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Motivated by recent results in the statistical physics of spin glasses, westudy 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 aconstrained version of the 2-spin Hamiltonian with external field for which itwas recently shown (in the absence of any structural constraint and in theasymptotic regime) that the geometry of the energy landscape becomes trivialabove a certain threshold $\lambda > \lambda_c\in (0,1)$. Building on this ideawe 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$m\gtrsim k^2 (1-\lambda)^2/\lambda^2$, $\lambda \gtrsim k^{-1/2}$ providedthat this vector satisfies a sufficiently strong incoherence condition, thusretrieving the linear regime for an external field $(1-\lambda)/\lambda\lesssim k^{-1/2}$. Our proof relies on an interpolation between the linear andquadratic settings, as well as on standard convex geometry arguments.

In the recovery of sparse vectors from quadratic measurements, the presence of linear terms breaks the square root bottleneck