We consider regression of a max-affine model that produces a piecewise linear model by combining affine models via the max function. The max-affine model ubiquitously arises in applications in signal processing and statistics including multiclass classification, auction problems, and convex regression. It also generalizes phase retrieval and learning rectifier linear unit activation functions. We present a non-asymptotic convergence analysis of gradient descent (GD) and mini-batch stochastic gradient descent (SGD) for max-affine regression when the model is observed at random locations following the sub-Gaussianity and an anti-concentration with additive sub-Gaussian noise. Under these assumptions, a suitably initialized GD and SGD converge linearly to a neighborhood of the ground truth specified by the corresponding error bound. We provide numerical results that corroborate the theoretical finding. Importantly, SGD not only converges faster in run time with fewer observations than alternating minimization and GD in the noiseless scenario but also outperforms them in low-sample scenarios with noise.

通过最大线性模型的回归，我们对具有最大函数的组合线性模型进行了研究，这种模型在信号处理、统计学和其他多个领域中普遍存在，并且在相位恢复、学习整流线性单元激活函数等方面具有扩展性。我们提供了渐进收敛性分析，并证明了梯度下降和随机梯度下降在最大线性回归问题中的收敛性。数值结果验证了理论发现，尤其是在带有噪声的低样本情况下，随机梯度下降不仅在运行时间上收敛更快，而且性能优于交替最小化和梯度下降。

最大仿射回归的一阶方法