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Fréchet Geodesic Boosting
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Proposes Fréchet geodesic boosting (FGBoost), a gradient-boosting method that fits regression trees to geodesics between current predictions and observations, for outputs in geodesic (Hadamard-type) metric spaces such as distributions, networks, and compositional data.
Problem
Gradient boosting relies on addition, subtraction and scalar multiplication, which non-Euclidean outputs (distributions, networks, SPD matrices, compositions) lack, so standard boosting cannot model regression onto such outputs.
Approach
Replace residuals with geodesics from the current prediction to the observation, define geodesic addition via a geodesic transport map, and fit trees with Fréchet-mean leaf geodesics under a product metric on geodesic pairs; theory gives strong convexity and uniform convergence in Hadamard spaces.
Key results
- On simulated distributional outputs at n=100, FGBoost achieves AMSPE 0.034 vs 0.041 for IFR and 0.053 for GFR; at n=2000 it reaches 0.015 vs 0.026 for FRF.
- On human mortality data FGBoost achieves MSPE 20.35 vs 22.02 for FRF; on taxi network data 8.30 vs 9.50 for FRF.
- On emotional well-being compositional data FGBoost achieves MSPE 0.2074 vs 0.4112 for SDR and 0.4163 for GFR (over 50% reduction).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use FGBoost instead of the strongest listed baseline (IFR) | Distribution simulation · Distribution simulation, n=100 | AMSPE: 0.041 -> 0.034 | Helped |
| use FGBoost instead of the strongest listed baseline (FRF) | Distribution simulation · Distribution simulation, n=2000 | AMSPE: 0.026 -> 0.015 | Helped |
| use FGBoost instead of the strongest listed baseline (FRF) | Network simulation · Network simulation, n=100 | AMSPE: 13.820 -> 13.644 | Helped |
| use FGBoost instead of the strongest listed baseline (FRF) | Network simulation · Network simulation, n=2000 | AMSPE: 8.308 -> 4.996 | Helped |
| use FGBoost instead of the strongest listed baseline (FRF) | FGBoost · Human Mortality | MSPE: 22.02 -> 20.35 | Helped |
| use FGBoost instead of the strongest listed baseline (FRF) | FGBoost · Taxi Network | MSPE: 9.50 -> 8.30 | Helped |
| use FGBoost instead of the strongest listed baseline (FRF) | Compositional simulation · Compositional simulation, n=100 | AMSPE: 0.0088 -> 0.0104 | Hurt |
| use FGBoost instead of the strongest listed baseline (FRF) | Compositional simulation · Compositional simulation, n=200 | AMSPE: 0.0076 -> 0.0074 | Helped |
| use FGBoost instead of the strongest listed baseline (GFR) | Compositional simulation · Compositional simulation, n=1000 | AMSPE: 0.0083 -> 0.0038 | Helped |
| use FGBoost instead of the strongest listed baseline (GFR) | FGBoost · Emotional well-being | MSPE: 0.4163 (0.0007) -> 0.2074 (0.0005) | Helped |
Surprising findings
- On compositional simulations at n=100, GFR and FRF slightly outperform FGBoost.
- Vectorized XGBoost and SketchBoost baselines underperform FGBoost on graph Laplacian outputs, indicating geometry-aware modeling matters.
Limitations
- Theory restricted to Hadamard (global NPC) spaces; requires boundedness and a geodesic transport map assumption.
- Predictors are restricted to Euclidean; extension to non-Euclidean predictors left to future work.
- Distributional simulations estimate outputs from 100 samples per distribution, introducing bias.
Experimental setup
Datasets: simulated distributional, network, and compositional outputs UN World Population Prospects 2024 life tables NYC yellow taxi trip records New Jersey unemployed worker survey
Benchmarks: simulated distribution outputs simulated network outputs simulated compositional outputs human mortality data NYC taxi network data emotional well-being data
Metrics: mean squared prediction error (MSPE) average MSPE over Monte Carlo runs
Baselines: global Frechet regression (GFR) sufficient dimension reduction (SDR) single index Frechet regression (IFR) Frechet random forest (FRF) random forest weighted local linear Frechet regression (RFWLLFR) XGBoost SketchBoost
Builds on: gradient boosting XGBoost Frechet regression Tree SHAP
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