arXiv subjects

Jianfeng Lu

Publications and source records attributed to Jianfeng Lu.

At least 19 recordsLinked to original sources

arXiv2026

Space-Time HWI Inequalities and the Wasserstein Current for Kinetic Equations

We prove a space-time analogue of the HWI inequality of Otto-Villani for controlled kinetic equations with a semiconvex confining potential. It bounds the time-averaged relative entropy by spatial and conditional velocity transport distances, the velocity Fisher information, an intrinsic action of the path, and a boundary term given by the time endpoint values of the Wasserstein current. The key step is a differential comparison inequality for the Wasserstein current, which captures the transfer of dissipation from velocity to position. Along the Langevin semigroup, this inequality yields a kinetic evolution variational inequality involving the second time derivative of the squared spatial Wasserstein distance and a damping term. Under a compatible spatial Talagrand's transport inequality, we obtain a boundary-free space-time logarithmic Sobolev inequality on every interval $[0,T]$, $T>0$, with the optimal $T^{-2}$ short-time scaling of the constant. As a consequence, we also establish entropy decay for every friction coefficient and every initial density of finite entropy, with optimal cubic order at short times and exponential decay at long times.

math.AP↗

arXiv2026

Rényi mixing bounds for kinetic samplers via $L^2$ hypocoercivity

We study convergence in Rényi divergence of order two for underdamped Langevin dynamics, the bouncy particle sampler, and the Zigzag sampler, for a target $e^{-U}$ on $\mathbb R^d$ that satisfies a logarithmic Sobolev inequality, with $\nabla^2U$ bounded below by a possibly negative constant and, for the Zigzag sampler, also bounded above. For initial laws whose Rényi divergence from equilibrium is at most $R$, we show that it decays exponentially, at a rate that depends on $R$. With the friction or refreshment rate chosen according to $R$, the time needed to reach a fixed accuracy in total variation or in Rényi divergence is of order $\sqrt{R\log R}$ at fixed dimension and target parameters, improving the linear dependence obtained by directly converting exponential $L^2$ decay. The key idea is to use the logarithmic Sobolev inequality for the position target in place of the usual spatial spectral-gap estimate in the $L^2$ hypocoercivity argument. For the bouncy particle and Zigzag samplers, this order is sharp for the Rényi divergence on a one-dimensional Gaussian target, uniformly in the refreshment rate. For the Zigzag sampler in dimension at least two, Gaussian examples show that the mixing-time bound is sharp up to logarithmic factors in the worst case over strongly convex targets, in both the initial divergence and the upper Hessian bound; in particular, strong convexity alone does not give a uniform mixing bound.

math.PR↗

arXiv2026

Entropic hypocoercivity via the Polchinski flow: an application to the lattice sine-Gordon model

We develop a multiscale criterion for exponential decay of relative entropy in the canonical stochastic quantization of lattice field theories, the underdamped Langevin dynamics of the field and its momentum. For smooth bounded perturbations of Gaussian fields, the criterion requires the uniform operator-norm bound on the smoothed interaction Hessian along the Polchinski flow of Bauerschmidt and Bodineau to have a small $L^2$ norm in the scale variable. The decay constants depend only on this $L^2$ norm and the friction. The proof measures the transfer of momentum dissipation to position on the same scales that decompose entropy. A phase-space extension of the Polchinski flow gives Gaussian coarse-graining that commutes with free kinetic evolution. Integrating position--momentum cross terms over these scales yields a corrector bounded by entropy. A force defect accounts for changes in the averaged interaction force away from equilibrium; its evolution in scale controls the errors in the kinetic estimate. For the massive two-dimensional lattice sine--Gordon model with $0<β<4π$ and small dimensionless local coupling, the criterion gives entropy decay for all finite-entropy initial laws, uniformly in the lattice spacing and the volume. For friction proportional to the mass $m$, the rate has optimal order $m$.

