A new locally linear embedding scheme in light of Hessian eigenmap

2021/12/16 by Liren Lin, Lin, Liren, Chih‐Wei Chen +1
Computer Science · Physics and Astronomy · #62-07 #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Neural Networks and Applications #Numerical Analysis (math.NA) #Theoretical and Computational Physics #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2112.09086

openalex publication_date 2021/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide a new interpretation of Hessian locally linear embedding (HLLE), revealing that it is essentially a variant way to implement the same idea of locally linear embedding (LLE). Based on the new interpretation, a substantial simplification can be made, in which the idea of "Hessian" is replaced by rather arbitrary weights. Moreover, we show by numerical examples that HLLE may produce projection-like results when the dimension of the target space is larger than that of the data manifold, and hence one further modification concerning the manifold dimension is suggested. Combining all the observations, we finally achieve a new LLE-type method, which is called tangential LLE (TLLE). It is simpler and more robust than HLLE.

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