Solved: 4. (Kernel Ridge Regression) We assume a model of form y = f(x), where

Question

4. (Kernel Ridge Regression) We assume a model of form y = f(x), where our task is to learn f, the regression function. Here, y is the response variable which we assume is an element of R, and the x are the Rd covariates or predictors. Pairs (x¹, y¹) € Rd x R are observed, and we seek f such that y = f(x¹) for i = 1,...,n. Given an RKHS H with kernel K, we can estimate f by solving an optimization problem over the RKHS, 1 f = arg min = [(y¹ − f(x¹))² + |||f|||· SEH 2 By the Representer Theorem, we know that f(-) = Σα;Κ(·,π΄). j=1 Let y = [y¹,,y"]¹₁a = [α₁,,an]¹ € R" and K € R"x" with Kij Then we obtain KB. a := arg min || - K3||2 + BER" Let K(x, 2) = (1 + Σ;_1 ª;²;)² be the polynomial kernel of degree 2. (a) What is the corresponding RKHS? (b) Write down the expression of f in this case. = K(x², x³).