math-ph↗

arXiv2026

Size-independent Logarithmic Sobolev Inequalities for Quantum Gibbs Samplers

We propose a new method for proving $L^2$ and modified logarithmic Sobolev inequalities with constants independent of system size for quantum Gibbs samplers whose Lindblad generators satisfy Kubo-Martin-Schwinger (KMS) detailed balance. We first establish the $L^2$ logarithmic Sobolev inequality by transporting it along a temperature-dependent path from the infinite-temperature product state to the target Gibbs state while maintaining a size-independent decay rate for the sandwiched Rényi-2 divergence. A tensor-power argument, combined with the quantum asymptotic equipartition property, then transfers this decay bound to relative entropy, yielding the modified logarithmic Sobolev inequality. This approach provides a general framework for proving size-independent logarithmic Sobolev inequalities for quantum Gibbs samplers. We apply it to two regimes: one-dimensional quantum spin chains at every finite temperature, and finite-range quantum spin systems at sufficiently high temperature in any fixed dimension.

quant-ph↗

arXiv2026

Entropy hypocoercivity and query complexity for the bouncy particle sampler

We study convergence in relative entropy of the bouncy particle sampler with Gaussian velocity refreshment, for a target $e^{-U}$ on $\mathbb R^d$ with $U$ $m$-strongly convex and $L$-smooth, and condition number $κ=L/m$. For initial laws with bounded density relative to equilibrium, we show that the relative entropy decays exponentially up to an arbitrary additive error $δ$, with a rate that depends only logarithmically on the density bound and on $δ$. From a Gaussian cold start and with a suitably chosen refreshment rate, the relative entropy falls below $δ$ after time $\widetilde O(\sqrt{d/m})$. Combined with exact simulation by windowed thinning, this gives an expected gradient-query complexity of $\widetilde O(d\sqrtκ)$, improving the $\widetilde O(d^2\sqrtκ)$ bound based on $L^2$ hypocoercivity.

math.PR↗

arXiv2026

Sharp dimensional analysis of midpoint methods for Langevin sampling

We study deterministic and randomized midpoint discretizations of Langevin dynamics for a target $π\propto e^{-V}$, where $0 \prec αI\preceq\nabla^2V\preceqβI$ and $κ=β/α$. To achieve $\sqrtα\,W_2\leqslant\varepsilon$, we show that deterministic Heun uses at most $\widetilde O(κ^{4/3}d^{1/3}\varepsilon^{-2/3})$ gradient queries, and underdamped exponential midpoint uses $\widetilde O(κ^{5/4}d^{1/4}\varepsilon^{-1/2})$. The proofs exploit cancellation at stationarity and smoothing using techniques from Malliavin calculus, outperforming previous upper bounds based on standard couplings. At bounded condition number, a lower bound matches the $d$ and $\varepsilon$ powers of both deterministic methods. To contrast, for the randomized midpoint methods and Poisson midpoint with at least two grid points (both overdamped and underdamped variants), a simple Gaussian calculation yields a lower bound $d^{1/3}\varepsilon^{-1/3}$ to get an $\varepsilon$-close sample despite starting at a benign initialization. This shows surprisingly that in high dimensions, deterministic discretizations can outperform their random counterparts.

math.ST↗

arXiv2026

High-accuracy simulation of Picard HMC, part I: Gaussian cloud correction

We study the problem of sampling from a continuous density $π\propto \exp(-V)$ on $\mathbb R^d$, where $V\in C^2(\mathbb R^d)$ has a $β$-Lipschitz gradient and $π$ satisfies a logarithmic Sobolev inequality with constant $α^{-1}$, and write $κ= β/α$. We introduce the Gaussian cloud sampler, which achieves total variation accuracy $\varepsilon$ using $\widetilde O(κd^{1/5}\,\text{polylog}(1/\varepsilon))$ gradient queries in expectation. The algorithm uses first-order rejection sampling (FORS) to correct the law of smoothed Picard HMC trajectories. To do so, we represent the iterates of the ideal Picard iteration via Gaussian clouds, whose centers are never evaluated, and we develop a suite of likelihood correction gadgets which only use samples from this indirect cloud representation.

