Given input data $x_t\in \mathbb{R}^n$ and output data $y_t\in\mathbb{R}^m$, the closed form solution to $\min_A \sum_t \|y_t - Ax_t\|^2_2$ is given by $A = (XX^T)^{-1}XY^T$ where $x_t$ form the columns of $X$ and similarly for $Y$. This is the solution to least-squares regression.
Suppose instead of one set of inputs, we had $k$ sets of inputs $x^1_t,\ldots,x^k_t$ from vector spaces $\mathbb{R}^{n_1},\ldots,\mathbb{R}^{n_k}$ and we wanted to find the multilinear map $A$ from $\mathbb{R}^{n_1}\times\ldots\times\mathbb{R}^{n_k}$ to $\mathbb{R}^m$ that minimizes
$$\sum_t \|y_t - A(x^1_t,\ldots,x^k_t)\|^2_2$$
Is there a closed form solution to this? Trying the usual method (write out multilinear map in index notation, take derivatives and set to zero) it looks like
$$A = \sum_t y_t \otimes (X^1X^{1T})^{-1}x^1_t \otimes \ldots \otimes (X^kX^{kT})^{-1}x^k_t$$
might work, where $\otimes$ denotes outer product, but my algebra could be wrong. It does, at least, reduce to the least squares regression case when $k=1$.