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Lingjiong Zhu

Publications and source records attributed to Lingjiong Zhu.

At least 19 recordsLinked to original sources
arXiv2026

Fluctuations, Moderate and Large Deviations about Hydrodynamic Limits of Space-Time Hawkes Processes

We study Gaussian fluctuations, moderate and large deviations around the high-density hydrodynamic limit of critical and nearly critical space-time Hawkes processes initialized by a prescribed Poisson field of past events, with no subsequent immigration. Under a finite mean reproduction delay and centered displacements with finite spatial covariance, a positive cluster-transform estimate yields a distribution-valued central limit theorem (CLT) and a moderate deviation principle (MDP) throughout the moderate range. For finite nonzero mass of the initial profile, we obtain a full large deviation principle (LDP) for the space-time event measure in the vague topology; its good rate function is a quadratic branching-control energy with a prescribed initial trace and a finite-mass domain. The lower bound uses an explicit approximation by smooth reproduction controls, and the upper bound uses a microscopic exponential martingale and a global generator approximation. We distinguish exact-mean centering from PDE centering and compute deterministic bias corrections under second delay and fourth spatial moments. For a factorized heavy-tail power-law delay, we establish the CLT, MDP, and full measure-valued LDP for the space-time measure, with finite nonzero mass of the initial profile for the LDP. Its fractional control energy retains the source generated by the prescribed past. A source-preserving regularization proves approximation of every finite-action measure. Compact-kernel bounds for initial profiles of infinite mass and the degenerate case of a zero initial profile are stated separately.

math.PR↗
arXiv2026

Underdamped Anchored Langevin Dynamics

Classical Langevin Monte Carlo algorithms, such as the unadjusted Langevin algorithm (ULA), are based on discretization schemes of the overdamped Langevin dynamics and are widely used for sampling in machine learning because they scale to high dimensions and large datasets. However, they require differentiable log-densities, and are less efficient for sampling heavy-tailed distributions. Although various approaches have been proposed in the literature for sampling non-differentiable and heavy-tailed targets, all of them are developed in the overdamped setting only. We propose a novel momentum-based underdamped anchored Langevin dynamics (UALD), and underdamped anchored Langevin Monte Carlo (UALMC) algorithm based on a discretization scheme of UALD. Our paper is the first to provide a kinetic (momentum-based) Langevin algorithm for non-differentiable and heavy-tailed targets that is free of asymptotic bias. The method replaces the original potential with a smooth reference potential and modifies the underdamped Langevin dynamics via multiplicative scaling in the drift term. We establish non-asymptotic guarantees in the 2-Wasserstein distance to the target distribution and show that the resulting iteration complexity has better dependence on dimension and accuracy level compared to overdamped-based anchored Langevin algorithms. We provide numerical experiments to illustrate the theory and practical performance of our proposed approach.

cs.LG↗
arXiv2026

Hydrodynamic and Stochastic Limits of Space-Time Hawkes Processes

We study the large-scale behavior of critical and nearly critical space-time Hawkes processes initialized by a prescribed Poisson field of past events. The density of this initial field separates three macroscopic regimes. In the high-density regime, fluctuations average out and the event measure converges to a hydrodynamic limit which is a deterministic diffusion profile. At the fluctuation threshold, random branching remains visible and produces a measure-valued stochastic limit. Below the threshold, the rescaled event measure vanishes locally even though its mean generally remains nontrivial, with rare large clusters accounting for the surviving mean. For reproduction delays with finite mean, the critical limits are described by heat flow and the occupation measure of a branching diffusion. A nearly critical perturbation introduces linear decay without changing the three-regime structure. Heavy-tailed delays lead to a different balance between time and space: the mean follows a fractional diffusion law, the stochastic limit is a superprocess occupation measure on time-space, characterized by a nonlinear fractional equation, and the past persists as a temporal memory term. We also determine how the scaling of the Poisson past must be matched to the observation scales and show that, in one spatial dimension, a shorter observation window recovers stochastic behavior hidden in the sparse regime. The results allow bounded initial profiles with possibly infinite total mass and describe the limits directly at the level of locally finite event measures.

