Research map / Learning theory and algorithms / Kernels and regression theory
Sample complexity and effective dimension for regression on manifolds
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a bandlimited kernel derived from the manifold Laplacian | Regression on a compact Riemannian manifold · Manifold bandlimited regression | normalized L2 error: 4 (√p + 2 √(log 4/δ))/√n σ | Helped |
| Use the manifold heat kernel for exponentially smooth functions | Regression on a compact Riemannian manifold · Manifold heat-kernel regression | normalized L2 error: (√(2 α) + 6 √(e^{-Ω²t/2}/vol(M))) ||f*||Hheat + (9/2) (√p + 2 √(log 4/δ))/√n σ | Helped |
| Exploit manifold spectral structure rather than ambient dimension | Regression on manifold-supported data · Manifold regression | effective dimension: p(Ω) = 3 √m V_m/(2π)^m vol(M) Ω^m | Helped |
| Choose p so that (p/n)σ² ≈ t_{p+1}r² | General RKHS regression · Noisy regression over an RKHS norm ball | minimax error rate: √(p/n) σ | Helped |
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