Nonasymptotic Guarantees for Spiked Matrix Recovery with Generative Priors
Nonasymptotic Guarantees for Spiked Matrix Recovery with Generative Priors
复制标题
DOI：
--
复制
发表时间：
2020-11
期刊：
影响因子：
--
通讯作者：
Jorio Cocola;Paul Hand;V. Voroninski
中科院分区：
文献类型：
--
作者：
Jorio Cocola;Paul Hand;V. Voroninski
统计学和机器学习中的许多问题都需要从噪声数据中重建一阶信号矩阵。在排名第一的组件上执行额外的先验信息通常是保证良好恢复性能的关键。低秩分量的先验之一是稀疏性，这就产生了稀疏主成分分析问题。不幸的是，有强有力的证据表明，这个问题受到计算与统计差距的影响，这可能是根本的。在这项工作中，我们研究了一种替代先验，其中低秩分量在训练生成网络的范围内。我们提供了一个非渐近分析与最优的样本复杂度，高达对数因子，排名一矩阵恢复下的扩展高斯网络先验。具体来说，我们为非线性最小二乘目标建立了一个有利的全局优化环境，前提是样本数量与生成模型的输入维数相同。这一结果表明，在有限数据、非渐近状态下，生成先验对结构化秩一矩阵恢复没有计算与统计的差距。我们在Wishart和Wigner尖峰矩阵模型的情况下提出了这种分析。
Many problems in statistics and machine learning require the reconstruction of a rank-one signal matrix from noisy data. Enforcing additional prior information on the rank-one component is often key to guaranteeing good recovery performance. One such prior on the low-rank component is sparsity, giving rise to the sparse principal component analysis problem. Unfortunately, there is strong evidence that this problem suffers from a computational-to-statistical gap, which may be fundamental. In this work, we study an alternative prior where the low-rank component is in the range of a trained generative network. We provide a non-asymptotic analysis with optimal sample complexity, up to logarithmic factors, for rank-one matrix recovery under an expansive-Gaussian network prior. Specifically, we establish a favorable global optimization landscape for a nonlinear least squares objective, provided the number of samples is on the order of the dimensionality of the input to the generative model. This result suggests that generative priors have no computational-to-statistical gap for structured rank-one matrix recovery in the finite data, nonasymptotic regime. We present this analysis in the case of both the Wishart and Wigner spiked matrix models.