math.PR↗
arXiv2026

Accelerating Non-Smooth and Heavy-Tailed Sampling

Anchored Langevin dynamics (ALD) is useful for non-smooth sampling where the density of the target distribution is possibly non-differentiable and heavy-tailed; reflected anchored Langevin dynamics (RALD) can sample possibly non-differentiable target density on a constrained domain. In this paper, we propose and study non-reversible anchored Langevin dynamics (NALD) for sampling possibly non-differentiable and heavy-tailed target density in the Euclidean space and the non-reversible reflected anchored Langevin dynamics (NRALD) for sampling possibly non-differentiable target density in the constrained space. Our construction adds a circulation drift generated by a possibly state-dependent divergence-free skew-symmetric matrix field and a stream potential. It preserves the target distribution without requiring derivatives of target density, admits a random-time-change representation, and applies both on the whole Euclidean space and on bounded domains with normal reflection. By breaking reversibility, we show that NALD and NRALD can converge to their target distributions faster than their reversible counterparts via finite-time non-asymptotic convergence analysis, a large deviations analysis and asymptotic variance reduction. Numerical experiments demonstrate the efficiency of the proposed algorithms.

stat.ML↗
arXiv2026

Reflected Anchored Langevin Algorithms

First order Langevin algorithms for constrained sampling in machine learning, such as projected Langevin Monte Carlo which are based on discretizations of reflected Langevin dynamics, require differentiable log densities that limits their applicability. This paper introduces reflected anchored Langevin dynamics (RALD), a reflected diffusion that converges to non-differentiable targets on constrained domains. The method uses a smooth anchored reference potential and multiplies the drift and noise covariance of its reflected Langevin dynamics by the same state dependent scaling factor. Its Euler-Maruyama discretization with projection gives reflected anchored Langevin Monte Carlo (RALMC) algorithm. We prove explicit convergence bounds and iteration complexity for RALMC in the 2-Wasserstein distance to the target distribution. Numerical experiments are provided to illustrate the theoretical predictions and the empirical performance of the method.

stat.ML↗
arXiv2026

A Hawkes Microfoundation for Multitype Inverse Gaussian Subordinators

We provide an event-level Hawkes microfoundation for a multitype inverse-Gaussian stochastic clock. We show that the event counts and integrated intensities of nearly critical multivariate linear Hawkes processes converge jointly to a multivariate pure-jump subordinator when reproduction delays have finite mean and immigration is balanced so that rare large families remain visible. The dependence among the limiting coordinates is inherited from microscopic cross-excitation, and the limit admits a Brownian additive-field first-passage representation. We call this process the multitype inverse-Gaussian subordinator. Its diagonal case consists of independent classical inverse-Gaussian subordinators, with the univariate model as a further special case. The square-root specialization recovers the inverse-Gaussian clock obtained at the boundary of hyper-rough square-root models by Abi Jaber--Attal--Rosenbaum (\textit{Ann. Appl. Probab.} \textbf{36}(4): 3635--3660, 2026). Our cluster proof exposes the underlying mechanism: finite-variance near-critical branching creates rare but macroscopic families, while their internal timing disappears on the observation scale, so each family becomes a jump. We also clarify the relation with the multivariate Hawkes scaling theory of Xu (arXiv:2412.14459): its diagonal-atom condition yields the diagonal specialization, whereas the genuinely coupled limit requires a separate uniqueness argument. In addition, under a common tilted-stability condition, we replace the high-intensity assumption in that theory by a weaker accumulated-activity condition and provide the Riccati identification needed when the potential has an atom. We strengthen convergence of the count and compensator and extend the scalar conclusion to finite-variance age-dependent branching clusters.

math.PR↗
arXiv2026

Global Convergence of Third-Order Langevin Dynamics for Non-Convex Optimization via Simulated Annealing

We study global convergence guarantees of third-order Langevin dynamics for non-convex optimization via simulated annealing with fixed friction and decreasing noise. An explicit three-block distorted entropy transfers dissipation from the noisy auxiliary variable to the full state. Under dissipativity, regularity, and low-temperature functional-inequality assumptions, logarithmic cooling drives the objective values to the global minimum in probability at the barrier-controlled kinetic rate. For the exact-force-integral and midpoint three-stage discretizations, polynomially decreasing steps preserve this rate on the physical time scale. The cubic local endpoint estimate gives a less restrictive sufficient step-size condition than the available frozen-force kinetic result. A comparison with the one-gradient UBU integrator shows how its centered stochastic local error leads, under the same strong-coupling analysis, to a smaller sufficient iteration exponent. Numerical experiments are conducted to illustrate our theory. For a double well objective, third-order Langevin terminal-success point estimates are higher than UBU at both a common horizon and an equal gradient budget. For a high-dimensional nonconvex neural-network objective using synthetic data, independently tuned UBU and third-order Langevin schemes both outperform overdamped Langevin dynamics; the third-order Langevin point estimate is higher. For the same neural-network objective on real data, we show the same point-estimate ordering for best-basin probability and post-quench test accuracy. Numerical code and associated experiment results are publicly available at https://github.com/gagawjbytw/simulated-annealing-third-order-langevin.

