The variance of nonparametric estimators is typically insensitive to the regularity of the object being estimated. We establish such a property for the spectra of graph Laplacian matrices at a fixed bandwidth $h>0$. Specifically, given $n$ i.i.d. samples from a probability measure $\mu$ on a Polish metric space, we compare the eigenvalues of the empirical weighted Laplacian operator $\Delta_{\mu_n}^h$ to those of the population counterpart $\Delta_\mu^h$ under a spectral gap condition, bounding the relative error by $1/\sqrt{nv_\mu(h)}$ for eigenvalues of order smaller than $h^{-2}$, where $v_\mu(h)$ is the smallest mass of a ball of radius $h$. This bound requires very weak regularity conditions on $\mu$: it is satisfied if $\mu$ belongs to the class of coarse PI measures that we introduce. This class contains measures on metric graphs, spaces with sufficiently regular boundaries, corners, or branch points, together with discretizations or thickenings of these at scale $O(h)$. Even for measures having densities of regularity $s>2$ on manifolds (the only known case so far), our bound improves on the state-of-the-art by shaving off logarithmic factors.
We study the long-time behavior of the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) between a probability measure $\rho$ and a target measure $\mu$, where the underlying kernel is given by a Coulomb potential. For $L^\infty$ target densities $\mu$, we establish the existence of global weak so...
Antonin Chodron de Courcel, Matthew Rosenzweig· 0 citations
We provide limit theory for the trace of the squared sample correlation matrix $\mathbf R$, constructed from $n$ observations of a $p$-dimensional random vector with iid components. If the entries have finite fourth moment and $p$ and $n$ grow proportionally, it is known that $\operatorname{tr}({\mathbf R}^2)$ satisfie...
The set of $n\times n$ correlation matrices, known as the elliptope, has volume decaying at the super-exponential rate $\exp\{-\tfrac14 n^2\log n\}$. We characterize where this vanishing volume concentrates. A uniform draw is entrywise close to the identity yet globally far from it and nearly singular: its maximum abso...
Let $\mu$ be an $n$-AD-regular measure in $\mathbb{R}^d$. Chousionis, Garnett, Le and Tolsa [CGLT] proved that $\mu$ is uniformly $n$-rectifiable if and only if the square function built from the density differences $\Delta_\mu(x,r)=\mu(B(x,r))/r^n-\mu(B(x,2r))/(2r)^n$ satisfies a Carleson condition. In this paper we s...
It is proved that membership in the approximation space $k_t$ is equivalent to polynomial decay of the best $n$-term approximation error, which is equivalent to polynomial decay of the best $n$-term approximation error.
Abhishake Rastogi, T. Bubba, T. Helin et al.· 0 citations
This work develops an approach to the spectral norm of the matrix product $\Pi U_V$, based on entropy estimates for level sets of vectors $x\in V$, and shows that matching results hold for other random models with negatively associated entries.
K. Tikhomirov· arXiv.org· 0 citations
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