arXiv subjects

Jiaoyang Huang

Publications and source records attributed to Jiaoyang Huang.

At least 37 records · Page 2Linked to original sources

arXiv2025

Gaussian Waves and Edge Eigenvectors of Random Regular Graphs

Backhausz and Szegedy (2019) demonstrated that the almost eigenvectors of random regular graphs converge to Gaussian waves with variance $0\leq σ^2\leq 1$. In this paper, we present an alternative proof of this result for the edge eigenvectors of random regular graphs, establishing that the variance must be $σ^2=1$. Furthermore, we show that the eigenvalues and eigenvectors are asymptotically independent. Our approach introduces a simple framework linking the weak convergence of the imaginary part of the Green's function to the convergence of eigenvectors, which may be of independent interest.

math.PR↗

arXiv2025

Spectrum of Random $d$-regular Graphs Up to the Edge

Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\geq3$. We prove that, with probability $1-N^{-1+{\varepsilon}}$ for any ${\varepsilon} >0$, the following two properties hold as $N \to \infty$ provided that $d\geq3$: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten-McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in $N$, i.e. $λ_2, |λ_N|\leq 2+N^{-c}$. (ii) All eigenvectors of random $d$-regular graphs are completely delocalized.

math.PR↗

arXiv2025

Ramanujan Property and Edge Universality of Random Regular Graphs

We consider the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $d\geq 3$ and denote its eigenvalues as $λ_1=d/\sqrt{d-1}\geq λ_2\geqλ_3\cdots\geq λ_N$. We establish the following two results as $N\rightarrow \infty$. (i) With high probability, all eigenvalues are optimally rigid, up to an additional $N^{{\rm o}(1)}$ factor. Specifically, the fluctuations of bulk eigenvalues are bounded by $N^{-1+{\rm o}(1)}$, and the fluctuations of edge eigenvalues are bounded by $N^{-2/3+{\rm o}(1)}$. (ii) Edge universality holds for random $d$-regular graphs. That is, the distributions of $λ_2$ and $-λ_N$ converge to the Tracy-Widom$_1$ distribution associated with the Gaussian Orthogonal Ensemble. As a consequence, for sufficiently large $N$, approximately $69\%$ of $d$-regular graphs on $N$ vertices are Ramanujan, meaning $\max\{λ_2,|λ_N|\}\leq 2$.

math.PR↗

arXiv2024

Uncertainty of high-dimensional genetic data prediction with polygenic risk scores

In many predictive tasks, there are a large number of true predictors with weak signals, leading to substantial uncertainties in prediction outcomes. The polygenic risk score (PRS) is an example of such a scenario, where many genetic variants are used as predictors for complex traits, each contributing only a small amount of information. Although PRS has been a standard tool in genetic predictions, its uncertainty remains largely unexplored. In this paper, we aim to establish the asymptotic normality of PRS in high-dimensional predictions without sparsity constraints. We investigate the popular marginal and ridge-type estimators in PRS applications, developing central limit theorems for both individual-level predicted values (e.g., genetically predicted human height) and cohort-level prediction accuracy measures (e.g., overall predictive $R$-squared in the testing dataset). Our results demonstrate that ignoring the prediction-induced uncertainty can lead to substantial underestimation of the true variance of PRS-based estimators, which in turn may cause overconfidence in the accuracy of confidence intervals and hypothesis testing. These findings provide key insights omitted by existing first-order asymptotic studies of high-dimensional sparsity-free predictions, which often focus solely on the point limits of predictive risks. We develop novel and flexible second-order random matrix theory results to assess the asymptotic normality of functionals with a general covariance matrix, without assuming Gaussian distributions for the data. We evaluate our theoretical results through extensive numerical analyses using real data from the UK Biobank. Our analysis underscores the importance of incorporating uncertainty assessments at both the individual and cohort levels when applying and interpreting PRS.

stat.ME↗

arXiv2024

The Spectral Distribution of Random Graphs with Given Degree Sequences

In this article, we study random graphs with a given degree sequence $d_1, d_2, \cdots, d_n$ from the configuration model. We show that under mild assumptions of the degree sequence, the spectral distribution of the normalized Laplacian matrix of such random graph converges in distribution to the semicircle distribution as the number of vertices $n\rightarrow \infty$. This extends work by McKay (1981) and Tran, Vu and Wang (2013) which studied random regular graphs ($d_1=d_2=\cdots=d_n=d$). Furthermore, we extend the assumption to show that a slightly more general condition is equivalent to the weak convergence to semicircle distribution. The equivalence is also illustrated by numerical simulations.

