Handling an infinite number of inequality constraints in infinite-dimensional spaces occurs in many fields, from global optimization to optimal transport. These problems have been tackled individually in several previous articles through kernel Sum-Of-Squares (kSoS) approximations. We propose here a unified theorem to prove convergence guarantees for these schemes. Pointwise inequalities are turned into equalities within a class of nonnegative kSoS functions. Assuming further that the functions appearing in the problem are smooth, focusing on pointwise equality constraints enables the use of scattering inequalities to mitigate the curse of dimensionality in sampling the constraints. Our approach is illustrated in learning vector fields with side information, here the invariance of a set.

翻译：处理无穷维空间中的无穷多个不等式约束在众多领域中均有出现，从全局优化到最优传输。此前多篇文献通过核平方和（kSoS）逼近分别解决了这些问题。本文提出一个统一定理来证明这些方案的收敛性保证。逐点不等式被转化为非负kSoS函数类中的等式约束。进一步假设问题中出现的函数是光滑的，则聚焦于逐点等式约束可利用散射不等式来缓解约束采样中的维数灾难。我们通过具有辅助信息（即集合不变性）的向量场学习来阐述该方法。