We study the fundamental problem of fixed design {\em multidimensional segmented regression}: Given noisy samples from a function $f$, promised to be piecewise linear on an unknown set of $k$ rectangles, we want to recover $f$ up to a desired accuracy in mean-squared error. We provide the first sample and computationally efficient algorithm for this problem in any fixed dimension. Our algorithm relies on a simple iterative merging approach, which is novel in the multidimensional setting. Our experimental evaluation on both synthetic and real datasets shows that our algorithm is competitive and in some cases outperforms state-of-the-art heuristics. Code of our implementation is available at \url{https://github.com/avoloshinov/multidimensional-segmented-regression}.

中文翻译：

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多维分段回归的高效算法

我们研究了固定设计{\em 多维分段回归}的基本问题：给定来自函数 $f$ 的噪声样本，承诺在一组未知的 $k$ 矩形上分段线性，我们希望将 $f$ 恢复到均方误差的期望精度。我们在任何固定维度上为这个问题提供了第一个样本和计算效率高的算法。我们的算法依赖于一种简单的迭代合并方法，这在多维设置中是新颖的。我们对合成数据集和真实数据集的实验评估表明，我们的算法具有竞争力，并且在某些情况下优于最先进的启发式算法。我们的实现代码可在 \url{https://github.com/avoloshinov/multidimensional-segmented-regression} 获得。