math.PR↗

arXiv2024

Extremal Eigenvalues of Random Kernel Matrices with Polynomial Scaling

We study the spectral norm of random kernel matrices with polynomial scaling, where the number of samples scales polynomially with the data dimension. In this regime, Lu and Yau (2022) proved that the empirical spectral distribution converges to the additive free convolution of a semicircle law and a Marcenko-Pastur law. We demonstrate that the random kernel matrix can be decomposed into a "bulk" part and a low-rank part. The spectral norm of the "bulk" part almost surely converges to the edge of the limiting spectrum. In the special case where the random kernel matrices correspond to the inner products of random tensors, the empirical spectral distribution converges to the Marcenko-Pastur law. We prove that the largest and smallest eigenvalues converge to the corresponding spectral edges of the Marcenko-Pastur law.

math.PR↗

arXiv2024

Efficient, Multimodal, and Derivative-Free Bayesian Inference With Fisher-Rao Gradient Flows

In this paper, we study efficient approximate sampling for probability distributions known up to normalization constants. We specifically focus on a problem class arising in Bayesian inference for large-scale inverse problems in science and engineering applications. The computational challenges we address with the proposed methodology are: (i) the need for repeated evaluations of expensive forward models; (ii) the potential existence of multiple modes; and (iii) the fact that gradient of, or adjoint solver for, the forward model might not be feasible. While existing Bayesian inference methods meet some of these challenges individually, we propose a framework that tackles all three systematically. Our approach builds upon the Fisher-Rao gradient flow in probability space, yielding a dynamical system for probability densities that converges towards the target distribution at a uniform exponential rate. This rapid convergence is advantageous for the computational burden outlined in (i). We apply Gaussian mixture approximations with operator splitting techniques to simulate the flow numerically; the resulting approximation can capture multiple modes thus addressing (ii). Furthermore, we employ the Kalman methodology to facilitate a derivative-free update of these Gaussian components and their respective weights, addressing the issue in (iii). The proposed methodology results in an efficient derivative-free sampler flexible enough to handle multi-modal distributions: Gaussian Mixture Kalman Inversion (GMKI). The effectiveness of GMKI is demonstrated both theoretically and numerically in several experiments with multimodal target distributions, including proof-of-concept and two-dimensional examples, as well as a large-scale application: recovering the Navier-Stokes initial condition from solution data at positive times.

cs.LG↗

arXiv2024

Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance

Sampling a probability distribution with an unknown normalization constant is a fundamental problem in computational science and engineering. This task may be cast as an optimization problem over all probability measures, and an initial distribution can be evolved to the desired minimizer dynamically via gradient flows. Mean-field models, whose law is governed by the gradient flow in the space of probability measures, may also be identified; particle approximations of these mean-field models form the basis of algorithms. The gradient flow approach is also the basis of algorithms for variational inference, in which the optimization is performed over a parameterized family of probability distributions such as Gaussians, and the underlying gradient flow is restricted to the parameterized family. By choosing different energy functionals and metrics for the gradient flow, different algorithms with different convergence properties arise. In this paper, we concentrate on the Kullback-Leibler divergence after showing that, up to scaling, it has the unique property that the gradient flows resulting from this choice of energy do not depend on the normalization constant. For the metrics, we focus on variants of the Fisher-Rao, Wasserstein, and Stein metrics; we introduce the affine invariance property for gradient flows, and their corresponding mean-field models, determine whether a given metric leads to affine invariance, and modify it to make it affine invariant if it does not. We study the resulting gradient flows in both probability density space and Gaussian space. The flow in the Gaussian space may be understood as a Gaussian approximation of the flow. We demonstrate that the Gaussian approximation based on the metric and through moment closure coincide, establish connections between them, and study their long-time convergence properties showing the advantages of affine invariance.

