Many problems in computer science reduce to the recovery of an $n$-sparse measure from its (generalized) moments. Sparse measure recovery has been the research focus in super-resolution, tensor decomposition, and learning neural networks. The existing methods use either convex relaxations or overparameterization for recovery. Here, we propose recovery with non-convex optimization without overparameterization. Our algorithm is a (sub)gradient descent method optimizing a non-convex energy function studied in physics. We establish the global convergence of gradient descent on the energy function. This result enables us to solve super-resolution in $O(n^2)$ time, which significantly improves upon $O(n^3)$ time for solving convex relaxations. For a particular neural network, we prove the global convergence of subgradient descent on the population loss without overparameterization. The studied network has zero-one activations, and inputs drawn from the unit sphere.

翻译：许多计算机科学问题可归结为从其（广义）矩中恢复一个$n$-稀疏测度。稀疏测度恢复一直是超分辨率、张量分解和神经网络学习领域的研究重点。现有方法利用凸松弛或过参数化进行恢复。本文提出一种无需过参数化的非凸优化恢复方法。我们的算法是一种（次）梯度下降方法，优化了物理学中研究的非凸能量函数。我们建立了梯度下降在该能量函数上的全局收敛性。这一结果使我们能在$O(n^2)$时间内解决超分辨率问题，相较于求解凸松弛所需的$O(n^3)$时间有显著改进。针对特定神经网络，我们证明了无需过参数化的情况下，次梯度下降在总体损失上的全局收敛性。所研究的网络采用0-1激活函数，且输入采样自单位球面。