A numerical study of the spectral gap
We present a numerical study of the spectral gap of the Dirichlet Laplacian, γ(K) = λ2(K) − λ1(K), of a planar convex region K. Besides providing supporting numerical evidence for the long-standing gap conjecture that γ(K) ≥ 3π2/d2(K), where d(K) denotes the diameter of K, our study suggests new types of bounds and several conjectures regarding the dependence of the gap not only on the diameter, but also on the perimeter and the area. One of these conjectures is a stronger version of the gap conjecture mentioned above. A similar study is carried out for the quotient of the first two Dirichlet eigenvalues.
In a recent paper, the authors introduced the notion of sample width for binary classifiers defined on the set of real numbers. It was shown that the performance of such classifiers could be quantified in terms of this sample width. This paper considers how to adapt the idea of sample width so that it can be applied in cases where the classifiers are defined on some finite metric space. We discuss how to employ a greedy set-covering heuristic to bound generalization error. Then, by relating the learning problem to one involving certain graph-theoretic parameters, we obtain generalization error bounds that depend on the sample width and on measures of `density' of the underlying metric space.