Precise expressions for random projections: Low-rank approximation and randomized Newton

It is often desirable to reduce the dimensionality of a large dataset by projecting it onto a lowdimensional subspace. Matrix sketching has emerged as a powerful technique for performing such dimensionality reduction very efficiently. Even though there is an extensive literature on the worst-case performance of sketching, existing guarantees are typically very different from what is observed in practice. We exploit recent developments in the spectral analysis of random matrices to develop novel techniques that provide provably accurate expressions for the expected value of random projection matrices obtained via sketching. These expressions can be used to characterize the performance of dimensionality reduction in a variety of common machine learning tasks, ranging from low-rank approximation to iterative stochastic optimization. Our results apply to several popular sketching methods, including Gaussian and Rademacher sketches, and they enable precise analysis of these methods in terms of spectral properties of the data. Empirical results show that the expressions we derive reflect the practical performance of these sketching methods, down to lower-order effects and even constant factors. * This version of the paper includes a correction to the assumptions in a technical result, Theorem 2. The previous claim relied on a formulation of the Hanson-Wright inequality given by [Zaj20, Corollary 2.8], which turns out to be false. This was not essential for our main results, so none of the other claims are affected by this change. The conference version of this paper, i.e., [DLLM20], does not include the correction, so we recommend to cite this arXiv version when referencing Theorem 2.