Research map / Learning theory and algorithms / Kernels and regression theory
Relating Leverage Scores and Density using Regularized Christoffel Functions
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a translation-invariant kernel with the stated localization condition and regularization approaching zero | Regularized Christoffel function · Continuity points z with p(z) > 0 | Christoffel function: C_\lambda(z) \underset{\lambda\to 0, \,\lambda > 0}{\sim} p(z) D\left( \frac{\lambda}{p(z)} \right) | Mixed |
| Use a Sobolev-related kernel with spectral density \hat{q}=1/P^\gamma | Regularized Christoffel function · Continuity points z with p(z) > 0 | Christoffel function: C_\lambda(z) \sim \lambda^{\frac{d}{2s\gamma}} p(z)^{1 - \frac{d}{2s\gamma}} \textstyle\frac{1}{q_0} | Helped |
| Increase the population density at a point | Regularized Christoffel function · Any z, fixed λ and kernel | Christoffel function | Helped |
| Increase the kernel by adding a positive definite kernel | Regularized Christoffel function · Any z, fixed λ and density | Christoffel function | Hurt |
| Evaluate outside the support of the population density | Regularized Christoffel function · Points with a neighborhood of zero population mass | Christoffel function: C_\lambda(z) = O(\sqrt{\lambda}D(\sqrt{\lambda})); C_\lambda(z) = O(\lambda) | Helped |
| Use a Gaussian kernel rather than the analyzed localized-kernel class | Regularized Christoffel function · Piecewise constant density; discontinuities | localization of density dependence: D(\lambda) is of the order of -1 / \log(\lambda) | Hurt |
| Use the plug-in estimate with λ² n^{1/2} sufficiently large | Plug-in regularized Christoffel function estimate · Monte Carlo approximation | estimation quality: \lambda^2 n^{1/2} | Helped |
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