Data-Driven Methods for Learning Sparse Graphical Models (November 30, 2017)
Somayeh Sojoudi (EECS and Mechanical Engineering, UC Berkeley)
Learning models from data has a significant impact on many disciplines, including computer vision, medical imaging, social networks, neuroscience and signal processing. In the network inference problem, one may model the relationships between the network components through an underlying inverse covariance matrix. Learning this graphical model is often challenged by the fact that only a small number of samples are available. Despite the popularity of graphical lasso for solving this problem, there is not much known about the properties of this statistical method as an optimization algorithm. In this talk, we will develop new notions of sign-consistent matrices and inverse-consistent matrices to obtain key properties of graphical lasso. In particular, we will prove that although the complexity of solving graphical lasso is high, the sparsity pattern of its solution has a simple formula if a sparse graphical model is sought. Besides graphical lasso, there are several techniques for learning graphical models. We will design an optimization-based mathematical framework to study the performance of various techniques. We will illustrate our results in different case studies.
License: CC BY-NC-SA 4.0
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🎯 Key Takeaways for quick navigation: 00:00 🗣️ The video introduces data-driven methods for learning sparse graphical models. 01:29 📊 Graphical models are used to model interactions in various systems, such as transportation networks, neuroscience, and power systems. 05:13 💡 Solving large-scale graphical model problems efficiently is essential, as problems can involve millions of parameters. 06:10 🧩 Graphical Lasso is an optimization framework for learning sparse Gaussian graphical models, promoting interpretability. 09:50 ⚙️ Efficient methods for solving the Graphical Lasso problem are discussed, including block coordinate descent. 14:38 🔄 Graphical Lasso and thresholding methods can provide similar graph structures under specific conditions. 17:01 🧐 Conditions for equivalence between Graphical Lasso and thresholding methods are defined. 19:24 📈 Closed-form solutions for Graphical Lasso are presented for both exact and approximate solutions. 23:51 🧠 Application examples using functional MRI data and traffic data are used to validate the presented methods. 26:28 🌀 Extensions of the methods to chordal graphs (graphs containing cycles of four or more vertices) are discussed. 26:41 📊 Chordal graphs become non-chordal when certain edges are removed, and quarter completion is a method to make a non-chordal graph chordal. 27:23 📚 A clique in a graph is a subset of vertices where every pair of vertices is adjacent, and they are essential for defining tree width in graph theory. 28:18 🌳 Tree width of a graph is related to the size of the largest clique in its quarter completions, providing insights into graph structure. 29:53 🧮 Conditions for applying graphical lasso and maximum determinant matrix completion depend on the size of the network, indicating limitations for smaller networks. 30:34 📈 In some cases, graphical lasso can be reduced to a maximum determinant matrix completion problem, allowing efficient computation of the Gaussian graphical model. 32:05 🔄 Block-coordinate descent can be used to speed up graphical lasso optimization by minimizing blocks of variables iteratively. 34:45 🤖 Machine learning methods for learning Gaussian graphical models require extra information about the graph structure, which may not be available in some applications. 36:33 📉 Cross-validation can help in selecting the best method for learning graph structures and determining the optimal number of edges, mitigating overfitting issues. 38:24 🧠 The approach used in functional MRI data can combine multiple individual networks to find a union graph, revealing insights into brain connectivity. 47:23 🧩 Cross-validation ML error curves can provide an upper bound on the best number of edges that methods can find in graph learning tasks.