arXiv subjects

Jianfei Xue

Publications and source records attributed to Jianfei Xue.

9 recordsLinked to original sources

arXiv2026

Sublinear Time Quantum Algorithm for Attention Approximation

Given the query, key and value matrices $Q, K, V\in \mathbb{R}^{n\times d}$, the attention module is defined as $\mathrm{Att}(Q, K, V)=D^{-1}AV$ where $A=\exp(QK^\top/\sqrt{d})$ with $\exp(\cdot)$ applied entrywise, $D=\mathrm{diag}(A{\bf 1}_n)$. The attention module is the backbone of modern transformers and large language models, but explicitly forming the softmax matrix $D^{-1}A$ incurs $Ω(n^2)$ time, motivating numerous approximation schemes that reduce runtime to $\widetilde O(nd)$ via sparsity or low-rank factorization. We propose a quantum data structure that approximates any row of $\mathrm{Att}(Q, K, V)$ using only row queries to $Q, K, V$. Our algorithm preprocesses these matrices in $\widetilde{O}\left( ε^{-1} n^{0.5} \left( s_λ^{2.5} + s_λ^{1.5} d + α^{0.5} d \right) \right)$ time, where $ε$ is the target accuracy, $s_λ$ is the $λ$-statistical dimension of the exponential kernel defined by $Q$ and $K$, and $α$ measures the row distortion of $V$ that is at most $d/{\rm srank}(V)$, the stable rank of $V$. Each row query can be answered in $\widetilde{O}(s_λ^2 + s_λd)$ time. To our knowledge, this is the first quantum data structure that approximates rows of the attention matrix in sublinear time with respect to $n$. Our approach relies on a quantum Nyström approximation of the exponential kernel, quantum multivariate mean estimation for computing $D$, and quantum leverage score sampling for the multiplication with $V$.

quant-ph↗

arXiv2025

Differential Privacy for Euclidean Jordan Algebra with Applications to Private Symmetric Cone Programming

In this paper, we study differentially private mechanisms for functions whose outputs lie in a Euclidean Jordan algebra. Euclidean Jordan algebras capture many important mathematical structures and form the foundation of linear programming, second-order cone programming, and semidefinite programming. Our main contribution is a generic Gaussian mechanism for such functions, with sensitivity measured in $\ell_2$, $\ell_1$, and $\ell_\infty$ norms. Notably, this framework includes the important case where the function outputs are symmetric matrices, and sensitivity is measured in the Frobenius, nuclear, or spectral norm. We further derive private algorithms for solving symmetric cone programs under various settings, using a combination of the multiplicative weights update method and our generic Gaussian mechanism. As an application, we present differentially private algorithms for semidefinite programming, resolving a major open question posed by [Hsu, Roth, Roughgarden, and Ullman, ICALP 2014].

math.OC↗

arXiv2022

Condensation, boundary conditions, and effects of slow sites in zero-range systems

We consider the space-time scaling limit of the particle mass in zero-range particle systems on a $1$D discrete torus $\mathbb{Z}/N\mathbb{Z}$ with a finite number of defects. We focus on two classes of increasing jump rates $g$, when $g(n)\sim n^α$, for $0<α\leq 1$, and when $g$ is a bounded function. In such a model, a particle at a regular site $k$ jumps equally likely to a neighbor with rate $g(n)$, depending only on the number of particles $n$ at $k$. At a defect site $k_{j,N}$, however, the jump rate is slowed down to $λ_j^{-1}N^{-β_j}g(n)$ when $g(n)\sim n^α$, and to $λ_j^{-1}g(n)$ when $g$ is bounded. Here, $N$ is a scaling parameter where the grid spacing is seen as $1/N$ and time is speeded up by $N^2$. Starting from initial measures with $O(N)$ relative entropy with respect to an invariant measure, we show the hydrodynamic limit and characterize boundary behaviors at the macroscopic defect sites $x_j = \lim_{N\uparrow \infty} k_{j, N}/N$, for all defect strengths. For rates $g(n)\sim n^α$, at critical or super-critical slow sites ($β_j=α$ or $β_j>α$), associated Dirichlet boundary conditions arise as a result of interactions with evolving atom masses or condensation at the defects. Differently, when $g$ is bounded, at any slow site ($λ_j>1$), we find the hydrodynamic density must be bounded above by a threshold value reflecting the strength of the defect. Moreover, due to interactions with masses of atoms stored at the slow sites, the associated boundary conditions bounce between being periodic and Dirichlet.

