Selecting the independent coordinates of manifolds with large aspect ratios
This addresses a pervasive and complex issue in manifold learning for researchers and practitioners dealing with high-dimensional data, offering a theoretically grounded solution with low computational overhead.
The paper tackles the problem of manifold embedding algorithms failing on data manifolds with large aspect ratios, such as long, thin strips, by proposing a bicriterial Independent Eigencoordinate Selection (IES) algorithm that selects smooth embeddings using few eigenvectors, achieving success on synthetic and large real data.
Many manifold embedding algorithms fail apparently when the data manifold has a large aspect ratio (such as a long, thin strip). Here, we formulate success and failure in terms of finding a smooth embedding, showing also that the problem is pervasive and more complex than previously recognized. Mathematically, success is possible under very broad conditions, provided that embedding is done by carefully selected eigenfunctions of the Laplace-Beltrami operator $Δ$. Hence, we propose a bicriterial Independent Eigencoordinate Selection (IES) algorithm that selects smooth embeddings with few eigenvectors. The algorithm is grounded in theory, has low computational overhead, and is successful on synthetic and large real data.