Determine whether graph-Laplacian eigenvalue estimators can be optimally debiased
Determine whether a debiasing method for graph-Laplacian-based estimators of the eigenvalues of the limiting differential operator can achieve rates faster than $1/\sqrt{n v_\mu(h)}$ under smoothness assumptions, and prove whether such debiasing is impossible for general coarse PI measures so that $1/\sqrt{n v_\mu(h)}$ is minimax on that class.
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The rates derived in this paper suggest that the rate $1/\sqrt{nv_\mu(h)}$ is not minimax for eigenvalue estimation, and that graph Laplacian-based estimators could be debiased. We conjecture that such a debiasing method requires some smoothness, and is therefore impossible for a general coarse PI measure, suggesting that the rate $1/\sqrt{nv_\mu(h)}$ becomes minimax on this larger class. This would constitute another advantage of considering the class of coarse PI measures, which would be in this sense the ``right'' class on which to study the estimator $\lambda_{\mu_n}h$. We leave this interesting question to future work.