We prove an upper bound on the covering number of real algebraic varieties, images of polynomial maps and semialgebraic sets. The bound remarkably improves the best known general bound by Yomdin-Comte, and its proof is much more straightforward. As a consequence, our result gives new bounds on the volume of the tubular neighborhood of the image of a polynomial map and a semialgebraic set, where results for varieties by Lotz and Basu-Lerario are not directly applicable. We apply our theory to three main application domains. Firstly, we derive a near-optimal bound on the covering number of low rank CP tensors. Secondly, we prove a bound on the sketching dimension for (general) polynomial optimization problems. Lastly, we deduce generalization error bounds for deep neural networks with rational or ReLU activations, improving or matching the best known results in the literature.

翻译：本文证明了实代数簇、多项式映射像以及半代数集在覆盖数上的一个上界。该界显著改进了Yomdin–Comte已知的最佳一般上界，且其证明更为直接。作为推论，我们的结果为多项式映射像及半代数集的管状邻域体积提供了新的估计，而此前Lotz与Basu–Lerario关于代数簇的结果无法直接应用于此类情形。我们将此理论应用于三个主要领域：首先，给出了低秩CP张量覆盖数的近最优界；其次，证明了（一般）多项式优化问题的草图化维数上界；最后，推导了具有有理或ReLU激活函数的深度神经网络的泛化误差界，改进或持平了现有文献中的最佳结果。