We prove an upper bound on the covering number of real algebraic varieties, images of polynomial maps and semialgebraic sets, which describe the set of low rank tensors and structured tensor networks. 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. In this talk, I will discuss the connection between this new result and tensor-based statistical learning methods. Time permitting, I will discuss applications of our theory to two main domains — sketching for (general) polynomial optimization problems and generalization error bounds for deep neural networks with rational or ReLU activations.