The Kronecker product is an important matrix operation with a wide range of applications in supporting fast linear transforms, including signal processing, graph theory, quantum computing and deep learning. In this work, we introduce a generalization of the fast Johnson-Lindenstrauss projection for embedding vectors with Kronecker product structure, the *Kronecker fast Johnson-Lindenstrauss transform* (KFJLT). The KFJLT drastically reduces the embedding cost to an exponential factor of the standard fast Johnson-Lindenstrauss transform (FJLT)’s cost when applied to vectors with Kronecker structure, by avoiding explicitly forming the full Kronecker products. We prove that this computational gain comes with only a small price in embedding power: given $N = \prod_{k=1}^d n_k$, consider a finite set of $p$ points in a tensor product of $d$ constituent Euclidean spaces $\bigotimes_{k=d}^{1}\mathbb{R}^{n_k} \subset \mathbb{R}^{N}$. With high probability, a random KFJLT matrix of dimension $N \times m$ embeds the set of points up to multiplicative distortion $(1\pm \varepsilon)$ provided by $m \gtrsim \varepsilon^{-2} \cdot \log^{2d - 1} (p) \cdot \log N$. We conclude by describing a direct application of the KFJLT to the efficient solution of large-scale Kronecker-structured least squares problems for fitting the CP tensor decomposition.

Type

Publication

arXiv

Date

Sep 2019

Tags

Citation

R. Jin, T. G. Kolda, R. Ward.
**Faster Johnson-Lindenstrauss Transforms via Kronecker Products**.
arXiv:1909.04801,
2019.
http://arxiv.org/abs/1909.04801

cs.IT, cs.NA, math.IT, math.NA, math.PR

```
@misc{JiKoWa19,
author = {Ruhui Jin and Tamara G. Kolda and Rachel Ward},
title = {Faster {Johnson-Lindenstrauss} Transforms via {Kronecker} Products},
month = {September},
year = {2019},
eprint = {1909.04801},
eprintclass = {cs.IT},
}
```