Uniform Approximations for Randomized Hadamard Transforms with Applications
This provides theoretical justification for the empirical success of RHTs in high-dimensional machine learning applications, though it is incremental as it extends prior work to new regimes.
The paper tackled the lack of theoretical guarantees for Randomized Hadamard Transforms (RHTs) in high-dimensional regimes, such as kernel approximation, by proving uniform convergence results comparable to Gaussian matrices, leading to improved guarantees for kernel approximation and distance estimation.
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.