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Contrastive Functional Principal Component Analysis
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Proposes Contrastive Functional PCA (CFPCA), which finds low-dimensional functional components enriched in a foreground group relative to a background group by eigendecomposing the difference of foreground and background covariance operators.
Problem
Standard FPCA captures dominant shared variation and fails to isolate low-dimensional structures specific to or enriched in a foreground (case/treatment) functional dataset relative to a background (control) dataset.
Approach
Extends contrastive PCA (cPCA) and contrastive latent variable models to functional data: leading eigenfunctions of C_x − αC_y (foreground minus scaled background covariance operator) give foreground-specific components; discretized via an n×n matrix eigenproblem, with consistency rates derived under a CLVM-style model.
Key results
- In simulations, CFPCA's first and second component bias was about 0.04-0.14, versus 0.72-1.39 for FPCA on foreground alone and 1.01-1.35 for FPCA on the union.
- On gait cycle data, CFPCA reached silhouette score 0.51 and Davies-Bouldin 0.66, versus 0.06/4.13 for FPCA on braced subjects only.
- On stock data, CFPCA reached silhouette 0.33 and DB 1.09, versus -0.008/8.33 for FPCA on technology stocks; bias decreases at roughly n^(-1/2) rate in simulation.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CFPCA instead of FPCA on foreground data | Foreground-specific component estimation · Simulations 1–4 | bias: Simulation 1 PC1: 0.72 -> 0.08; PC2: 0.43 -> 0.08; Simulation 2 PC1: 1.32 -> 0.14; PC2: 1.13 -> 0.14; Simulation 3 PC1: | Helped |
| Use CFPCA with unbraced subjects as background instead of FPCA on braced subjects | Gait Cycle · Gait Cycle | Silhouette score: 0.06 -> 0.51 | Helped |
| Use CFPCA with unbraced subjects as background instead of FPCA on braced subjects | Gait Cycle · Gait Cycle | Davies-Bouldin index: 4.13 -> 0.66 | Helped |
| Use CFPCA with non-technology stocks as background instead of FPCA on technology stocks | Technology stock analysis · Stock Market | Silhouette score: −0.008 -> 0.33 | Helped |
| Use CFPCA with non-technology stocks as background instead of FPCA on technology stocks | Technology stock analysis · Stock Market | Davies-Bouldin index: 8.33 -> 1.09 | Helped |
Limitations
- Current implementation is limited to time series data and linear methods; no kernel/nonlinear version.
- Choice of contrast parameter alpha and number of components L is heuristic; no automatic tuning, and cross-validation only applies when subgroups exist.
- Dimension reduction alone gives no statistical significance; hypothesis testing is needed.
Experimental setup
Datasets: Berkeley growth study gait cycle data (Shorter et al. 2008; Helwig et al.) Yahoo Finance daily stock closing prices (yfinance)
Benchmarks: simulated functional data with known components Berkeley growth gait cycle stock market
Metrics: bias (L2 norm of eigenfunction difference) Silhouette score Davies-Bouldin index eigenvalue error
Baselines: FPCA on foreground FPCA on foreground plus background union
Builds on: contrastive principal component analysis contrastive latent variable model functional principal component analysis probabilistic contrastive principal component analysis
Closest papers by meaning
- Disentangling Interpretable Factors with Supervised Independent Subspace Principal Component Analysis (NeurIPS 2024 · similarity 0.84)
- Contrastive dimension reduction: when and how? (NeurIPS 2024 · similarity 0.83)
- A New Computationally Efficient Algorithm to solve Feature Selection for Functional Data Classification in High-dimensional Spaces (ICML 2024 · similarity 0.82)
- Separating common from salient patterns with Contrastive Representation Learning (ICLR 2024 · similarity 0.82)
- Unconstrained Stochastic CCA: Unifying Multiview and Self-Supervised Learning (ICLR 2024 · similarity 0.82)
- PCA++: How Uniformity Induces Robustness to Background Noise in Contrastive Learning (NeurIPS 2025 · similarity 0.82)
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