We propose to approximate a (possibly discontinuous) multivariate function f (x) on a compact set by the partial minimizer arg miny p(x, y) of an appropriate polynomial p whose construction can be cast in a univariate sum of squares (SOS) framework, resulting in a highly structured convex semidefinite program. In a number of non-trivial cases (e.g. when f is a piecewise polynomial) we prove that the approximation is exact with a low-degree polynomial p. Our approach has three distinguishing features: (i) It is mesh-free and does not require the knowledge of the discontinuity locations. (ii) It is model-free in the sense that we only assume that the function to be approximated is available through samples (point evaluations). (iii) The size of the semidefinite program is independent of the ambient dimension and depends linearly on the number of samples. We also analyze the sample complexity of the approach, proving a generalization error bound in a probabilistic setting. This allows for a comparison with machine learning approaches.

翻译：我们提出通过适当多项式p的局部极小化器arg min_y p(x, y)来逼近紧集上（可能不连续）的多元函数f(x)，该多项式的构造可转化为单变量平方和（SOS）框架，从而形成一个高度结构化的凸半定规划问题。在若干非平凡情形下（例如当f为分段多项式时），我们证明该逼近可通过低次多项式p实现精确重建。本方法具有三个显著特征：（i）无需网格剖分且不要求已知间断点位置；（ii）无模型依赖，仅需假设待逼近函数可通过样本（点值评估）获取；（iii）半定规划的规模与空间维数无关，且随样本数量线性增长。我们还分析了该方法的样本复杂度，在概率框架下证明了泛化误差界。这为与机器学习方法的比较提供了依据。