math.NA↗
arXiv2026

Sharp hypocoercive convergence estimates for underdamped Langevin dynamics with specular reflection

We study the underdamped (kinetic) Langevin dynamics confined to a bounded domain $Ω\subset\mathbb{R}^d$ by specular reflection of the velocity at the boundary. This process is the natural momentum-based analogue of the normally reflected overdamped Langevin diffusion, and it is used in practice for constrained sampling. Assuming only that the position marginal $μ_x\propto e^{-U}$ satisfies a Poincaré inequality on $Ω$ with constant $m>0$, $\nabla^2U\succeq-K\,\mathrm{Id}$ and $Ω$ is a convex domain, we prove that the law converges to the Gibbs measure exponentially fast in $L^2$, with an explicit rate that scales like $\sqrt m$, which is optimal when $U$ is convex. Since the normally reflected overdamped dynamics converges exactly at rate $m$, this establishes a square-root acceleration for constrained sampling in the small-gap regime when $m$ is small, matching the acceleration known in the unconstrained case. The explicit rate is the same in the unconstrained setting of Fan--Li--Lu. The proof adapts the modified $L^2$ hypocoercivity method of Dolbeault--Mouhot--Schmeiser with the gap-shifted corrector of Fan--Li--Lu. The specular symmetry makes the transport operator antisymmetric, and that the corrector automatically selects the Neumann realization of the overdamped generator, which is precisely the boundary condition that keeps every auxiliary function inside the specular class. The Bochner identity used in the whole-space argument is replaced by a weighted Reilly formula, whose boundary contribution involves the second fundamental form of $\partialΩ$ and is nonnegative for convex $Ω$. Finally, we extend our results to the setting where the domain $Ω$ is non-convex. We obtain an explicit contraction rate that depends on the domain.

math.PR↗
arXiv2026

Moderate Deviations for Nonlinear Hawkes Processes

A Hawkes process is a simple point process whose intensity depends on its history; the resulting dynamics are generally non-Markovian. We establish a sample-path moderate deviation principle for a nonlinear Hawkes process in the full moderate regime. Since a Poisson cluster representation is unavailable for nonlinear Hawkes processes, we use the past configuration as a Markov state and construct a potential, or Poisson corrector, for the centered stochastic intensity. A monotone Poisson coupling shows that the add-one increment of the corrector is uniformly bounded. The centered counting process is consequently the sum of a martingale with bounded jumps and an exponentially negligible boundary term. Exponential stabilization of the predictable quadratic variation follows from the process-level large deviation principle for nonlinear Hawkes processes. The martingale moderate deviation theorem then yields the result for every scale between the central-limit and large-deviation scales. The same construction gives a response formula for the asymptotic variance and, in particular, verifies that the variance dominates the stationary mean intensity in the self-exciting case.

math.PR↗
arXiv2026

BRIDLE: Generalized Self-supervised Learning with Quantization

Self-supervised learning has been a powerful approach for learning meaningful representations from unlabeled data across various domains, reducing the reliance on large labeled datasets. Inspired by BERT's success in capturing deep bidirectional contexts in natural language processing, similar frameworks have been adapted to other modalities such as audio, with models like BEATs extending the bidirectional training paradigm to audio signals using vector quantization (VQ). However, these frameworks face challenges, notably their dependence on a single codebook for quantization, which may not capture the complex, multifaceted nature of signals. In addition, inefficiencies in codebook utilization lead to underutilized code vectors. To address these limitations, we introduce BRIDLE (Bidirectional Residual Quantization Interleaved Discrete Learning Encoder), a self-supervised encoder pretraining framework that incorporates residual quantization (RQ) into the bidirectional training process, and is generalized for pretraining with audio, image, and video. Using multiple hierarchical codebooks, RQ enables fine-grained discretization in the latent space, enhancing representation quality. BRIDLE involves an interleaved training procedure between the encoder and tokenizer. We evaluate BRIDLE on audio understanding tasks using classification benchmarks, achieving state-of-the-art results, and demonstrate competitive performance on image classification and video classification tasks, showing consistent improvements over traditional VQ methods in downstream performance.

