用非参数方法在低维空间保持层次数据的连续性。
Hyperboloid GPLVM for Discovering Continuous Hierarchies via Nonparametric Estimation
- 基于高斯过程构建双曲面隐变量模型，实现无参连续嵌入。
- 三种变体分别使用原始点、稀疏点和贝叶斯估计，提升灵活性。
- 适用于需要保持层次结构连续性的高维数据可视化与分析。
降维技术能有效表示复杂高维数据。近年来的降维方法关注双曲几何以忠实还原层次化数据的低维表示，但现有方法多依赖邻域嵌入，常破坏层次结构的连续性。本文提出双曲面高斯过程隐变量模型（hGP-LVM），通过非参数估计隐式保持层次数据的连续性。采用高斯过程生成建模，实现有效的层次嵌入并缓解欠定的超参数调优问题。本文提出三种变体：基于原始点、稀疏点和贝叶斯估计。通过引入黎曼优化与高斯过程隐变量模型的主动近似方案，建立其学习算法。针对贝叶斯推断，进一步引入重参数化技巧实现隐变量的贝叶斯学习。最后，将hGP-LVM应用于多个数据集，验证其在低维空间中表示高维层次结构的能力。
原文摘要 · Abstract (English)
Dimensionality reduction (DR) offers a useful representation of complex high-dimensional data. Recent DR methods focus on hyperbolic geometry to derive a faithful low-dimensional representation of hierarchical data. However, existing methods are based on neighbor embedding, frequently ruining the continual relation of the hierarchies. This paper presents hyperboloid Gaussian process (GP) latent variable models (hGP-LVMs) to embed high-dimensional hierarchical data with implicit continuity via nonparametric estimation. We adopt generative modeling using the GP, which brings effective hierarchical embedding and executes ill-posed hyperparameter tuning. This paper presents three variants that employ original point, sparse point, and Bayesian estimations. We establish their learning algorithms by incorporating the Riemannian optimization and active approximation scheme of GP-LVM. For Bayesian inference, we further introduce the reparameterization trick to realize Bayesian latent variable learning. In the last part of this paper, we apply hGP-LVMs to several datasets and show their ability to represent high-dimensional hierarchies in low-dimensional spaces.
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