO operator (∆) L
The divergence ( ∇ ·) on a manifold
The gradient ( ∇ ) on a manifold 1. Computing the gradient in euclidean space 2. Project the gradient onto the tangent space
THE GRADIENT IN EUCLIDEAN SPACE
Project the gradient onto the tangent space 1. Determining the neighbourhood 2. Extracting local information
Project the gradient onto the tangent space 3. Constructing alignment matrix 4. Computing the maps
Solve the poisson equations
Total steps
Negative results The gradient just disappeared… And the whole program went wrong from the very first step of solving the heat equation… Honest Reasons: 1. Our laziness 2. Lack of time for debugging/trial-and-error (because of 1) Well Known Reasons: 1. Noise around the manifold 2. High curvature of the manifold 3. High intrinsic dimension of the manifold 4. Presence of many manifolds with little data per manifold
Conclusions In this project, we implemented a framework –- although it is a failure -- to compute the geodesic distance used in MDS on manifold based on heat method. We mainly focused on how to implement the heat method on point cloud. The intrinsic reasons for negative result in manifold learning are analyzed after we get the negative results.
References [1]Keenan Crane, Clarisse Weischedel, Max Wardetzky, "Geodesics in Heat: A New Approach to Computing Distance Based on Heat Flow“ [2]Mikhail Belkin, Partha Niyogi, "Laplacian Eigenmaps for Dimensionality Reduction and Data Representation“ [3]SayanMukherjee,QiangWu,EstimationofGradientsandCoordinateCovariat ioninClassification” [4]Tianhao Zhang, Jie Yang, Deli Zhao, Xinliang Ge, "Linear local tangent space alignment and application to face recognition“ [5]YoshuaBengio,Martin Monperrus,"Non-Local Manifold Tangent Learning"

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