Abstract: We propose a fast algorithm for ridge regression when the number of features is much larger than the number of observations ($p\gg n$). The standard way to solve ridge regression in this setting works in the dual space and gives a running time of $O(n^{2}p)$. Our algorithm (SRHT-DRR) runs in time $O(np\log (n))$ and works by preconditioning the design matrix by a Randomized Walsh-Hadamard Transform with a subsequent subsampling of features. We provide risk bounds for our SRHT-DRR algorithm in the fixed design setting and show experimental results on synthetic and real datasets.