High-dimensional real-world systems can often be well characterized by a small number of simultaneous low-complexity interactions. The analysis of variance (ANOVA) decomposition and the anchored decomposition are typical techniques to find sparse additive decompositions of functions. In this paper, we are interested in a setting, where these decompositions are not directly spare, but become so after an appropriate basis transform. Noting that the sparsity of those additive function decompositions is equivalent to the fact that most of its mixed partial derivatives vanish, we can exploit a connection to the underlying function graphs to determine an orthogonal transform that realizes the appropriate basis change. This is done in three steps: we apply singular value decomposition to minimize the number of vertices of the function graph, and joint block diagonalization techniques of families of matrices followed by sparse minimization based on relaxations of the zero ''norm'' for minimizing the number of edges. For the latter one, we propose and analyze minimization techniques over the manifold of special orthogonal matrices. Various numerical examples illustrate the reliability of our approach for functions having, after a basis transform, a sparse additive decomposition into summands with at most two variables.

翻译：高维现实世界系统通常可以通过少量同时发生的低复杂度相互作用来很好地表征。方差分析分解和锚定分解是寻找函数稀疏加性分解的典型技术。本文关注的一种场景是，这些分解并非直接稀疏，而是在适当的基变换后才变得稀疏。注意到这些加性函数分解的稀疏性等价于其大多数混合偏导数消失，我们可以利用与底层函数图的联系来确定实现适当基变换的正交变换。这通过三个步骤完成：应用奇异值分解以最小化函数图的顶点数量；采用矩阵族的联合块对角化技术；随后基于零"范数"的松弛进行稀疏最小化以最小化边缘数量。对于后者，我们提出并分析了特殊正交矩阵流形上的最小化技术。各种数值示例说明了我们的方法对于经过基变换后具有至多两个变量的加性分解项的函数的可靠性。