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

Date

Sep 2020

Tags

Links

Citation

R. Jin, T. G. Kolda, R. Ward.
**Faster Johnson-Lindenstrauss Transforms via Kronecker Products**.
*Information and Inference*, accepted for publication,
2020.
http://arxiv.org/abs/1909.04801

Johnson-Lindenstrauss embedding, fast Johnson-Lindenstrauss transform (FJLT),Kronecker structure, concentration inequality, restricted isometry property

```
@article{JiKoWaXX,
author = {Ruhui Jin and Tamara G. Kolda and Rachel Ward},
title = {Faster {Johnson-Lindenstrauss} Transforms via {Kronecker} Products},
journal = {Information and Inference},
month = {September},
year = {2020},
note = {accepted for publication},
eprint = {1909.04801},
}
```