cs.LG↗
arXiv2026

Improved Analysis for Hessian-free High-resolution Monte Carlo Sampling

Hessian-free high-resolution (HFHR) dynamics augments underdamped Langevin dynamics (ULD) with reversible position diffusion for sampling problems that arise in machine learning. We establish an explicit quantitative contraction rate for HFHR dynamics under a position Poincaré inequality, weighted Hessian and Laplacian bounds, and a compact Sobolev embedding, where the potential function is not necessarily convex. An adapted time-augmented Poincaré inequality yields an explicit rate that improves upon the contraction rate of the underdamped Langevin dynamics. We also give a weak-solution construction and a self-contained spectral proof of the divergence lemma underlying the argument. For HFHR Monte Carlo (HFHRMC) algorithm, which is based on a discretization scheme of HFHR dynamics, we use a path-space Girsanov argument to obtain a non-asymptotic convergence bound and an explicit iteration complexity in total variation distance. The bounds hold for every $α\geq0$ and $γ>0$ and remain regular at the ULD endpoint. Optimizing the iteration complexity bound yields a positive, accuracy-dependent position-diffusion parameter at finite accuracy, while its leading high-accuracy order coincides with that of the optimized ULD endpoint. Our iteration complexity bound improves upon the existing work on HFHR algorithms. Numerical experiments including Bayesian learning problems on real data are provided to illustrate the effect of positive $α$ and its benefit.

stat.ML↗
arXiv2026

DIGing--SGLD: Decentralized and Scalable Langevin Sampling over Time--Varying Networks

Sampling from a target distribution induced by training data is central to Bayesian learning, with Stochastic Gradient Langevin Dynamics (SGLD) serving as a key tool for scalable posterior sampling and decentralized variants enabling learning when data are distributed across a network of agents. This paper introduces DIGing-SGLD, a decentralized SGLD algorithm designed for scalable Bayesian learning in multi-agent systems operating over time-varying networks. Existing decentralized SGLD methods are restricted to static network topologies, and many exhibit steady-state sampling bias caused by network effects, even when full batches are used. DIGing-SGLD overcomes these limitations by integrating Langevin-based sampling with the gradient-tracking mechanism of the DIGing algorithm, originally developed for decentralized optimization over time-varying networks, thereby enabling efficient and bias-free sampling without a central coordinator. To our knowledge, we provide the first finite-time non-asymptotic Wasserstein convergence guarantees for decentralized SGLD-based sampling over time-varying networks, with explicit constants. Under standard strong convexity and smoothness assumptions, DIGing-SGLD achieves geometric convergence to an $O(\sqrtη)$ neighborhood of the target distribution, where $η$ is the stepsize, with dependence on the target accuracy matching the best-known rates for centralized and static-network SGLD algorithms using constant stepsize. Numerical experiments on Bayesian linear and logistic regression validate the theoretical results and demonstrate the strong empirical performance of DIGing-SGLD under dynamically evolving network conditions.

math.OC↗
arXiv2026

An Eyring--Kramers Law for the Hypoelliptic Third-Order Langevin Diffusion

We prove an Eyring--Kramers law for metastable transitions of the hypoelliptic third-order Langevin diffusion in the low-temperature limit. This diffusion is a three-level Markovian lifting of Langevin dynamics: the Brownian noise acts only on the highest auxiliary variable and reaches the position variable through a third-order H"ormander chain. For a double-well potential with a unique index-one transition saddle, we determine both the Arrhenius exponential scale and the sharp prefactor of the mean transition time. The prefactor is governed by the unique positive unstable rate of the deterministic linearization at the saddle, equivalently the positive root of a cubic polynomial. Our proof combines a weak-capacity framework with a saddle-adapted boundary layer, an explicit Gaussian current calculation, committor localization, and intrawell flatness. Under matched kinetic normalizations, the resulting metastable prefactor is strictly smaller than its underdamped counterpart. A numerical experiment for a one-dimensional double well illustrates the Arrhenius scaling and the predicted prefactor comparison.

math.PR↗
arXiv2026

Microstructural Foundation for the Rough Hawkes--Heston Model

Hawkes-based microstructural foundations for rough volatility, leverage, and rough Heston-type limits were developed by El Euch et al. (2018, Finance Stoch., 22(2), 241--280) and connected to the affine rough Heston framework of El Euch and Rosenbaum (2019, Math. Finance, 29(1), 3--38). The rough Hawkes--Heston model with common price--volatility jumps of Bondi et al. (2024, Math. Finance, 34(4), 1197--1241) extends this framework by adding state-dependent common jumps to rough affine volatility. We provide a microstructural foundation for its variance and common-jump mechanism by constructing a Poisson-embedded marked Hawkes order-flow model. Ordinary arrivals generate rough continuous volatility and leverage through a nearly unstable heavy-tailed Hawkes mechanism, while rare marked arrivals represent common shock events that produce simultaneous price jumps and volatility excitation. Under the nearly unstable scaling and the reduced-form admissibility conditions, the complete rescaled price/variance/jump system converges along the full sequence to the unique complete canonical rough Hawkes--Heston weak solution. The Hawkes renewal structure yields a Mittag--Leffler Volterra representation, which is then rewritten in Riemann--Liouville fractional form. The limiting coefficients are expressed explicitly in terms of the microscopic parameters. The construction provides a microstructural foundation for the variance and common-jump mechanism of the rough Hawkes--Heston model. Numerical experiments illustrate the convergence of our microstructural foundation to the rough Hawkes-Heston model.