stat.ML↗

arXiv2024

Asymptotics of Symmetric Polynomials: A Dynamical Point of view

In this paper we study the asymptotic behavior of the (skew) Macdonald and Jack symmetric polynomials as the number of variables grows to infinity. We characterize their limits in terms of certain variational problems. As an intermediate step, we establish a large deviation principle for the $θ$ analogue of non-intersecting Bernoulli random walks. When $θ=1$, these walks are equivalent to random Lozenges tilings of strip domains, where the variational principle (with general domains and boundary conditions) has been proven in the seminal work by Cohn, Kenyon, and Propp. Our result gives a new argument of this variational principle, and also extends it to non-intersecting $θ$-Bernoulli random walks for any $θ\in (0,\infty)$. Remarkably, the rate functions remain identical, differing only by a factor of $1/θ$.

math.PR↗

arXiv2024

Optimal Eigenvalue Rigidity of Random Regular Graphs

Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\geq 3$, and denote the eigenvalues as $λ_1=d/\sqrt{d-1}\geq λ_2\geqλ_3\cdots\geq λ_N$. We prove that the optimal (up to an extra $N^{{\rm o}_N(1)}$ factor, where ${\rm o}_N(1)$ can be arbitrarily small) eigenvalue rigidity holds. More precisely, denote $γ_i$ as the classical location of the $i$-th eigenvalue under the Kesten-Mckay law in decreasing order. Then with probability $1-N^{-1+{\rm o}_N(1)}$, \begin{align*} |λ_i-γ_i|\leq \frac{N^{{\rm o}_N(1)}}{N^{2/3} (\min\{i,N-i+1\})^{1/3}},\quad \text{ for all } i\in \{2,3,\cdots,N\}. \end{align*} In particular, the fluctuations of extreme eigenvalues are bounded by $N^{-2/3+{\rm o}_N(1)}$. This gives the same order of fluctuation as for the eigenvalues of matrices from the Gaussian Orthogonal Ensemble.

math.PR↗

arXiv2024

Sampling via Gradient Flows in the Space of Probability Measures

Sampling a target probability distribution with an unknown normalization constant is a fundamental challenge in computational science and engineering. Recent work shows that algorithms derived by considering gradient flows in the space of probability measures open up new avenues for algorithm development. This paper makes three contributions to this sampling approach by scrutinizing the design components of such gradient flows. Any instantiation of a gradient flow for sampling needs an energy functional and a metric to determine the flow, as well as numerical approximations of the flow to derive algorithms. Our first contribution is to show that the Kullback-Leibler divergence, as an energy functional, has the unique property (among all f-divergences) that gradient flows resulting from it do not depend on the normalization constant of the target distribution. Our second contribution is to study the choice of metric from the perspective of invariance. The Fisher-Rao metric is known as the unique choice (up to scaling) that is diffeomorphism invariant. As a computationally tractable alternative, we introduce a relaxed, affine invariance property for the metrics and gradient flows. In particular, we construct various affine invariant Wasserstein and Stein gradient flows. Affine invariant gradient flows are shown to behave more favorably than their non-affine-invariant counterparts when sampling highly anisotropic distributions, in theory and by using particle methods. Our third contribution is to study, and develop efficient algorithms based on Gaussian approximations of the gradient flows; this leads to an alternative to particle methods. We establish connections between various Gaussian approximate gradient flows, discuss their relation to gradient methods arising from parametric variational inference, and study their convergence properties both theoretically and numerically.

stat.ML↗

arXiv2024

Dynamical Loop Equation

We introduce dynamical versions of loop (or Dyson-Schwinger) equations for large families of two--dimensional interacting particle systems, including Dyson Brownian motion, Nonintersecting Bernoulli/Poisson random walks, $β$--corners processes, uniform and Jack-deformed measures on Gelfand-Tsetlin patterns, Macdonald processes, and $(q,κ)$-distributions on lozenge tilings. Under technical assumptions, we show that the dynamical loop equations lead to Gaussian field type fluctuations. As an application, we compute the limit shape for $(q,κ)$--distributions on lozenge tilings and prove that their height fluctuations converge to the Gaussian Free Field in an appropriate complex structure.