math.PR↗

arXiv2021

Limit shapes for Gibbs partitions of sets

This study extends a prior investigation of limit shapes for partitions of integers, which was based on analysis of sums of geometric random variables. Here we compute limit shapes for grand canonical Gibbs ensembles of partitions of sets, which lead to the sums of Poisson random variables. Under mild monotonicity assumptions, we study all possible scenarios arising from different asymptotic behaviors of the energy, and also compute local limit shape profiles for cases in which the limit shape is a step function.

math-ph↗

arXiv2020

On hydrodynamic limits in Sinai-type random environments

We investigate the hydrodynamical behavior of a system of random walks with zero-range interactions moving in a common `Sinai-type' random environment on a one dimensional torus. The hydrodynamic equation found is a quasilinear SPDE with a `rough' random drift term coming from a scaling of the random environment and a homogenization of the particle interaction. Part of the motivation for this work is to understand how the space-time limit of the particle mass relates to that of the known single particle Brox diffusion limit. In this respect, given the hydrodynamic limit shown, we describe formal connections through a two scale limit.

math.PR↗

arXiv2019

On Entropy Minimization and Convergence

We examine the minimization of information entropy for measures on the phase space of bounded domains, subject to constraints that are averages of grand canonical distributions. We describe the set of all such constraints and show that it equals the set of averages of all probability measures absolutely continuous with respect to the standard measure on the phase space (with the exception of the measure concentrated on the empty configuration). We also investigate how the set of constrains relates to the domain of the microcanonical thermodynamic limit entropy. We then show that, for fixed constraints, the parameters of the corresponding grand canonical distribution converge, as volume increases, to the corresponding parameters (derivatives, when they exist) of the thermodynamic limit entropy. The results hold when the energy is the sum of any stable, tempered interaction potential that satisfies the Gibbs variational principle (e.g.~Lennard-Jones) and the kinetic energy. The same tools and the strict convexity of the thermodynamic limit pressure for continuous systems (valid whenever the Gibbs variational principle holds) give solid foundation to the folklore local homeomorphism between thermodynamic and macroscopic quantities.

math-ph↗

arXiv2018

On Hydrodynamic Limits of Young Diagrams

We consider a family of stochastic models of evolving two-dimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. `Static' scaling limits of the shape functions, under these Gibbs measures, have been shown by several over the years. The purpose of this article is to study corresponding `dynamical' limits of which less is understood. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types parabolic PDEs, depending on the energy structure.

math-ph↗

arXiv2016

Macroscopic Non-uniqueness and Limits of Hamiltonian Dynamics

We construct explicit examples of spontaneous energy generation and non-uniqueness for the compressible Euler system, with and without pressure, by taking limits of Hamiltonian dynamics as the number of molecules increases to infinity. The examples come from rescalings of well-posed, deterministic systems of molecules that either collide elastically or interact via singular pair potentials.

math-ph↗

arXiv2014

On Hydrodynamic Equations at the Limit of Infinitely Many Molecules

We show that weak convergence of point measures and $(2+ε)$-moment conditions imply hydrodynamic equations at the limit of infinitely many interacting molecules. The conditions are satisfied whenever the solutions of the classical equations for $N$ interacting molecules obey uniform in $N$ bounds. As an example, we show that this holds when the initial conditions are bounded and that the molecule interaction, a certain $N$-rescaling of potentials that include all $r^{-p}$ for $1<p$, is weak enough at the initial time. In this case the hydrodynamic equations coincide with the macroscopic equations of Maxwell.

math-ph↗