# Recycling Randomness with Structure for Sublinear time Kernel Expansions

arXiv: https://arxiv.org/abs/1605.09049 · HTML: https://mlsanity.com/paper/1605.09049/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 May 2016.

## Built on by (most cited first, 9 of 9)

- [On the Expressive Power of Self-Attention Matrices](https://mlsanity.com/paper/2106.03764/) (2021)
- [Hybrid Random Features](https://mlsanity.com/paper/2110.04367/) (2021)
- [On Learning the Transformer Kernel](https://mlsanity.com/paper/2110.08323/) (2021)
- [Random Fourier Signature Features](https://mlsanity.com/paper/2311.12214/) (2023)
- [Spectraformer: A Unified Random Feature Framework for Transformer](https://mlsanity.com/paper/2405.15310/) (2024)
- [Error Estimation for Random Fourier Features](https://mlsanity.com/paper/2302.11174/) (2023)
- [Feature maps for the Laplacian kernel and its generalizations](https://mlsanity.com/paper/2502.15575/) (2025)
- [Manifold Random Features](https://mlsanity.com/paper/2602.03797/) (2026)
- [Random Fourier Features in Near-Linear Time](https://mlsanity.com/paper/2610.14751/) (2026)

## Abstract (arXiv)

We propose a scheme for recycling Gaussian random vectors into structured matrices to approximate various kernel functions in sublinear time via random embeddings. Our framework includes the Fastfood construction as a special case, but also extends to Circulant, Toeplitz and Hankel matrices, and the broader family of structured matrices that are characterized by the concept of low-displacement rank. We introduce notions of coherence and graph-theoretic structural constants that control the approximation quality, and prove unbiasedness and low-variance properties of random feature maps that arise within our framework. For the case of low-displacement matrices, we show how the degree of structure and randomness can be controlled to reduce statistical variance at the cost of increased computation and storage requirements. Empirical results strongly support our theory and justify the use of a broader family of structured matrices for scaling up kernel methods using random features.