math.ST↗

arXiv2026

State-space models through the lens of ensemble control

State-space models (SSMs) are effective architectures for sequential modeling, but a rigorous theoretical understanding of their training dynamics is still lacking. We formulate continuous-time SSM parameter-path optimization as an ensemble optimal control problem: each input sequence generates a corresponding state trajectory in the ensemble, while the trainable parameter path forms a common control shared by all trajectories. Within this formulation, model evaluation is represented by the forward state equation and backward sensitivity propagation by the corresponding adjoint equation. We show that the Hamiltonian gradient with respect to the control variable represents the negative first-variation density of the reduced ensemble objective. Using this representation, we analyze a Bregman mirror-descent scheme for the continuous-time ensemble objective; in Euclidean geometry this reduces to functional projected gradient descent. State-adjoint and Hamiltonian-gradient stability yield two-sided relative curvature of the reduced objective and relative convexity under sufficient regularization. We further show that in the relatively convex regime the optimal-control problem admits a minimizer, which is unique under relative strong convexity. For SSMs, the explicit stability-generated defect $κ_{\mathrm{stab}}^{\mathrm{SSM}}$ gives an $O(1/k)$ objective rate at $τ=κ_{\mathrm{stab}}^{\mathrm{SSM}}$ and geometric convergence to the unique minimizer for $τ>κ_{\mathrm{stab}}^{\mathrm{SSM}}$, under the stated stepsize condition.

math.OC↗

arXiv2026

Smoothed Picard Hamiltonian Monte Carlo

We develop a new low-accuracy sampler, called smoothed Picard Hamiltonian Monte Carlo, which combines Gaussian smoothing, Picard iteration, and higher-order discretization. For a log-concave target $π\propto \exp(-V)$ in dimension $d$ satisfying $0 \prec αI \preceq \nabla^2 V \preceq βI$, with condition number $κ:= β/α$, smoothed Picard HMC returns a sample with $\sqrt α\,W_2(\cdot,π) \le \varepsilon$ using $\widetilde O(κ^2 + κ^{7/6} d^{1/6}/\varepsilon^{1/3})$ gradient queries. We also prove stronger $W_q$ bounds, and then develop an algorithmic framework, the recursive warm start generator, to upgrade these $W_q$ bounds to stronger divergence guarantees. This produces a warm start for the proximal bouncy particle sampler, introduced in a companion work, leading to a high-accuracy log-concave sampler with complexity $\widetilde O((κ^{7/6} d^{1/6} + κ^{1/2} d^{1/4})\mathrm{polylog}(1/\varepsilon))$.

math.ST↗

arXiv2026

Sharp Gaussian divergence and Bernstein-Markov inequalities

We prove two dimension-free estimates in Gaussian space. The first is an optimal Meyer-type inequality for the Gaussian divergence: for $p\geq2$, \[ \|δV\|_{L^p(γ_d)} \leq \sqrt p\,|\mathbb E V| +Cp\,\|DV\|_{L^p(γ_d;\mathrm{HS}_d)}\, . \] Here $\mathrm{HS}_d$ is the space of $d\times d$ matrices with its Hilbert-Schmidt norm. Its main ingredient is a dimension-free weak-type $(1,1)$ bound for the second-order transform $D^2\mathcal N^{-1}$, where $\mathcal N := -Δ+x\cdot D$ is the Ornstein-Uhlenbeck operator. We also include a direct change-of-variables proof of the weaker bounded derivative divergence inequality in the appendix. The second result is a Gaussian Bernstein-Markov inequality. If $P$ is a polynomial, $p\geq2$, and $m_p := (\mathbb E|g|^p)^{1/p}$ for a standard Gaussian $g$, then \[ \|DP\|_{L^p(γ_d;\mathbb R^d)} \leq\frac{2\sqrt{2e\, \mathrm{deg}(P)}}{m_p}\,\|P\|_{L^p(γ_d)}\, . \] The factor $2$ is unnecessary when $p$ is even, or $P$ is even or odd. Thus, the constant is of order $\sqrt{\mathrm{deg}(P)/p}$ throughout the range $p\geq2$.

math.PR↗

arXiv2026

CDFCI: High-Performance Parallel Software for Quantum Many-Body Eigenvalue Problems

CDFCI is a shared-memory parallel numerical program for computing low-lying eigenpairs of large-scale, non-relativistic fermionic Hamiltonians. The software is designed to handle a broad class of many-body quantum models, including both ab initio electronic structure Hamiltonians and lattice-based Hamiltonians arising in condensed matter physics. CDFCI combines an efficient coordinate-descent-based selected configuration interaction algorithm with dedicated parallelization strategies, achieving high performance on modern multi-core architectures. Benchmark results on representative quantum chemistry and condensed matter test cases demonstrate that CDFCI attains state-of-the-art accuracy with competitive performance compared to established selected configuration interaction (such as CIPSI or SHCI) and DMRG implementations. The software is open-source, extensively documented, and provides a Python interface for seamless integration with PySCF and other many-body simulation workflows.

