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Enhancing Sufficient Dimension Reduction via Hellinger Correlation
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Proposes a single-index sufficient dimension reduction method that maximizes the estimated Hellinger correlation between a projected predictor and the response, refining initial estimates from SIR, SAVE, DR, MAVE (and DCOV, HSIC, BCOV) with simulated annealing and downhill simplex.
Problem
Estimating the central subspace in single-index sufficient dimension reduction without relying on inverse-regression distributional assumptions or the independence of projected predictors assumed by prior dependence-measure SDR methods.
Approach
Use the Hellinger correlation (a copula-based f-divergence dependence measure that is invariant to monotone transformations and characterizes independence) as the objective for SDR direction search, proving population identifiability and consistency without requiring eta0'X independent of eta1'X.
Key results
- On normal predictors with n=100, SIR-HC reduces the subspace error in Model I from 0.9546 (SIR) to 0.037, and MAVE-HC gives 0.0379 vs 0.0654 for MAVE.
- On the UCI real estate data, SIR-HC and MAVE-HC both achieve test MSE 0.243, versus 0.313 for SIR and 0.485 for MAVE.
- Under non-normal predictors with n=400 in Model II, MAVE-HC reaches 0.004 error versus 0.0148 for MAVE.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply Hellinger correlation optimization initialized with SIR | SIR, Model I with normal predictors · Model I, normal predictors, n=100 | Δ subspace error: 0.9546 -> 0.037 | Helped |
| Apply Hellinger correlation optimization initialized with MAVE | MAVE, Model I with normal predictors · Model I, normal predictors, n=100 | Δ subspace error: 0.0654 -> 0.0379 | Helped |
| Apply Hellinger correlation optimization initialized with MAVE | MAVE, Model II with normal predictors · Model II, normal predictors, n=100 | Δ subspace error: 0.0699 -> 0.0602 | Helped |
| Apply Hellinger correlation optimization initialized with DR | DR, Model III with normal predictors · Model III, normal predictors, n=100 | Δ subspace error: 0.4165 -> 0.0172 | Helped |
| Apply Hellinger correlation optimization initialized with MAVE | MAVE, Model III with normal predictors · Model III, normal predictors, n=100 | Δ subspace error: 0.0455 -> 0.018 | Helped |
| Apply Hellinger correlation optimization initialized with SIR | SIR, Model I with non-normal predictors · Model I, non-normal predictors, n=100 | Δ subspace error: 0.6971 -> 0.2674 | Helped |
| Apply Hellinger correlation optimization initialized with MAVE | MAVE, Model II with non-normal predictors · Model II, non-normal predictors, n=100 | Δ subspace error: 0.0686 -> 0.0377 | Helped |
| Apply Hellinger correlation optimization initialized with DR | DR, Model III with non-normal predictors · Model III, non-normal predictors, n=100 | Δ subspace error: 0.0286 -> 0.0379 | Hurt |
| Apply Hellinger correlation optimization initialized with MAVE | MAVE, non-sparse Model I · Model I, non-sparse direction, n=100 | Δ subspace error: 0.0534 -> 0.0328 | Helped |
| Apply Hellinger correlation optimization initialized with MAVE | MAVE, non-sparse Model II · Model II, non-sparse direction, n=100 | Δ subspace error: 0.0725 -> 0.0741 | Hurt |
| Apply Hellinger correlation optimization initialized with DR | DR, non-sparse Model III · Model III, non-sparse direction, n=100 | Δ subspace error: 0.194 -> 0.1012 | Mixed |
| Apply Hellinger correlation optimization initialized with SIR | SIR, real estate valuation · Real estate valuation test set | MSE: 0.313 -> 0.243 | Helped |
Surprising findings
- SIR fails on the U-shaped Model I, yet SIR-HC recovers the subspace, showing refinement can overcome a poor initializer.
- In the non-sparse Model II experiment, SAVE-HC, DR-HC and MAVE-HC at n=400 do not improve over their initializers, and MAVE-HC is worse than MAVE in Model II (0.0594 vs 0.0378).
Limitations
- Restricted to single-index models (structural dimension d=1) with continuous X and Y; multi-index extension left to future work.
- Computational cost O(n^2 p k) per optimization with heuristic global search (simulated annealing) and no convergence rate analysis.
- Real data evaluation uses a single random train/test split and local polynomial regression as the downstream model.
Experimental setup
Datasets: UCI Real Estate Valuation dataset (414 observations)
Benchmarks: simulated single-index models I-III UCI real estate valuation test MSE
Metrics: projection matrix spectral norm distance between subspaces test MSE
Baselines: SIR SAVE DR MAVE DCOV HSIC BCOV
Builds on: Hellinger correlation SIR SAVE DR MAVE
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