Research map / Learning theory and algorithms / Sample complexity
Total Variation Classes Beyond 1d: Minimax Rates, and the Limitations of Linear Smoothers
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace TV denoising with any linear smoother (e.g., Laplacian smoothing, eigenmaps, mean) on TV classes | discrete TV class T_d(C_n), d>=2 · canonical scaling C_n ~ n^{1-1/d} | worst-case MSE | Hurt |
| Use TV denoising instead of the mean estimator | discrete TV class, d=2 with C_n ~ sqrt(n) · canonical TV scaling | MSE rate | Helped |
| Use Laplacian smoothing or Laplacian eigenmaps on Sobolev classes | discrete Sobolev class S_d(C_n') · d=1,2,3 (LS) and all d (LE) | worst-case MSE | No clear effect |
| Use TV denoising on the smaller inscribed Sobolev ball for d>=3 | TV denoiser on discrete Sobolev class, d=3 · linear signal theta_i proportional to sum of coordinates | empirical MSE | No clear effect |
| Empirical comparison of Laplacian smoothing vs TV denoising on cameraman image | image denoising · cameraman image | MSE reduction: about 35% reduction in MSE | Helped |
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