physics.comp-ph↗

arXiv2026

Mean square error analysis of stochastic gradient and variance-reduced sampling algorithms

This paper considers mean square error (MSE) analysis for stochastic gradient sampling algorithms applied to underdamped Langevin dynamics under a global convexity assumption. A novel discrete Poisson equation framework is developed to bound the time-averaged sampling error. For the Stochastic Gradient UBU (SG-UBU) sampler, we derive an explicit MSE bound and establish that the numerical bias exhibits first-order convergence with respect to the step size $h$, with the leading error coefficient proportional to the variance of the stochastic gradient. The analysis is further extended to variance-reduced algorithms for finite-sum potentials, specifically the SVRG-UBU and SAGA-UBU methods. For these algorithms, we identify a phase transition phenomenon whereby the convergence rate of the numerical bias shifts from first to second order as the step size decreases below a critical threshold. Theoretical findings are validated by numerical experiments. In addition, we compare the computational cost that mini-batch SG-UBU and SVRG-UBU require to reach a prescribed accuracy, which indicates when variance reduction is worthwhile.

math.NA↗

arXiv2026

Accelerated High-Accuracy Sampling from a Warm Start via the Proximal Bouncy Particle Sampler

We study the problem of sampling from $μ(\mathrm{d}x)\propto e^{-V(x)}\,\mathrm{d}x$ on $\mathbb{R}^d$, where $V$ is $α$-strongly convex and $β$-smooth, and write $κ:=β/α$. We design and analyze the Proximal Bouncy Particle Sampler (Proximal BPS), a new sampler that combines ideas from the proximal sampler and the bouncy particle sampler. From a warm start initialization with $ O(1) $ Rényi divergence w.r.t. $μ$, Proximal BPS returns a sample whose law is $\varepsilon$-close to $μ$ in total variation distance using $\widetilde O(\sqrtκ\,d^{1/4} \,\mathrm{polylog}(1/\varepsilon))$ gradient queries in expectation.

math.ST↗

arXiv2026

Mirror Descent for Deterministic Optimal Control

We study an explicit mirror-descent method for finite-horizon deterministic optimal control problems. The method is motivated by Pontryagin's maximum principle: at each iteration, one solves the state and adjoint equations and updates the control by maximizing a first-order approximation of the regularized Hamiltonian penalized by a Bregman divergence. In the Euclidean case, the update reduces to a projected gradient step in the control variable. Under global smoothness assumptions and uniform convexity of the mirror map, we prove a relative smoothness estimate for the cost functional and derive an energy dissipation inequality for sufficiently small step sizes. Under an additional concavity assumption on the unregularized Hamiltonian and convexity of the terminal cost, we establish relative convexity of the regularized objective. These estimates yield an $O(1/n)$ convergence rate in the unregularized convex case and a geometric rate when the control regularization parameter is positive. Numerical examples illustrate the behavior of the method in linear-quadratic, degenerate convex, and nonlinear high-dimensional settings.

math.OC↗

arXiv2026

Windowed thinning and query complexity for the bouncy particle and Zigzag samplers

Let $μ(d x)\propto e^{-U(x)} d x$ on $\R^d$, where $U$ is $m$-strongly convex and $L$-smooth, and denote by $κ=L/m$ the condition number. We consider windowed thinning, an exact simulation method for the bouncy particle sampler and the coordinate Zigzag process. The method divides a trajectory into deterministic windows and uses a gradient evaluation at the beginning of each window to construct a tractable local envelope for the event rate. Combining this construction with quantitative mixing estimates and finite-time bounds on the expected numbers of bounces and flips yields query complexity guarantees from a Gaussian cold start. For total-variation error $\varepsilon$, the expected query counts are $O(κ^{1/2}d\,(d\logκ+\log\frac1\varepsilon))$ gradient queries for the bouncy particle sampler and $O(κd^{1/4}(d\logκ+\log\frac1\varepsilon))$ full-gradient equivalents for Zigzag, where $d$ coordinate-partial queries count as one equivalent.

