In the book Elements of Statistical Learning, section 2.8.3 describes Basis Functions, citing an example of a radial basis function as $f_{\theta}(x) = \sum_{m=1}^M \beta_M \sigma(\alpha_m'x + b_m)$, with $\sigma(x)$ as the activation function. This makes sense to me, but I am confused as to how the model is actually being fit. The given form of $f_{\theta}(x)$ that can be used to generate a prediction from $x$, but how are the parameters estimated?

Or, is the parameter fitting a separate question entirely? My intuition is that the RSS, $\sum_{i=1}^N (y_i - f(x_i))^2$ can still be minimized for parameter fitting. Would that be correct?

Sorry for the naive question. I am just trying to understand whether the class of the restricted estimator is linked in any way with the objective function we are trying to minimize?


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