用低秩分解加速大规模图学习，提升效率与精度。
Leveraging Low-rank Factorizations of Conditional Correlation Matrices in Graph Learning
- 通过低秩分解条件相关矩阵，降低图学习的计算复杂度。
- 在合成与真实数据上均实现高维场景下的高效性能表现。
- 适合处理节点数多、数据维度高的图学习任务。
本文研究从节点采集的数据中学习无向图的问题。在图信号处理框架下，图拓扑可关联于数据条件相关矩阵的支持集。传统方法面临变量数量平方级的计算开销，尤其在高维情况下难以处理。为此，提出一种基于条件相关矩阵低秩分解的图学习框架，并推导适用于该结构的黎曼优化工具。进一步将该方法具体化为低秩约束的GLasso算法，即高斯图模型的正则化最大似然估计。在合成数据和真实数据上的实验表明，该方法能在维度与性能之间实现高效权衡。
原文摘要 · Abstract (English)
This paper addresses the problem of learning an undirected graph from data gathered at each nodes. Within the graph signal processing framework, the topology of such graph can be linked to the support of the conditional correlation matrix of the data. The corresponding graph learning problem then scales to the squares of the number of variables (nodes), which is usually problematic at large dimension. To tackle this issue, we propose a graph learning framework that leverages a low-rank factorization of the conditional correlation matrix. In order to solve for the resulting optimization problems, we derive tools required to apply Riemannian optimization techniques for this particular structure. The proposal is then particularized to a low-rank constrained counterpart of the GLasso algorithm, i.e., the penalized maximum likelihood estimation of a Gaussian graphical model. Experiments on synthetic and real data evidence that a very efficient dimension-versus-performance trade-off can be achieved with this approach.
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