Sparse recovery from quadratic measurements with external field

Motivated by recent results in the statistical physics of spin glasses, we study the recovery of a sparse vector $x_0\in S^{n-1}$, where $S^{n-1}$ denotes the n-dimensional unit sphere, ${\Vert x_0\Vert }_{{\ell }_0}=k<n$, from m quadratic measurements of the form $(1-\lambda )〈A_i,x_0{x}_0^{⊺}〉+\lambda 〈c_i,x_0〉$ where $A_i,c_i$ 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 shown (in the absence of any structural constraint and in the asymptotic regime) in [1] that the geometry of the energy landscape becomes trivial above a certain threshold $\lambda >{\lambda }_c\in (0,1)$. Building on this idea, we characterize the recovery of $x_0$ as a function of $\lambda \in [0,1]$. We show that recovery of the vector $x_0$ can be guaranteed as soon as $m\gtrsim k^2{(1-\lambda )}^2/{\lambda }^2\vee k$, $\lambda >1/2$ provided that this vector satisfies a sufficiently strong incoherence condition, thus retrieving the linear regime for an external field $(1-\lambda )/\lambda \lesssim k^{-1/2}$. A similar result (with a slightly deteriorating sample complexity) can be shown for weaker fields. Our proof relies on an interpolation between the linear and quadratic settings, as well as on standard convex geometry arguments.