q-fin.MF↗
arXiv2026

Weak Equilibrium Measures and Capacity--Hitting Identities for the Hypoelliptic Third-Order Langevin Diffusion

We construct weak equilibrium measures and weak capacities for the hypoelliptic third-order Langevin diffusion motivated by an accelerated sampling algorithm (Mou et al. (2021) \textit{J. Mach. Learn. Res.}, \textbf{22}(42), 1--41). In this process, the Brownian noise acts only in the highest-order auxiliary variable and reaches the physical variables through a step-three Hörmander chain, so the standard uniformly elliptic boundary-flux theory is not directly applicable at characteristic points of phase-space balls. We prove an elliptic-regularization stability theorem for the corresponding hitting laws and then define the weak equilibrium measure and weak capacity. The proof combines the boundary-hitting stability strategy of Lee--Ramil--Seo (2026, \textit{arXiv:2503.12610v2}) with localized hypoelliptic heat-kernel estimates (Pigato (2022) \textit{Stoch. Process. Appl.}, \textbf{145}, 117--142) adapted to the third-order chain. We obtain the bounded-domain weak capacity--hitting identity and a Lyapunov drift argument in the spirit of Lee--Ramil--Seo that yields positive Harris recurrence and extends the construction to a whole-space weak equilibrium measure, and whole-space capacity--hitting identity.

math.PR↗
arXiv2026

Variance Reduction for Stochastic Gradient Generalized Non-reversible Langevin Monte Carlo Algorithms

We study the leading-order fluctuation of stochastic gradient Euler-Maruyama estimators for generalized non-reversible Langevin dynamics. Under structural assumptions tailored to the small-stepsize central limit theorem and under an unbiased stochastic gradient oracle, we prove that the empirical average over a horizon of order the inverse squared stepsize satisfies a central limit theorem in the vanishing-stepsize regime. The limiting variance is characterized through the Poisson equation of the limiting full-gradient diffusion. We then rewrite this constant in an operator form that links it to the continuous-time asymptotic variance and, under standard operator-theoretic assumptions, derive a sufficient condition under which an anti-symmetric perturbation strictly reduces the leading-order fluctuation constant relative to the reversible baseline. We also identify bounded smooth predictive observables that re directly covered by the main theorem. As a separate Gaussian calculation beyond the bounded-test-function regime, we obtain closed-form formulas for quadratic Hamiltonians and linear observables. The framework covers non-reversible Langevin dynamics and augmented-state examples including Hessian-free high-resolution dynamics and a positive-definite subclass of gradient-adjusted underdamped Langevin dynamics that allow stochastic gradients. Numerical experiments on basic examples and Bayesian linear regression using synthetic data, and Bayesian logistic regression using real data support the predicted Gaussian fluctuations and show that the non-reversible schemes consistently reduce the root mean squared error (RMSE) relative to their reversible baselines.

stat.ML↗
arXiv2026

VIX options in Bergomi models

We present a study of the leading-order asymptotics for VIX option prices in Bergomi models in the short-maturity and small volatility-of-volatility regimes. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered for one-factor, two-factor Bergomi and $N$-factor models. The leading-order asymptotics are obtained in closed-form, which are translated into predictions for the small-maturity asymptotics of the VIX implied volatility. Numerical illustrations are provided to illustrate the efficiency of the closed-form asymptotic formulas.

q-fin.PR↗
arXiv2026

Large deviations for the mean-field limit of Hawkes processes

Hawkes processes are a class of simple point processes whose intensity depends on the past history, and is in general non-Markovian. Limit theorems for Hawkes processes in various asymptotic regimes have been studied in the literature. In this paper, we study a multidimensional nonlinear Hawkes process in the asymptotic regime when the dimension goes to infinity, whose mean-field limit is a time-inhomogeneous Poisson process, and our main result is a large deviation principle for the mean-field limit.

math.PR↗