math.PR↗

arXiv2023

Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble

We consider uniformly random lozenge tilings of simply connected polygons subject to a technical assumption on their limit shape. We show that the edge statistics around any point on the arctic boundary, that is not a cusp or tangency location, converge to the Airy line ensemble. Our proof proceeds by locally comparing these edge statistics with those for a random tiling of a hexagon, which are well understood. To realize this comparison, we require a nearly optimal concentration estimate for the tiling height function, which we establish by exhibiting a certain Markov chain on the set of all tilings that preserves such concentration estimates under its dynamics.

math.PR↗

arXiv2023

Pearcey universality at cusps of polygonal lozenge tiling

We study uniformly random lozenge tilings of general simply connected polygons. Under a technical assumption that is presumably generic with respect to polygon shapes, we show that the local statistics around a cusp point of the arctic curve converge to the Pearcey process. This verifies the widely predicted universality of edge statistics in the cusp case. Together with the smooth and tangent cases proved in Aggarwal-Huang and Aggarwal-Gorin, these are believed to be the three types of edge statistics that can arise in a generic polygon. Our proof is via a local coupling of the random tiling with non-intersecting Bernoulli random walks (NBRW). To leverage this coupling, we establish an optimal concentration estimate for the tiling height function around the cusp. As another step and also a result of potential independent interest, we show that the local statistics of NBRW around a cusp converge to the Pearcey process when the initial configuration consists of two parts with proper density growth, via careful asymptotic analysis of the determinantal formula.

math.PR↗

arXiv2023

Edge Rigidity of Dyson Brownian Motion with General Initial Data

In this paper, we study the edge behavior of Dyson Brownian motion with general $β$. Specifically, we consider the scenario where the averaged initial density near the edge, on the scale $η_*$, is lower bounded by a square root profile. Under this assumption, we establish that the fluctuations of extreme particles are bounded by $(\log n)^{{\rm O}(1)}n^{-2/3}$ after time $C\sqrt{η_*}$. Our result improves previous edge rigidity results from [1,24] which require both lower and upper bounds of the averaged initial density. Additionally, combining with [24], our rigidity estimates are used to prove that the distribution of extreme particles converges to the Tracy-Widom $β$ distribution in short time.

math.PR↗

arXiv2023

Local Statistics and Concentration for Non-intersecting Brownian Bridges With Smooth Boundary Data

In this paper we consider non-intersecting Brownian bridges, under fairly general upper and lower boundaries, and starting and ending data. Under the assumption that these boundary data induce a smooth limit shape (without empty facets), we establish two results. The first is a nearly optimal concentration bound for the Brownian bridges in this model. The second is that the bulk local statistics of these bridges along any fixed time converge to the sine process.

math.PR↗

arXiv2023

Edge Universality of Random Regular Graphs of Growing Degrees

We consider the statistics of extreme eigenvalues of random $d$-regular graphs, with $N^{\mathfrak c}\leq d\leq N^{1/3-{\mathfrak c}}$ for arbitrarily small ${\mathfrak c}>0$. We prove that in this regime, the fluctuations of extreme eigenvalues are given by the Tracy-Widom distribution. As a consequence, about 69% of $d$-regular graphs have all nontrivial eigenvalues bounded in absolute value by $2\sqrt{d-1}$.

math.PR↗

arXiv2023

How Does Information Bottleneck Help Deep Learning?

Numerous deep learning algorithms have been inspired by and understood via the notion of information bottleneck, where unnecessary information is (often implicitly) minimized while task-relevant information is maximized. However, a rigorous argument for justifying why it is desirable to control information bottlenecks has been elusive. In this paper, we provide the first rigorous learning theory for justifying the benefit of information bottleneck in deep learning by mathematically relating information bottleneck to generalization errors. Our theory proves that controlling information bottleneck is one way to control generalization errors in deep learning, although it is not the only or necessary way. We investigate the merit of our new mathematical findings with experiments across a range of architectures and learning settings. In many cases, generalization errors are shown to correlate with the degree of information bottleneck: i.e., the amount of the unnecessary information at hidden layers. This paper provides a theoretical foundation for current and future methods through the lens of information bottleneck. Our new generalization bounds scale with the degree of information bottleneck, unlike the previous bounds that scale with the number of parameters, VC dimension, Rademacher complexity, stability or robustness. Our code is publicly available at: https://github.com/xu-ji/information-bottleneck

cs.LG↗