This presentation covers two recent contributions to variable selection in functional linear regression. For the scalar-on-function setting, we propose SOFIA (Scalar-On-Function Integrated Adaptive Lasso). We assume the functional covariates are in a Hilbert space while the coefficient functions belong to a specific subspace of it, such as a reproducing kernel Hilbert space. In this way, we impose a controlled level of smoothness or periodicity on the coefficients. The method satisfies a functional oracle property even when the number of predictors exceeds the sample size.
For the function-on-scalar setting, we propose MIP-FoSR, a mixed-integer programming framework that performs simultaneous variable selection and outlier detection. It extends the "mean-shift outlier model" to the functional setting, and uses grouped binary indicators on basis-expansion coefficients to impose explicit bounds on the number of selected predictors and detected outliers. We establish an equivalence with a functional sparse trimming problem, derive a finite-sample breakdown point, and prove a functional robust strong oracle property.