Research map / Learning theory and algorithms / Matrix and tensor methods
Fun with Flags: Robust Principal Directions via Flag Manifolds
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Unifies PCA variants (robust, dual, geodesic/tangent) under flag manifolds and solves the resulting flag-formulated problems via iteratively reweighted optimization on the Stiefel manifold.
Problem
Robust and dual principal component analysis, including PCA on Riemannian manifolds (PGA / tangent PCA), lack a common formulation and efficient optimization, and existing methods are fragmented and sensitive to outliers.
Approach
Recasts variance-maximizing and reconstruction-error PCA variants (robust L_p, dual PCA, tangent PCA) as optimization over flag manifolds of nested subspaces, and shows these can be solved as iteratively reweighted weighted-PCA problems on the Stiefel manifold; flag type interpolates between L1 and L2 behaviors.
Key results
- On 5-D uniform data, fRPCA(1,...,k) reached objective 54.66 in 0.19s versus 54.69 for L1-RPCA, with the flag method being faster.
- On 2D Hands with outliers, fTDPCP variants achieved the best outlier-detection AUC, with near-perfect outlier detection at the highest contamination levels.
- On UCMercedLandUse, novel fDPCP(1,40) gave a slight but consistent AUC improvement over L2-DPCP, RPCA and PCA baselines.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use flagified fRPCA(1,...,k) instead of L1-RPCA | fRPCA(1,...,k) · Synthetic Euclidean data (R5, k=2) | time: 70.66 -> 0.19 | Helped |
| Use flagified fRPCA(1,...,k) instead of L1-RPCA | fRPCA(1,...,k) · Synthetic Euclidean data (R5, k=2) | objective: 54.69 -> 54.66 | No clear effect |
| Use flagified fWPCA(k) instead of L2-WPCA | fWPCA(k) · Synthetic Euclidean data (R5, k=2) | time: 0.24 -> 0.45 | Hurt |
| Use flagified fWPCA(k) instead of L2-WPCA | fWPCA(k) · Synthetic Euclidean data (R5, k=2) | objective: 42.89 -> 42.92 | No clear effect |
| Use flagified fDPCP(k) instead of L2-DPCP | fDPCP(k) · Synthetic Euclidean data (R5, k=2) | objective: 34.83 -> 34.66 | Helped |
| Use flagified fDPCP(k) instead of L2-DPCP | fDPCP(k) · Synthetic Euclidean data (R5, k=2) | time: 0.26 -> 0.38 | Hurt |
| Use robust fDPCP(1,40) for outlier detection | fDPCP(1,40) · UCMercedLandUse | AUC | Helped |
| Use robust flag methods for outlier detection | fWT PCA and fRT PCA · Cropped YaleFaceDB-B | AUC | Helped |
| Use the dual flag method for outlier detection | fDPCP · Cropped YaleFaceDB-B | AUC | Helped |
| Use fT DPCP(1,...,k) for outlier detection | Tangent flag methods · Gr(2,4) | AUC | Helped |
| Use the dual tangent flag method for outlier detection | fT DPCP · 2D Hands | AUC | Helped |
Surprising findings
- The novel fRPCA(1,...,k) was much faster than L1-RPCA while reaching essentially the same objective value.
- Euclidean PCA variants produce lower AUC than their tangent counterparts on Gr(2,4) synthetic outlier data.
Limitations
- Cannot handle non-linear flags, so nested spheres and hyperbolic spaces are not covered.
- Barycentric subspace analysis is not included.
- Convergence is proven only for fDPCP under an assumption; other variants are left to future work.
Experimental setup
Datasets: UCMercedLandUse Cropped YaleFaceDB-B Caltech101 2D Hands synthetic data on Gr(2,4) and 4-sphere
Benchmarks: UCMercedLandUse outlier detection Cropped YaleFaceDB-B outlier detection synthetic Gr(2,4) outlier detection 2D Hands Kendall pre-shape outlier detection
Metrics: AUC objective value run time ROC curves reconstruction error
Baselines: L1-RPCA bit-flipping L2-WPCA alternating scheme DPCP-IRLS (L2-DPCP) Stiefel CGD Flag RTR tangent PCA
Builds on: PCA L1-RPCA DPCP principal geodesic analysis flagified PCA (fPCA)
Closest papers by meaning
- A Flag Decomposition for Hierarchical Datasets (CVPR 2025 · similarity 0.90)
- Manifold denoising by Nonlinear Robust Principal Component Analysis (NeurIPS 2019 · similarity 0.89)
- Intrinsic Grassmann Averages for Online Linear and Robust Subspace Learning (CVPR 2017 · similarity 0.88)
- The Flag Median and FlagIRLS (CVPR 2022 · similarity 0.88)
- A Geometric Analysis of PCA (NeurIPS 2025 · similarity 0.85)
- StelLA: Subspace Learning in Low-rank Adaptation using Stiefel Manifold (NeurIPS 2025 · similarity 0.84)
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