  We study the least squares regression problem \begin{align*} \min_{\Theta \in
\mathcal{S}_{\odot D,R}} \|A\Theta-b\|_2, \end{align*} where
$\mathcal{S}_{\odot D,R}$ is the set of $\Theta$ for which $\Theta =
\sum_{r=1}^{R} \theta_1^{(r)} \circ \cdots \circ \theta_D^{(r)}$ for vectors
$\theta_d^{(r)} \in \mathbb{R}^{p_d}$ for all $r \in [R]$ and $d \in [D]$, and
$\circ$ denotes the outer product of vectors. That is, $\Theta$ is a
low-dimensional, low-rank tensor. This is motivated by the fact that the number
of parameters in $\Theta$ is only $R \cdot \sum_{d=1}^D p_d$, which is
significantly smaller than the $\prod_{d=1}^{D} p_d$ number of parameters in
ordinary least squares regression. We consider the above CP decomposition model
of tensors $\Theta$, as well as the Tucker decomposition. For both models we
show how to apply data dimensionality reduction techniques based on {\it
sparse} random projections $\Phi \in \mathbb{R}^{m \times n}$, with $m \ll n$,
to reduce the problem to a much smaller problem $\min_{\Theta} \|\Phi A \Theta
- \Phi b\|_2$, for which if $\Theta'$ is a near-optimum to the smaller problem,
then it is also a near optimum to the original problem. We obtain significantly
smaller dimension and sparsity in $\Phi$ than is possible for ordinary least
squares regression, and we also provide a number of numerical simulations
supporting our theory.
