  Motivated by recent results in the statistical physics of spin glasses, we
study the recovery of a sparse vector $\mathbf{x}_0\in \mathbb{S}^{n-1}$,
$\|\mathbf{x}_0\|_{\ell_0} = k<n$, from $m$ quadratic measurements of the form
$ (1-\lambda)\langle \mathbf{A}_i, \mathbf{x}_0\mathbf{x}_0^T \rangle + \lambda
\langle\mathbf{c}_i,\mathbf{x}_0 \rangle $ where $\mathbf{A}_{i},
\mathbf{c}_{i}$ have i.i.d Gaussian entries. This can be related to a
constrained version of the 2-spin Hamiltonian with external field for which it
was recently shown (in the absence of any structural constraint and in the
asymptotic regime) that the geometry of the energy landscape becomes trivial
above a certain threshold $\lambda > \lambda_c\in (0,1)$. Building on this idea
we study the evolution of the so-called square root bottleneck for $\lambda\in
[0,1]$ in the setting of the sparse rank one matrix recovery/sensing problem.
We show that recovery of the vector $\mathbf{x}_0$ can be guaranteed as soon as
$m\gtrsim k^2 (1-\lambda)^2/\lambda^2$, $\lambda \gtrsim k^{-1/2}$ provided
that this vector satisfies a sufficiently strong incoherence condition, thus
retrieving the linear regime for an external field $(1-\lambda)/\lambda
\lesssim k^{-1/2}$. Our proof relies on an interpolation between the linear and
quadratic settings, as well as on standard convex geometry arguments.
