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 bound by Yomdin-Comte, and its proof is much more straightforward. As a consequence, our result gives a bound on volume of the tubular neighborhood of a real variety, improving the results by Lotz and Basu-Lerario. 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：我们证明了实代数簇、多项式映射像及半代数集覆盖数的上界。该界显著优于Yomdin-Comte已知的最佳结果，且证明过程更为直接。作为推论，我们的结果给出了实代数簇管状邻域体积的界，改进了Lotz与Basu-Lerario的结论。我们将理论应用于三个主要领域：首先，推导了低秩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维度界；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，改进或匹配了文献中的已知最佳结果。