Sparse recovery from extreme eigenvalues deviation inequalities

Deviation Inequalities on Largest Eigenvalues

In these notes, we survey developments on the asymptotic behavior of the largest eigenvalues of random matrix and random growth models, and describe the corresponding known non-asymptotic exponential bounds. We then discuss some elementary and accessible tools from measure concentration and functional analysis to reach some of these quantitative inequalities at the correct small deviation rate ...

Finding Sparse Cuts via Cheeger Inequalities for Higher Eigenvalues

Cheeger’s fundamental inequality states that any edge-weighted graph has a vertex subset S such that its expansion (a.k.a. conductance of S or the sparsity of the cut (S, S̄)) is bounded as follows: φ(S) def = w(S, S̄) min{w(S), w(S̄)} 6 √ 2λ2, where w is the total edge weight of a subset or a cut and λ2 is the second smallest eigenvalue of the normalized Laplacian of the graph. We study three nat...

Large Shadows from Sparse Inequalities

The d-dimensional Goldfarb cube is a polytope with the property that all its 2 vertices appear on some shadow of it (projection onto a 2-dimensional plane). The Goldfarb cube is the solution set of a system of 2d linear inequalities with at most 3 variables per inequality. We show in this paper that the d-dimensional Klee-Minty cube — constructed from inequalities with at most 2 variables per i...

Isoperimetric Inequalities and Eigenvalues

Isoperimetric Inequalities and Eigenvalues

An upper bound is given on the minimum distance between i subsets of the same size of a regular graph in terms of the i-th largest eigenvalue in absolute value. This yields a bound on the diameter in terms of the i-th largest eigenvalue, for any integer i. Our bounds are shown to be asymptotically tight. A recent result by Quenell relating the diameter, the second eigenvalue, and the girth of a...