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CS359G Lecture 3: Cheeger’s inequality
In which we prove the easy case of Cheeger’s inequality. 1. Expansion and The Second Eigenvalue Let be an undirected -regular graph, its adjacency matrix, its normalized adjacency matrix, and be the...
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CS359G Lecture 4: Spectral Partitioning
In which we prove the difficult direction of Cheeger’s inequality. As in the past lectures, consider an undirected -regular graph , call its adjacency matrix, and its scaled adjacency matrix. Let be...
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The Cheeger inequality in manifolds
Readers of in theory have heard about Cheeger’s inequality a lot. It is a relation between the edge expansion (or, in graphs that are not regular, the conductance) of a graph and the second smallest...
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Proof of the Cheeger inequality in manifolds
Having (non-rigorously) defined the Laplacian operator in manifolds in the previous post, we turn to the proof of the Cheeger inequality in manifolds, which we restate below. Theorem 1 (Cheeger’s...
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CS294 Lecture 3: Cheeger Inequalities
In which we generalize the notion of normalized Laplacian to irregular graphs, we extend the basic spectral graph theory results from last lecture to irregular graphs, and we prove the easy direction...
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CS294 Lecture 4: Cheeger Inequalities cont’d
In which we finish the proof of Cheeger’s inequalities. It remains to prove the following statement. Lemma 1 Let be a vector with non-negative entries. Then there is a such that We will provide a...
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CS294 Lecture 6: Higher-order Cheeger Inequality
In which we state an analog of Cheeger’s inequalities for the -th smallest Laplacian eigenvalue, and we discuss the connection between this result and the analysis of spectral partitioning algorithms...
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CS294 Lecture 7: Higher order Cheeger inequality, cont’d
In which we state an analog of Cheeger’s inequalities for the -th smallest Laplacian eigenvalue, and we discuss the connection between this result and the analysis of spectral partitioning algorithms...
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Buser Inequalities in Graphs
As life is tentatively returning to normal, I would like to once again post technical material here. Before returning to online optimization, I would like to start with something from 2015 that we...
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