Nested Grassmannians for Dimensionality Reduction with Applications

Chun-Hao Yang1, Baba C. Vemuri2
1: National Taiwan University, 2: University of Florida
IPMI 2021 special issue
Publication date: 2022/03/01
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Abstract

In the recent past, nested structure of Riemannian manifolds has been studied in the context of dimensionality reduction as an alternative to the popular principal geodesic analysis (PGA) technique, for example, the principal nested spheres. In this paper, we propose a novel framework for constructing a nested sequence of homogeneous Riemannian manifolds. Common examples of homogeneous Riemannian manifolds include the spheres, the Stiefel manifolds, and the Grassmann manifolds. In particular, we focus on applying the proposed framework to the Grassmann manifolds, giving rise to the nested Grassmannians (NG). An important application in which Grassmann manifolds are encountered is planar shape analysis. Specifically, each planar (2D) shape can be represented as a point in the complex projective space which is a complex Grassmann manifold. Some salient features of our framework are: (i) it explicitly exploits the geometry of the homogeneous Riemannain manifolds and (ii) the nested lower-dimensional submanifolds need not be geodesic. With the proposed NG structure, we develop algorithms for the supervised and unsupervised dimensionality reduction problems respectively. The proposed algorithms are compared with PGA via simulation studies and real data experiments and are shown to achieve a higher ratio of expressed variance compared to PGA.
The code is available at https://github.com/cvgmi/NestedGrassmann

Keywords

Grassmann Manifolds · Dimensionality Reduction · Shape Analysis · Homogeneous Riemannian Manifolds

Bibtex @article{melba:2022:002:yang, title = "Nested Grassmannians for Dimensionality Reduction with Applications", author = "Yang, Chun-Hao and Vemuri, Baba C.", journal = "Machine Learning for Biomedical Imaging", volume = "1", issue = "IPMI 2021 special issue", year = "2022", issn = "2766-905X", url = "https://melba-journal.org/papers/2022:002.html" }
RISTY - JOUR AU - Yang, Chun-Hao AU - Vemuri, Baba C. PY - 2022 TI - Nested Grassmannians for Dimensionality Reduction with Applications T2 - Machine Learning for Biomedical Imaging VL - 1 IS - IPMI 2021 special issue SN - 2766-905X UR - https://www.melba-journal.org/papers/2022:002.html ER -

2022:002 cover