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 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=∏k=1dnk
Nearby in the stack
, consider a finite set of
p
points in a tensor product of
d
constituent Euclidean spaces
⨂k=d1Rnk⊂RN
. With high probability, a random KFJLT matrix of dimension
N×m
embeds the set of points up to multiplicative distortion
(1±ε)
provided by
m≳ε−2⋅log2d−1(p)⋅logN
. 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.