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Principle Component Trees and Their Persistent Homology
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Introduces Principal Component Trees (PCTs), a graph generalization of PCA and subspace clustering built by recursive hypothesis tests on pairwise angle distributions, and analyzes them with persistent-homology height and width curves.
Problem
Standard PCA learns a single subspace and subspace clustering learns disjoint unions of subspaces; neither captures mixtures of components that describe intersecting subspace structure in high-dimensional data, and there is no shape-based measure to compare subspace structure across datasets.
Approach
Each tree node is a principal component; children are added only when a Cramér-von Mises test on pairwise absolute-cosine angles detects disjoint subspace clusters, and an intersection test allows nodes to share ancestors. Height and width curves (from persistent homology-style filtrations) summarize tree shape.
Key results
- Fashion MNIST is more subspace clustered than MNIST Digits, with higher average height and width curves and corresponding areas under the curves.
- Classifier latent space is more subspace clustered than original data, while VAE latent space is less clustered, showing WT is more sensitive than HT in that comparison.
- Italian word embeddings are least subspace clustered and Russian most clustered among the three languages studied.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Principal Component Trees to analyze subspace clusteredness instead of PCA-style representations | MNIST Fashion vs MNIST Digits · MNIST | subspace clusteredness | Mixed |
| Use a classifier latent space rather than the original data or autoencoder latent spaces | MNIST neural network latent spaces · MNIST Digits | subspace clusteredness | Helped |
| Use a VAE latent space rather than the original data | MNIST variational autoencoder · MNIST Digits | subspace clusteredness | Hurt |
| Use UMAP embeddings rather than neural-network latent spaces | UMAP · MNIST Digits | subspace clusteredness | Helped |
Surprising findings
- Word embeddings of the same TED talks in different languages have markedly different subspace clusteredness, contrary to the expectation of structural similarity.
Limitations
- Tree construction relies on angle-distribution tests whose null distribution is estimated by Monte Carlo for N < 100D and requires whitening.
- Only binary splits are focused on in the paper, though the structure supports arbitrary children.
- Experiments are exploratory dataset-level comparisons rather than downstream task performance.
Experimental setup
Datasets: MNIST Fashion MNIST TED 2020 multilingual word embeddings
Benchmarks: MNIST Fashion MNIST multilingual word embeddings (TED 2020)
Metrics: area under height curve area under width curve Wilcoxon rank-sum p-value Cramér-von Mises p-value
Baselines: PCA UMAP
Builds on: PCA sparse subspace clustering graph filtration persistent homology Cramér-von Mises test
Closest papers by meaning
- The tree autoencoder model, with application to hierarchical data visualization (NeurIPS 2024 · similarity 0.87)
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- Heterogeneous Sufficient Dimension Reduction and Subspace Clustering (ICML 2025 · similarity 0.84)
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