Covering numbers are a powerful tool used in the development of approximation algorithms, randomized dimension reduction methods, smoothed complexity analysis, and others. In this paper we prove upper bounds on the covering number of numerous sets in Euclidean space, namely real algebraic varieties, images of polynomial maps and semialgebraic sets in terms of the number of variables and degrees of the polynomials involved. The bounds remarkably improve the best known general bound by Yosef Yomdin and Georges Comte (Tame Geometry with Application in Smooth Analysis, 2004), and our proof is much more straightforward. In particular, 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 Martin Lotz (In: Proceedings of the American Mathematical Society, 143.5:1875–1889, 2015) and Saugata Basu and Antonio Lerario (In: Mathematische Annalen, 1–31, 2022) are not directly applicable. We illustrate the power of the result on three computational applications. Firstly, we derive a near-optimal bound on the covering number of tensors with low canonical polyadic (CP) rank, quantifying their approximation properties and filling in an important missing piece of theory for tensor dimension reduction and reconstruction. Secondly, we prove a bound on dimensionality reduction of images of polynomial maps via randomized sketching, which has direct applications to large scale polynomial optimization. Finally, we deduce generalization error bounds for deep neural networks with rational or ReLU activation functions, improving or matching the best known results in the machine learning literature, while helping to quantify the impact of architecture choice on generalization error.
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