Motivated by recent results in the statistical physics of spin glasses, we study the recovery of a sparse vector , where denotes the -dimensional unit sphere, , from quadratic measurements of the form where 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 that the geometry of the energy landscape becomes trivial above a certain threshold . Building on this idea, we characterize the recovery of as a function of . We show that recovery of the vector can be guaranteed as soon as , provided that this vector satisfies a sufficiently strong incoherence condition, thus retrieving the linear regime for an external field . 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.

中文翻译：

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使用外场进行二次测量的稀疏恢复

受自旋玻璃统计物理学最新结果的启发，我们研究了稀疏向量 的恢复，其中 表示 维单位球体 ，来自具有独立同分布高斯项的形式的二次测量。这可能与所显示的具有外部场的 2-自旋哈密顿量的约束版本有关（在没有任何结构约束的情况下并且在渐进状态下），因为能量景观的几何形状在超过某个阈值时变得微不足道。基于这个想法，我们将 的恢复描述为 的函数。我们证明，只要矢量满足足够强的不相干条件，就可以保证矢量的恢复，从而恢复外场的线性状态。对于较弱的场可以显示类似的结果（样本复杂性略有恶化）。我们的证明依赖于线性和二次设置之间的插值，以及标准凸几何参数。