[Paper Review] Generalized Clustering and Multi-Manifold Learning with Geometric Structure Preservation

Lirong Wu, Zicheng Liu+4 more|arXiv (Cornell University)|Sep 21, 2020

Gait Recognition and Analysis4 citations

TL;DR

This paper proposes GCML, a unified framework for generalized clustering and multi-manifold learning that preserves both local intra-manifold and global inter-manifold geometric structures. By jointly optimizing a clustering loss with isometric and ranking losses, GCML achieves state-of-the-art clustering performance while maintaining high-fidelity latent space geometry across diverse data types including speech, text, and biological data.

ABSTRACT

Though manifold-based clustering has become a popular research topic, we observe that one important factor has been omitted by these works, namely that the defined clustering loss may corrupt the local and global structure of the latent space. In this paper, we propose a novel Generalized Clustering and Multi-manifold Learning (GCML) framework with geometric structure preservation for generalized data, i.e., not limited to 2-D image data and has a wide range of applications in speech, text, and biology domains. In the proposed framework, manifold clustering is done in the latent space guided by a clustering loss. To overcome the problem that the clustering-oriented loss may deteriorate the geometric structure of the latent space, an isometric loss is proposed for preserving intra-manifold structure locally and a ranking loss for inter-manifold structure globally. Extensive experimental results have shown that GCML exhibits superior performance to counterparts in terms of qualitative visualizations and quantitative metrics, which demonstrates the effectiveness of preserving geometric structure.

Motivation & Objective

• To address the gap in existing manifold-based clustering methods that fail to preserve geometric structure in the latent space despite using clustering losses.
• To develop a unified framework that simultaneously performs generalized clustering and multi-manifold learning with explicit preservation of local and global geometric structures.
• To resolve the contradiction between clustering objectives and local structure preservation, which can degrade the quality of learned representations.
• To demonstrate the effectiveness of the framework on generalized data beyond 2D images, including speech, text, and biological data.
• To show that geometric structure preservation enhances downstream task performance, validating the utility of the learned embeddings.

Proposed method

• Introduces learnable clustering centers as parameters to guide the clustering process in the latent space.
• Employs a clustering loss to promote inter-manifold discriminability and cluster separation.
• Proposes an isometric loss to preserve local geometric structure within each manifold by minimizing distance distortion between neighboring points.
• Introduces a ranking loss to preserve global inter-manifold structure by maintaining the relative distance ranking between clusters.
• Uses an autoencoder-based architecture with shared encoder and decoder for end-to-end training of the latent space.
• Applies alternating training and weight continuation strategies to stabilize optimization and improve convergence.

Experimental results

Research questions

• RQ1 Can clustering losses corrupt the local and global geometric structure of the latent space in manifold learning?
• RQ2 How can local geometric structure within manifolds and global discriminability between manifolds be preserved during clustering-driven representation learning?
• RQ3 To what extent does preserving geometric structure improve clustering performance and downstream task accuracy?
• RQ4 Is the proposed framework effective across generalized data types beyond 2D images, such as speech, text, and biological data?
• RQ5 How do the individual components (clustering loss, isometric loss, ranking loss, training strategies) contribute to the overall performance?

Key findings

• GCML achieves a clustering accuracy (CRA) of 1.0 on all seven datasets, indicating perfect preservation of inter-cluster distance ranking in the latent space.
• Visualizations and statistical analysis confirm that only GCML fully preserves the global rank ordering between clusters, while other methods fail to maintain inter-manifold structure.
• The isometric loss in GCML results in the narrowest and highest peak distribution of distance differences between input and latent space, indicating superior local isometry compared to baselines.
• On downstream tasks using frozen embeddings, GCML outperforms all baselines across all classifiers (LR, MLP, SVM, RFC) on MNIST-full and Fashion-MNIST.
• Ablation studies show that the clustering loss is essential for clustering performance, while the structure-oriented losses (isometric and ranking) significantly improve multi-manifold learning and overall robustness.
• The alternating training and weight continuation strategies further enhance performance, particularly on metrics like RRE, Trust, and CRA.