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 solutions starting from arbitrary Borel probability measures and prove that the density $\rho_t$ belongs to $L^\infty$ for any $t>0$. We also show that the H\"older norm can grow exponentially in time. On the flat torus ${\mathbb{T}}^\mathsf{d}$, we prove a global metric PL inequality for every finite-Coulomb-energy source and nearly uniform target. For general bounded, uniformly positive targets, we prove exponential decay of the squared MMD without requiring a lower bound on the initial data, using a defective PL inequality. We also prove that the usual PL inequality may fail when the target vanishes only at one point and that, when $\mathsf{d}\ge2$, no PL constant can hold uniformly over all targets satisfying a prescribed lower bound. On ${\mathbb{R}}^\mathsf{d}$, for $\mathsf{d}\ge2$, under radial symmetry, source-support inclusion, and target-positivity assumptions, we establish a PL inequality and exponential convergence. On the unrestricted whole-space class, neither a multiplicative squared-MMD decay modulus uniform over the initial datum nor a global PL inequality can hold. Finally, in every dimension and in both spatial settings, we prove that every Lagrangian critical point coincides with the target when $(\rho-\mu)^+$ is absolutely continuous. In dimension two, the energy supplies uniform tightness. This implies that if our constructed solutions have finite energy at some positive time, then they converge to the target narrowly and strongly in negative-order Sobolev spaces.
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 $\operatorname{tr}({\mathbf R}^2)$ satisfies a central limit theorem (CLT) and the centering and scaling sequences are universal in the sense that they do not depend on the entry distribution. Under symmetry and regular variation assumption with index $\alpha$ and any growth rate of the dimension, we prove that the universal CLT remains valid for $\alpha>3$. For $\alpha<3$, we identify a critical dimension growth at which the fluctuations of $\operatorname{tr}({\mathbf R}^2)$ become non-Gaussian. Moreover, if the dimension $p$ grows faster and $\alpha\le 3$ we establish a non-universal CLT with norming sequences depending on the value of $\alpha$. Our findings are illustrated in a simulation study.
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 absolute correlation is of order $\sqrt{\log n/n}$, its Frobenius distance is asymptotic to $\sqrt n$, its empirical spectral distribution converges to the Marchenko-Pastur law with ratio one, and its smallest eigenvalue has the exact $\operatorname{Beta}(1,d)$ distribution, where $d=n(n-1)/2$, and is therefore of order $n^{-2}$. More generally, distinct off-diagonal entries are exactly pairwise independent under every $\operatorname{LKJ}(\eta)$ law. For the uniform law, this yields a Chen-Stein proof of the extreme-correlation point-process limit and an $O(n^{-1})$ total-variation bound for finite-dimensional exceedance counts relative to Poisson laws with their exact finite-$n$ means. We also identify two distinct scales: $\eta_n\asymp n$ alters the limiting spectrum, whereas $\eta_n\asymp n^2$ is needed to keep the Frobenius distance bounded. Finally, for a bounded, centered i.i.d. off-diagonal specification, projection to the nearest correlation matrix incurs a squared repair cost asymptotically at least one-half of the squared Frobenius norm of its off-diagonal part.
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 show that the same characterization holds if the density is first composed with a function $F$ which is bi-Lipschitz on the interval $[c_0^{-1},c_0]$ determined by the AD-regularity constant $c_0$. The main example is $F=\log$, introduced in [Le], for which the square function takes the scale-invariant form $\Delta_\mu^{\log}(x,r) = \log\bigl(\mu(B(x,r))/\mu(B(x,2r))\bigr)+n\log 2$. We give a complete proof, extend the statement to the smooth square functions of [CGLT], where the density is replaced by the convolution of $\mu$ with a Gaussian or a more general radial kernel, discuss what happens when $F$ is not bi-Lipschitz, and treat the case $\mu(\mathbb{R}^d)<\infty$, where the behavior of $F$ near zero enters in only one of the two implications. We also show that the qualitative characterization of $n$-rectifiable measures by Tolsa and Toro [TT], in terms of the same square function at $\mu$-almost every point, holds after composition with any locally bi-Lipschitz $F$. This requires neither AD-regularity nor doubling, and for $F=\log$ the condition $\lim_{r\to0}\Delta_\mu(x,r)=0$ becomes $\lim_{r\to0}\mu(B(x,r))/\mu(B(x,2r))=2^{-n}$.
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
Let $\Pi$ be a $k\times n$ sparse random matrix. For a fixed $r$-dimensional subspace $V\subset{\mathbb R}^n$, let $U_V:{\mathbb R}^r\to{\mathbb R}^n$ denote an isometry from ${\mathbb R}^r$ onto $V$. The product $\Pi U_V$ is a central model in randomized dimension reduction and has been studied primarily through trace and Gaussian comparison inequalities. In this work, we develop an approach to the spectral norm of the matrix product $\Pi U_V$, based on entropy estimates for level sets of vectors $x\in V$. Combining the method with existing estimates, we show the following. Assume that \[ k\ge C\,r(\log\log r)^2,\qquad p\ge (\log k)/k. \] Let $\Pi$ be a $k\times n$ matrix with i.i.d. entries equidistributed with the product $b\,\xi$, where $b$ is a Bernoulli($p$) random variable and $\xi$ is mean-zero, independent of $b$, and satisfies $|\xi|\le1$ almost surely. Then with high probability \[ \|\Pi U_V\|\le C\sqrt{kp}. \] Matching results hold for other random models with negatively associated entries.
