PRIM-cipal components analysis
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910138270482432 |
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| author | Liu, Tianhao Díaz-Pachón, Daniel Andrés Rao, J. Sunil |
| author_facet | Liu, Tianhao Díaz-Pachón, Daniel Andrés Rao, J. Sunil |
| contents | Supervised No Free Lunch Theorems (NFLTs) are well studied, yet unsupervised NFLTs remain underexplored. For elliptical distributions, we prove that there exist two equally optimal, scientifically meaningful bump-hunting strategies that are exact opposites, with no universal winner. Specifically, peeling $k$ orthogonal dimensions from $\mathbb{R}^d$ ($d \ge k$), retaining an inter-quantile region of probability $1-α$ per peeled dimension, maximizes total variance and Frobenius norm when the $k$ smallest principal components (called pettiest components) are selected, and minimizes them when the selected dimensions are the $k$ leading principal components. These optima inspire PRIM-based bump-hunting algorithms either by minimizing variance or by minimizing volume, thereby motivating an NFLT. We test our results on the Fashion-MNIST database, showing that peeling the largest principal components captures multiplicity, while peeling the smallest principal components isolates popular styles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15538 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | PRIM-cipal components analysis Liu, Tianhao Díaz-Pachón, Daniel Andrés Rao, J. Sunil Machine Learning 62H25 (primary), 62H30 (secondary), 90C27 (secondary), 68T09 (secondary) G.3; I.5 Supervised No Free Lunch Theorems (NFLTs) are well studied, yet unsupervised NFLTs remain underexplored. For elliptical distributions, we prove that there exist two equally optimal, scientifically meaningful bump-hunting strategies that are exact opposites, with no universal winner. Specifically, peeling $k$ orthogonal dimensions from $\mathbb{R}^d$ ($d \ge k$), retaining an inter-quantile region of probability $1-α$ per peeled dimension, maximizes total variance and Frobenius norm when the $k$ smallest principal components (called pettiest components) are selected, and minimizes them when the selected dimensions are the $k$ leading principal components. These optima inspire PRIM-based bump-hunting algorithms either by minimizing variance or by minimizing volume, thereby motivating an NFLT. We test our results on the Fashion-MNIST database, showing that peeling the largest principal components captures multiplicity, while peeling the smallest principal components isolates popular styles. |
| title | PRIM-cipal components analysis |
| topic | Machine Learning 62H25 (primary), 62H30 (secondary), 90C27 (secondary), 68T09 (secondary) G.3; I.5 |
| url | https://arxiv.org/abs/2604.15538 |