On some stable linear functional regression estimators based on random projections

In this work, we develop two stable estimators for solving linear functional regression problems. It is well known that such a problem is an ill-posed stochastic inverse problem. Hence, a special interest has to be devoted to the stability issue in the design of an estimator for solving such a problem. Our proposed estimators are based on combining a stable least-squares technique and a random projection of the slope function \(\beta _0(\cdot )\in L^2(J),\) where J is a compact interval. Moreover, these estimators have the advantage of having a fairly good convergence rate with reasonable computational load, since the involved random projections are generally performed over a fairly small dimensional subspace of \(L^2(J).\) More precisely, the first estimator is given as a least-squares solution of a regularized minimization problem over a finite dimensional subspace of \(L^2(J).\) In particular, we give an upper bound for the empirical risk error as well as the convergence rate of this estimator. The second proposed stable LFR estimator is based on combining the least-squares technique with a dyadic decomposition of the i.i.d. samples of the stochastic process, associated with the LFR model. In particular, we provide an \(L^2\)-risk error of this second LFR estimator. Finally, we provide some numerical simulations on synthetic as well as on real data that illustrate the results of this work. These results indicate that our proposed estimators are competitive with some existing and popular LFR estimators.