math.NA↗

arXiv2026

HyperVision: A Channel-Adaptive Ground-Based Hyperspectral Vision Pre-trained Backbone

While hyperspectral imaging provides rich spatial-spectral information across hundreds of narrow wavelength bands for precise material identification, ground-based hyperspectral pre-trained backbones remain absent, constrained by varying spectral configurations across sensors, limited annotations and heterogeneous labeling schemes, and the limited scale and scene diversity of existing datasets. To address these challenges and enable universal perception, we propose HyperVision, the first ground-based hyperspectral pre-trained backbone. First, to handle varying spectral configurations, HyperVision adopts a channel-adaptive dynamic embedding mechanism to map heterogeneous inputs into a unified token space. Second, we develop an unsupervised representation learning framework. Specifically, to address limited annotations and heterogeneous labeling schemes, a multi-source pseudo-labeling method is introduced to fuse spatial structures from SAM2 and fine-grained spectral material information from HyperFree. Furthermore, to enrich scene diversity and compensate for limited dataset scale, a cross-modal knowledge distillation mechanism is utilized to transfer rich semantic representations from a pre-trained RGB vision model to our backbone. Pre-trained on a collection of 15k images from 26 diverse ground-based datasets, HyperVision demonstrates exceptional generalization. Requiring only efficient head-only adaptation without adjusting backbone parameters, it outperforms state-of-the-art task-specific methods across three downstream tasks under varying sensor configurations, yielding up to a 16.3% relative improvement in hyperspectral semantic segmentation $\mathrm{Acc}_{\mathrm{M}}$, a 2.1% relative gain in object tracking AUC, and a 35.5% reduction in salient object detection MAE. The source code and pre-trained models are available at https://github.com/lronkitty/HyperVision.

cs.CV↗

arXiv2026

Sharp hypocoercive convergence estimates for underdamped Langevin dynamics via the modified $L^2$ method

In this note, we consider the underdamped Langevin dynamics with invariant measure $μ(\mathrm{d}x\,\mathrm{d}v) \propto e^{-U(x)-|v|^2/2}\,\mathrm{d}x\,\mathrm{d}v$. Assume that the position marginal $μ_x(\mathrm{d}x)\propto e^{-U(x)}\,\mathrm{d}x$ satisfies a Poincaré inequality with constant $m>0$, and that $\nabla^2 U\ge -K\,\mathrm{Id}$ for some $K\ge 0$. We revisit the modified $L^2$ method of Dolbeault--Mouhot--Schmeiser, employing a shifted corrector \begin{equation*} A_α=(α- L_{\mathrm o})^{-1}(L_aΠ_v)^*, \qquad α\ge0, \end{equation*} where ${L}_{\mathrm{o}}=Δ_x-\nabla U\cdot\nabla_x$ is the overdamped generator, ${L}_a$ is the generator of the Hamiltonian flow, and $Π_v$ denotes averaging over the velocity variable. We establish an explicit hypocoercive $L^2$-convergence rate $Λ_{α,γ}$ for every shift $α\ge0$ and friction coefficient $γ>0$, and show that, for each fixed $γ$, the rate is maximized at $α=0$. Optimizing further over $γ$ gives \begin{equation*} Λ_{0,γ_*} = \frac{\sqrt m} {2\left(2+\sqrt{2+\frac{2K}{m}} +\sqrt{6+\frac{2K}{m}}\right)}\,, \qquad \text{at} \quad γ_*=\sqrt{6m+2K}. \end{equation*} For convex $U$, this recovers the optimal $O(\sqrt m)$ rate.

math.AP↗

arXiv2026

Exact and Efficient Representation of Totally Anti-Symmetric Functions

This paper concerns the long-standing question of representing (totally) anti-symmetric functions in high dimensions. We propose a new ansatz based on the composition of an odd function with a fixed set of anti-symmetric basis functions. We prove that this ansatz can exactly represent every anti-symmetric and continuous function and the number of basis functions has efficient scaling with respect to dimension (number of particles). The singular locus of the anti-symmetric basis functions is precisely identified.

math.CA↗