2023/11/08 by Titouan Vayer, Vayer, Titouan, Etienne Lasalle +5
Computer Science · Engineering · #Blind Source Separation Techniques #Distributed Sensor Networks and Detection Algorithms #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.2311.04673
openalex publication_date 2023/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the problem of learning a graph modeling the statistical relations of the d variables from a dataset with n samples X ∈ ℝn × 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 d2 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 sketch of the data, i.e. a low-dimensional vector of size m ≪ d2 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((d+2k)log(d)) 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.