  Supervised learning problems may become ill-posed when there is a lack of
information, resulting in unstable and non-unique solutions. However, instead
of solely relying on regularization, initializing an informative ill-posed
operator is akin to posing better questions to achieve more accurate answers.
The Fredholm integral equation of the first kind (FIFK) is a reliable ill-posed
operator that can integrate distributions and prior knowledge as input
information. By incorporating input distributions and prior knowledge, the FIFK
operator can address the limitations of using high-dimensional input
distributions by semi-supervised assumptions, leading to more precise
approximations of the integral operator. Additionally, the FIFK's incorporation
of probabilistic principles can further enhance the accuracy and effectiveness
of solutions. In cases of noisy operator equations and limited data, the FIFK's
flexibility in defining problems using prior information or cross-validation
with various kernel designs is especially advantageous. This capability allows
for detailed problem definitions and facilitates achieving high levels of
accuracy and stability in solutions. In our study, we examined the FIFK through
two different approaches. Firstly, we implemented a semi-supervised assumption
by using the same Fredholm operator kernel and data function kernel and
incorporating unlabeled information. Secondly, we used the MSDF method, which
involves selecting different kernels on both sides of the equation to define
when the mapping kernel is different from the data function kernel. To assess
the effectiveness of the FIFK and the proposed methods in solving ill-posed
problems, we conducted experiments on a real-world dataset. Our goal was to
compare the performance of these methods against the widely used least-squares
method and other comparable methods.
