The spectral method uses a decoupling argument recently introduced by Kaushik, Romberg, and Muthukumar to control nonlinear error terms to control nonlinear error terms.
Abstract
We study the problem of recovering latent inner products from a random geometric graph with anisotropic Gaussian latent points. More precisely, for an i.i.d. sample $x_1, \dots, x_n \sim N(0,\Sigma)$ where $\Sigma \in \mathbb{R}^{d \times d}$, an edge $(i,j)$ is present in the graph if and only if $\langle x_i, x_j \rangle \ge \zeta$ for a threshold $\zeta$. We assume the threshold $\zeta$ to be chosen such that the average edge density of the graph is of constant order. To address the undesired degree fluctuations amplified by the anisotropy of the latent points, we consider the doubly centered adjacency matrix of the graph, and estimate the latent inner products using a rank-$d$ spectral approximation of the doubly centered matrix. The estimator obtains a mean squared error with a rate involving the stable rank of the covariance matrix $\Sigma$. Notably, the rate of estimation matches the state of the art for the isotropic case $\Sigma = I_d$, and permits an ill-conditioned covariance matrix with a diverging condition number. The analysis of the spectral method proceeds via the entrywise Hermite expansion of the doubly centered adjacency matrix with respect to the latent inner products. Instead of the standard trace method, it uses a decoupling argument recently introduced by Kaushik, Romberg, and Muthukumar (2025) to control nonlinear error terms.
We study sparse threshold random geometric graphs generated by high-dimensional spherical or Gaussian latent vectors. Although each edge has marginal probability $p$, shared latent variables make the adjacency entries dependent. At the connectivity scale $np=\Omega(\log n)$, the spherical adjacency matrix satisfies, with high probability,$\|A-\mathbb E A\|_{\mathrm{op}}=O\left(\sqrt{np\log n}+np\tau\right)$, where $\tau$ is the cap threshold; an analogous estimate holds for Gaussian vectors after controlling radial fluctuations. This sharpens the spectral bound in Liu, Mohanty, Schramm, and Yang (2023) under weaker assumptions and strengthens the global-synchronization guarantee of Abdalla, Bandeira, and Invernizzi (2024) for the homogeneous Kuramoto model. The leading eigenspace also estimates the latent geometry. When $np\gg\log n$, vector and relative Gram-matrix errors vanish for$\log(1/p)\ll d\ll np\log(1/p)/\log n$ in the spherical model and $\log^2(1/p)\log n\ll d\ll np\log(1/p)/\log n$ in the Gaussian model, improving the recovery conditions of Li and Schramm (2023). For the Gaussian mixture block model introduced there, a polynomial-time semidefinite program gives, to our knowledge, the first exact-recovery guarantee at the connectivity scale in a moderate-separation regime. At much larger separation, fixed edge density creates isolated vertices and makes exact recovery impossible. Our reusable decoupling and matrix concentration framework avoids trace-moment methods and applies broadly to random graph models with latent vectors.
V. ManuelFernandez, Yizhe Zhu· arXiv.org· 1 citation
We study whether an observed graph can distinguish a random graph generated by latent geometry from one with independent edges. In the geometric model, vertex positions are independent and uniform on a high-dimensional sphere. Conditional on these positions, edges occur independently, with probabilities determined by a connection function $K$ of the inner products of their endpoints. The comparison model is an Erd\H{o}s-R\'enyi graph with the same edge density. A general spectral conjecture asserts that if the cubic spectral trace, which corresponds to the signed triangle mean, is sufficiently small in the sense that \[ n^3[\operatorname{tr}(\kappa^3)]^2\longrightarrow0, \] where $n$ is the number of vertices and $\kappa$ is the centered and standardized spherical kernel operator, then the total variation distance between the two graph distributions tends to zero. Consequently, the lower limit of the sum of the two error probabilities of any sequence of tests is at least one. We give a counterexample to the formulation allowing dimension-dependent connection functions without monotonicity or a common-sign condition on the spectrum. Our quadratic connection functions are uniformly bounded away from zero and one. Their cubic trace vanishes identically through cancellation between positive and negative eigenvalues, whereas their quartic trace is strictly positive. When $d=\max\{3,\lfloor n^{1/20}\rfloor\}$, a signed four-cycle test has a sum of error probabilities tending to zero, and the total variation distance instead tends to one. A perturbation making the cubic trace strictly nonzero still satisfies the stated cubic-trace condition and yields strong detection. The construction and detection result follow, respectively, from a finite-rank spectral decomposition of the spherical kernel and estimates of the mean and variance of the four-cycle statistic.
It is proved that sketching dimension m = O(k^{3/2}/\epsilon^2) suffices for subspace embedding with a Khatri-Rao sketching matrix with any fixed order $d$.
The spherical random geometric graph $G(n,d,p)$ is obtained by sampling $n$ independent points uniformly on the unit sphere $\mathbb{S}^{d-1}\subseteq\mathbb{R}^d$ and joining pairs of points which are sufficiently close, where the threshold is chosen so that the edge probability is $p$. The central question related to this model, and to a broad class of other models, is the following: when does the underlying geometry affect the resulting graph in a way which makes it distinguishable from the Erd\H{o}s--R\'enyi random graph $G(n,p)$, as measured in total variation distance? The precise answer to this question was conjectured by Bubeck, Ding, Eldan, and R\'acz, who predicted that $G(n,d,p)$ and $G(n,p)$ are indistinguishable precisely when $d \gg n^3p^3(\log p^{-1})^3$, and provided a test for distinguishing these models in the low-dimensional regime. Although this conjecture attracted considerable attention from researchers in probability, theoretical computer science, and high-dimensional statistics, it was previously fully proved only in the constant-density case. In this paper, we resolve the distinguishability conjecture in the broad range $1/3 \geq p \geq n^{-1/5} \text{polylog}(n)$. The key ingredient of our proof is a stronger statement which gives a precise asymptotic formula for the probability that $G(n,d,p)$ realizes a prescribed graph $H$: above the conjectured threshold, this probability is at most $(1+o(1))$ times the corresponding probability for $G(n,p)$, with the signed triangle count of $H$ appearing as the leading correction term.
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.
Let $X_1, X_2, \ldots, X_n$ be independent random vectors. For a directed graph $G=(V,E)$ with vertex set $V=\{1,2,\ldots,n\}$ and a collection of bivariate kernels $\{h_e:e\in E\}$, we consider \[ U=\sum_{e=(i,j)\in E} h_e(X_i,X_j). \] This framework generalizes incomplete U-statistics by allowing the random vectors to be non-identically distributed, the kernels to be asymmetric and edge-dependent, and the sampling structure to be specified by an arbitrary graph. We derive several concentration inequalities for $U-\mathbb{E}U$. The main proof strategy exploits edge-coloring results from graph theory and relates the tail behavior of $U$ to the chromatic index of $G$. This approach is elementary, transparent, and readily adaptable to broader settings, including U-statistics of order $m>2$ and statistics involving doubly indexed random vectors.
Z. Ke· 1 citation
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