  Randomized Hadamard Transforms (RHTs) have emerged as a computationally
efficient alternative to the use of dense unstructured random matrices across a
range of domains in computer science and machine learning. For several
applications such as dimensionality reduction and compressed sensing, the
theoretical guarantees for methods based on RHTs are comparable to approaches
using dense random matrices with i.i.d.\ entries. However, several such
applications are in the low-dimensional regime where the number of rows sampled
from the matrix is rather small. Prior arguments are not applicable to the
high-dimensional regime often found in machine learning applications like
kernel approximation. Given an ensemble of RHTs with Gaussian diagonals,
$\{M^i\}_{i = 1}^m$, and any $1$-Lipschitz function, $f: \mathbb{R} \to
\mathbb{R}$, we prove that the average of $f$ over the entries of $\{M^i v\}_{i
= 1}^m$ converges to its expectation uniformly over $\| v \| \leq 1$ at a rate
comparable to that obtained from using truly Gaussian matrices. We use our
inequality to then derive improved guarantees for two applications in the
high-dimensional regime: 1) kernel approximation and 2) distance estimation.
For kernel approximation, we prove the first \emph{uniform} approximation
guarantees for random features constructed through RHTs lending theoretical
justification to their empirical success while for distance estimation, our
convergence result implies data structures with improved runtime guarantees
over previous work by the authors. We believe our general inequality is likely
to find use in other applications.
