Precise expressions for random projections: Low-rank approximation and randomized Newton
This work addresses the problem of precise performance analysis for sketching methods in machine learning, offering incremental improvements in theoretical understanding for practitioners in data science and optimization.
The paper tackled the gap between theoretical worst-case guarantees and practical performance of matrix sketching for dimensionality reduction, providing provably accurate expressions for the expected value of random projection matrices that reflect empirical performance down to lower-order effects and constant factors.
It is often desirable to reduce the dimensionality of a large dataset by projecting it onto a low-dimensional 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.