# Optimal Iterative Sketching with the Subsampled Randomized Hadamard Transform

arXiv: https://arxiv.org/abs/2002.00864 · HTML: https://mlsanity.com/paper/2002.00864/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Feb 2020.

## Built on by (most cited first, 3 of 3)

- [Newton-LESS: Sparsification without Trade-offs for the Sketched Newton Update](https://mlsanity.com/paper/2107.07480/) (2021)
- [Adaptive Newton Sketch: Linear-time Optimization with Quadratic Convergence and Effective Hessian Dimensionality](https://mlsanity.com/paper/2105.07291/) (2021)
- [Training Quantized Neural Networks to Global Optimality via Semidefinite Programming](https://mlsanity.com/paper/2105.01420/) (2021)

## Abstract (arXiv)

Random projections or sketching are widely used in many algorithmic and learning contexts. Here we study the performance of iterative Hessian sketch for least-squares problems. By leveraging and extending recent results from random matrix theory on the limiting spectrum of matrices randomly projected with the subsampled randomized Hadamard transform, and truncated Haar matrices, we can study and compare the resulting algorithms to a level of precision that has not been possible before. Our technical contributions include a novel formula for the second moment of the inverse of projected matrices. We also find simple closed-form expressions for asymptotically optimal step-sizes and convergence rates. These show that the convergence rate for Haar and randomized Hadamard matrices are identical, and asymptotically improve upon Gaussian random projections. These techniques may be applied to other algorithms that employ randomized dimension reduction.