  We consider a statistical inverse learning problem, where we observe the
image of a function $f$ through a linear operator $A$ at i.i.d. random design
points $X_i$, superposed with an additive noise. The distribution of the design
points is unknown and can be very general. We analyze simultaneously the direct
(estimation of $Af$) and the inverse (estimation of $f$) learning problems. In
this general framework, we obtain strong and weak minimax optimal rates of
convergence (as the number of observations $n$ grows large) for a large class
of spectral regularization methods over regularity classes defined through
appropriate source conditions. This improves on or completes previous results
obtained in related settings. The optimality of the obtained rates is shown not
only in the exponent in $n$ but also in the explicit dependency of the constant
factor in the variance of the noise and the radius of the source condition set.

Machine Learning