  The $k$-tensor Ising model is an exponential family on a $p$-dimensional
binary hypercube for modeling dependent binary data, where the sufficient
statistic consists of all $k$-fold products of the observations, and the
parameter is an unknown $k$-fold tensor, designed to capture higher-order
interactions between the binary variables. In this paper, we describe an
approach based on a penalization technique that helps us recover the signed
support of the tensor parameter with high probability, assuming that no entry
of the true tensor is too close to zero. The method is based on an
$\ell_1$-regularized node-wise logistic regression, that recovers the signed
neighborhood of each node with high probability. Our analysis is carried out in
the high-dimensional regime, that allows the dimension $p$ of the Ising model,
as well as the interaction factor $k$ to potentially grow to $\infty$ with the
sample size $n$. We show that if the minimum interaction strength is not too
small, then consistent recovery of the entire signed support is possible if one
takes $n = \Omega((k!)^8 d^3 \log \binom{p-1}{k-1})$ samples, where $d$ denotes
the maximum degree of the hypernetwork in question. Our results are validated
in two simulation settings, and applied on a real neurobiological dataset
consisting of multi-array electro-physiological recordings from the mouse
visual cortex, to model higher-order interactions between the brain regions.
