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.

翻译：监督学习问题在信息不足时可能成为不适定问题，导致解的不稳定和非唯一性。然而，与其单纯依赖正则化，初始化一个信息丰富的适定算子更像是提出更好的问题以获得更精确的答案。第一类弗雷德霍姆积分方程（FIFK）是一种可靠的适定算子，能够整合分布和先验知识作为输入信息。通过引入输入分布和先验知识，FIFK算子可以借助半监督假设解决高维输入分布的限制，从而更精确地逼近积分算子。此外，FIFK中概率原理的融合能进一步提升解的准确性和有效性。在算子方程含噪声且数据有限的情况下，FIFK通过先验信息或不同核设计的交叉验证来定义问题的灵活性尤为突出。此能力允许对问题进行详细定义，并有助于在解中实现高水平的精度和稳定性。在本研究中，我们通过两种不同的方法审视了FIFK。首先，我们采用半监督假设，使用相同的弗雷德霍姆算子核和数据函数核，并纳入未标注信息。其次，我们使用MSDF方法，即对方程两侧选择不同的核，以定义映射核与数据函数核不同的情况。为评估FIFK及所提方法在解决不适定问题中的有效性，我们在真实数据集上进行了实验。我们的目标是比较这些方法与广泛使用的最小二乘法及其他可比方法的性能。