低秩矩阵恢复的景观：超越RIP及其在秩一测量中的应用

中文摘要 AI 辅助

我们研究通过低秩矩阵分解形式（核范数正则化最小二乘）的全局非凸景观，从线性测量中恢复低秩矩阵的问题。如果景观是良性的，即没有不良的局部最优解，那么实用且可扩展的算法能够计算出良好的统计估计。以往最先进的景观保证通常假设线性测量算子具有受限等距性质，即该算子在所有低秩矩阵上近似为等距。这一假设在许多应用中并不现实；特别是当单个测量矩阵本身是低秩的时，我们通常只能获得较差的上等距常数。为克服这一限制，我们在更弱的等距条件下建立了良性景观的新保证：我们不再要求算子在所有低秩矩阵上具有上等距性质，而仅要求其在低秩真值矩阵的线性低秩切空间上满足该性质。为说明该结果的实用性，我们将其应用于从随机秩一线性测量中恢复矩阵的问题；通过随机测量算子的高概率集中界，我们证明了具有统计上接近最优的样本复杂度和恢复误差的新景观保证。

英文摘要

We study the problem of low-rank matrix recovery from linear measurements via the global nonconvex landscape of a low-rank factored formulation of the matrix LASSO (nuclear-norm--regularized least-squares). If the landscape is benign, that is, has no bad local optima, then practical and scalable algorithms can compute good statistical estimates. Previous state-of-the-art landscape guarantees have typically assumed that the linear measurement operator has the restricted isometry property, that is, the operator is approximately an isometry over all low-rank matrices. This is an unrealistic assumption for many applications; in particular, when the individual measurement matrices are themselves low-rank, we typically have poor upper isometry constants. To overcome this, we establish new guarantees of a benign landscape under a weaker isometry condition: rather than requiring upper isometry over all low-rank matrices, we only require it over the linear low-rank tangent space to the low-rank ground truth matrix. To illustrate the utility of this result, we apply it to the problem of matrix recovery from random rank-one linear measurements; via high-probability concentration bounds on the random measurement operator, we prove a novel landscape guarantee with statistically near-optimal sample complexity and recovery error.

发表机构

• CERMICS, ENPC, Institut Polytechnique de Paris, CNRS（巴黎高等师范学院计算数学与控制系统实验室，巴黎高等师范工程学校，巴黎理工学院，法国国家科学研究中心）

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