Covering Number of Real Algebraic Varieties: Improved Bound and Applications

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.