We study the problems of testing and learning high-dimensional discrete convex sets. The simplest high-dimensional discrete domain where convexity is a non-trivial property is the ternary hypercube, $\{-1,0,1\}^n$. The goal of this work is to understand structural combinatorial properties of convex sets in this domain and to determine the complexity of the testing and learning problems. We obtain the following results. Structural: We prove nearly tight bounds on the edge boundary of convex sets in $\{0,\pm 1\}^n$, showing that the maximum edge boundary of a convex set is $\widetilde \Theta(n^{3/4}) \cdot 3^n$, or equivalently that every convex set has influence $\widetilde{O}(n^{3/4})$ and a convex set exists with influence $\Omega(n^{3/4})$. Learning and sample-based testing: We prove upper and lower bounds of $3^{\widetilde{O}(n^{3/4})}$ and $3^{\Omega(\sqrt{n})}$ for the task of learning convex sets under the uniform distribution from random examples. The analysis of the learning algorithm relies on our upper bound on the influence. Both the upper and lower bound also hold for the problem of sample-based testing with two-sided error. For sample-based testing with one-sided error we show that the sample-complexity is $3^{\Theta(n)}$. Testing with queries: We prove nearly matching upper and lower bounds of $3^{\widetilde{\Theta}(\sqrt{n})}$ for one-sided error testing of convex sets with non-adaptive queries.

翻译：我们研究高维离散凸集的测试与学习问题。在离散域中，凸性成为非平凡性质的最简单高维结构是三元超立方体 $\{-1,0,1\}^n$。本文旨在理解该域中凸集的结构组合性质，并确定测试与学习问题的复杂度。我们获得以下结果：结构性质：我们在 $\{0,\pm 1\}^n$ 中建立了凸集边界的几乎紧界，证明凸集的最大边边界为 $\widetilde \Theta(n^{3/4}) \cdot 3^n$，等价于每个凸集的影响度为 $\widetilde{O}(n^{3/4})$，且存在影响度为 $\Omega(n^{3/4})$ 的凸集。学习与基于样本的测试：对于在均匀分布下从随机样本学习凸集的任务，我们证明上下界分别为 $3^{\widetilde{O}(n^{3/4})}$ 和 $3^{\Omega(\sqrt{n})}$。学习算法的分析依赖于我们对影响度的上界。这两个界同样适用于具有双面误差的基于样本的测试问题。对于具有单面误差的基于样本的测试，我们证明样本复杂度为 $3^{\Theta(n)}$。基于查询的测试：对于使用非自适应查询的单面误差凸集测试，我们证明几乎匹配的上下界为 $3^{\widetilde{\Theta}(\sqrt{n})}$。