稀疏精度矩阵的压缩恢复
Compressive Recovery of Sparse Precision Matrices
AI总结：
针对高维下基于最大似然的精度矩阵估计存储成本过高的问题，提出从数据低维草图中估计稀疏精度矩阵的压缩方法，给出信息论保证并验证其在压缩数据上的良好性能。
摘要：
我们研究从包含$n$个样本的数据集$X \in \mathbb{R}^{n \times d}$中学习建模$d$个变量统计关系的图的问题。标准方法相当于寻找一个能充分解释数据的、代表高斯图模型的精度矩阵$\Theta$。然而，大多数基于最大似然的估计器通常需要存储经验协方差矩阵的$d^2$个值，这在高维场景下会变得难以实现。在本研究中，我们采用压缩视角，旨在从数据的“草图（sketch）”——即使用非线性随机特征从$X$精心构造的、维度为$m \ll d^2$的低维向量——中估计稀疏的$\Theta$。在对$\Theta$的谱（或其条件数）的特定假设下，我们证明可以从大小为$m=\Omega\left((d+2k)\log(d)\right)$的草图中估计该矩阵，其中$k$是底层图的最大边数。这些信息论保证受压缩感知理论启发，涉及限制等距性质和实例最优解码器。我们探究了使用基于graphical lasso（图套索，被视为一种特定去噪器）的迭代算法实现实际恢复的可能性。我们在合成数据集上对比了所提方法与graphical lasso，证明即使数据集被压缩，该方法仍具备良好性能。
英文摘要：
We consider the problem of learning a graph modeling the statistical relations of the $d$ variables from a dataset with $n$ samples $X \in \mathbb{R}^{n \times d}$. Standard approaches amount to searching for a precision matrix $Θ$ representative of a Gaussian graphical model that adequately explains the data. However, most maximum likelihood-based estimators usually require storing the $d^{2}$ values of the empirical covariance matrix, which can become prohibitive in a high-dimensional setting. In this work, we adopt a compressive viewpoint and aim to estimate a sparse $Θ$ from a \emph{sketch} of the data, i.e. a low-dimensional vector of size $m \ll d^{2}$ carefully designed from $X$ using non-linear random features. Under certain assumptions on the spectrum of $Θ$ (or its condition number), we show that it is possible to estimate it from a sketch of size $m=Ω\left((d+2k)\log(d)\right)$ where $k$ is the maximal number of edges of the underlying graph. These information-theoretic guarantees are inspired by compressed sensing theory and involve restricted isometry properties and instance optimal decoders. We investigate the possibility of achieving practical recovery with an iterative algorithm based on the graphical lasso, viewed as a specific denoiser. We compare our approach and graphical lasso on synthetic datasets, demonstrating its favorable performance even when the dataset is compressed.
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