Sparse recovery from quadratic measurements with external field

Abstract

Motivated by recent results in the statistical physics of spin glasses, we study the recovery of a sparse vector x0∈Sn−1, where Sn−1 denotes the n-dimensional unit sphere, ‖x0‖ℓ0=k<n, from m quadratic measurements of the form (1−λ)〈Ai,x0x0⊺〉+λ〈ci,x0〉 where Ai,ci 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 λ>λc∈(0,1). Building on this idea, we characterize the recovery of x0 as a function of λ∈[0,1]. We show that recovery of the vector x0 can be guaranteed as soon as m≳k2(1−λ)2/λ2∨k, λ>1/2 provided that this vector satisfies a sufficiently strong incoherence condition, thus retrieving the linear regime for an external field (1−λ)/λ≲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.