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 $\Theta$ 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 $\Theta$ 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 $\Theta$ (or its condition number), we show that it is possible to estimate it from a sketch of size $m=\Omega\left((d+2k)\log(d)\right)$ 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.

翻译：我们考虑从包含 $n$ 个样本的数据集 $X \in \mathbb{R}^{n \times d}$ 中学习一个图形模型的问题，该模型用于建模 $d$ 个变量之间的统计关系。标准方法相当于搜索一个能充分解释数据的高斯图模型所对应的精度矩阵 $\Theta$。然而，大多数基于最大似然估计的方法通常需要存储经验协方差矩阵的 $d^{2}$ 个值，在高维场景下这可能变得难以实现。在本工作中，我们采用压缩感知视角，旨在从数据的\textit{草图}中估计一个稀疏的 $\Theta$，即通过精心设计的非线性随机特征从 $X$ 中提取一个大小为 $m \ll d^{2}$ 的低维向量。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，涉及受限等距性质和实例最优解码器。我们研究了通过基于图形套索的迭代算法（将其视为特定去噪器）实现实际恢复的可能性。我们在合成数据集上比较了本方法与图形套索的性能，展示了即使数据集被压缩时该方法也具有优越的性能。