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 miny p(x,y)来逼近紧集上的（可能不连续）多元函数f(x)。该多项式的构造可归结为单变量平方和（SOS）框架，从而形成高度结构化的凸半定规划问题。在若干非平凡情形（例如f为分段多项式函数）中，我们证明使用低次多项式p即可实现精确逼近。本方法具有三个显著特征：(i) 无网格特性，无需预知间断位置；(ii) 无模型特性，仅需通过样本（点评估）获取待逼近函数；(iii) 半定规划的规模与空间维度无关，且与样本数量呈线性关系。我们还分析了方法的样本复杂度，在概率框架下证明了泛化误差界，从而可与机器学习方法进行对比。