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Infinite-σ Limits For Tikhonov Regularization

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Figure

Figure 1: (property 1) of Definition 6 says that paths of points within epi fs cannot end up belowepi f, while (property 2) says that at least one such path hits every point of epi f.
Figure 2: The function above, fs(x) = 2s x(x−s)+1, has different pointwise and epi-limits, havingvalues 1 and 0, respectively, at x = 0 and ∞ for all other x.
Figure 3: f(x) = .5(1−x)+150x(x−.25)(x−.3)(x−.75)(x−.95), a random data set drawn fromf(x) with added Gaussian noise, and data-based polynomial approximations to f.
Figure 4: As s → ∞, σ2 = s2 and λ = s−(2k+1), the solution to Gaussian RLS approaches the kthorder polynomial solution.

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