Sparse recovery from quadratic equations, part II: hardness and incoherence
Augustin Cosse
TL;DR
The work analyzes sparse recovery from quadratic measurements, addressing the square-root bottleneck by showing that, when the incoherence parameter $\mu_0$ is favorable, recovery is possible from $m$ measurements with $m \gtrsim k\mu_0^{-2} \vee \mu_0^{-4}$ (up to log factors) via empirical loss minimization; this extends to phase retrieval with subexponential sensing. It provides a computationally practical initialization and convergence guarantees for SPF and a truncated gradient descent variant, delivering explicit nonasymptotic error bounds in the presence of noise. Beyond algorithmic recovery, the paper develops a topological hardness certificate via the Overlap Gap Property in the fully incoherent and under-sampled regime ($m=o(k^2)$), supported by first/second moment analyses and moderate-deviation tails of chi-squared sums. Collectively, the results illuminate both when efficient algorithms can succeed and when topological barriers emerge, clarifying the role of incoherence, overparametrization, and initialization in quadratic-sensing problems. The findings have implications for structured sparse recovery and phase retrieval tasks in high dimensions, guiding algorithm design and setting expectations for sample complexity in practical settings.
Abstract
We study the square root bottleneck in the recovery of sparse vectors from quadratic equations. It is acknowledged that a sparse vector $ \mathbf x_0\in \mathbb{R}^n$, $\| \mathbf x_0\|_0 = k$ can in theory be recovered from as few as $O(k)$ generic quadratic equations but no polynomial time algorithm is known for this task unless $m = Ω(k^2)$. This bottleneck was in fact shown in previous work to be essentially related to the initialization of descent algorithms. Starting such algorithms sufficiently close to the planted signal is known to imply convergence to this signal. In this paper, we show that as soon as $m\gtrsim μ_0^{-2}k \vee μ_0^{-4}$ (up to log factors) where $μ_0 = \| \mathbf x_0\|_\infty/\| \mathbf x_0\|_2$, it is possible to recover a $k$-sparse vector $ \mathbf x_0\in \mathbb{R}^n$ from $m$ quadratic equations of the form $\langle \mathbf A_i, \mathbf x \mathbf x^\intercal\rangle = \langle \mathbf A_i, \mathbf x_0 \mathbf x_0^\intercal\rangle + \varepsilon_i $ by minimizing the classical empirical loss. The proof idea carries over to the phase retrieval setting for which it provides an original initialization that matches the current optimal sample complexity (see e.g. [Cai 2023]). In the maximally incoherent regime $μ_0^{-2}=k$, and for $m=o(k^2)$ we provide evidence for topological hardness by showing that a property known as the Overlap Gap Property (OGP), which originated in spin glass theory and is conjectured to be indicative of algorithmic intractability when optimizing over random structures, holds for a particular level of overparametrization. The key ingredient of the proof is a lower bound on the tail of chi-squared random variables which follows from the theory of moderate deviations.