Learning Positive Functions in a Hilbert Space
نویسنده
چکیده
Semidefinite Programming Formulation By representer theorem: f(x) = ∑n l=1 αlK (Xi, x) under the condition that the function has an SoS representation, i.e., f(x) = φ(x)>Qφ(x) for some Q 0. Define a d × n matrix Φ = [φ(X1) · · ·φ(Xn)] and an n × n diagonal matrix A = diag(α) = diag(α1, . . . , αn). We have Q = ΦAΦ>. Q is d × d, but has rank n, which can be much smaller than d. The constraint on PSDness of Q can be written as eig(Q) = eig(ΦAΦ) = eig(Φ √ A √ AΦ) = eig( √ AΦΦ } {{ }
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